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We present measurements of the three independent components of the second-order Raman scattering tensor for RbF and RbBr at 300 and 80 \ifmmode^\circ\else\textdegree\fi{}K, together with a brief analysis in terms of existing neutron data. The general features of the spectra are compared with those of the other members of the rubidium halide family.

The excitation wavelength dependence of the polarized Raman spectrum of p-dichlorobenzene single crystal has been measured. The Raman lines of the CC stretching vibration v2(ag) and the ring breathing vibration v4(ag) were found to show great intensity enhancements in the tensor component (aa) when the exciting light was shifted to shorter wavelength. With the assumption of the oriented gas model, the intermediate electronic states contributing to the Raman intensities were determined to be the 1B3u states of the molecule. The Raman lines of v2(ag) and v4(ag) were correlated, from the excitation wavelength dependence of the solution spectrum, to the 1B3u states corresponding to the 1B1u and 1E1u states of benzene, respectively. These correlations are explained in terms of the geometrical structures of the molecule in the electronic excited states.

A geometric symmetry group is defined as a point transformation of a Riemannian manifold combined with a transformation law for the field as a geometrical object. A covariant definition of invariance of an action integral is given. It is shown that geometric invariance groups can be determined from knowledge of a tensor S,βα which can be computed from the density function L in the action integral. A general form for conservation laws due to geometric symmetry is given. Results are applied to electromagnetic fields and it is shown that the Bessel‐Hagen conservation laws represent all of the possible conservation laws for electromagnetic fields arising from geometric symmetry.

Differential cross sections and vector and tensor analyzing powers of 15 MeV deuterons elastically scattered by 52Cr, 90Zr and 122Sn have been analyzed in terms of an optical potential containing a tensor term TR. Neglecting any tensor interactions the differential cross section and vector analyzing power data were well reproduced but only qualitative agreement was obtained with the corresponding tensor analyzing power measurements. The inclusion of a TR potential only slightly improved the fits to the tensor data. The quantity T21 was found to impose greater restrictions than T20 and T22 on the parameters of the TR potential. It appears that the conventional optical model has some difficulty in providing an overall description of the data.

It is shown that electron capture to β + decay ratios are insensitive to the induced tensor interaction.

Summary A systematic approach to general orthogonal co‐ordinates is presented which has the advantage of introducing tensor quantities and providing the distinction between covariant and contravariant vectors. The equations and quantities needed in classical problems are then determined in an elementary way and the equations of motion of a fluid in orthogonal coordinates derived as an example.

A new method of constructing models of cylindrical geological structures has been developed which allows the finite strain to be determined for each of a large number of small, initially square, elements. Each material layer is made up of uniformly thick, alternately colored laminations which are oriented vertically in one half of the model and horizontally in the other half. After the model has been deformed it is cut into slices perpendicular to the axis of the cylindrical structure, and the lamination patterns from the two model halves are superimposed graphically to reveal the strained shapes of the initially square elements. The three-dimensional finite-strain tensor of each element can be easily determined algebraically from measurements of its deformed shape, provided that the deformation involved rotation about no more than one principal direction. As an example the method is applied to a silicone putty model of a diapiric ridge which was formed by spinning the model in a centrifuge.

The electromagnetic scattering properties of a medium bounded by a plane surface and whose (scalar) dielectric constant is the sum of a nonrandom part assumed to vary only with depth and a small random part are investigated by means of an approximate, first-order perturbation method. It is shown that the equivalent dielectric constant to be used in calculating the average reflected power in the specular direction is a function of depth even in the case where the nonrandom part of the dielectric constant is independent of depth. Further, the equivalent dielectric constant is shown to be a tensor. Nonspecular scattering is also considered.

The asymmetry parameter η of the electric field gradient tensor having reflection symmetry is discussed for spn hybridized atoms. It is shown that in this case η is a function of a single variable, which depends on the degree of hybridization and the occupation numbers of the hybrids. The model provides a physical picture for the various possible values of η as well as closed expressions for the occupation numbers. Two specific cases, sp3 and sp2, are discussed in detail.

A relativistic, collisionless gas of gravitating particles all having the same proper mass (possibly equal to zero) is studied under the assumption that the oneparticle distribution function is locally ellipsoidal in momentum space with respect to some timelike vector field (observer). Liouville's equation implies that the distribution function depends only on a quadratic form in the 4- momenta, whose coefficients are a Killing tensor in the case of non- vanishing proper mass, and a conformal Killing tensor in the case of vanishing rest mass of the particles. It is suggested that cosmological models of Bianchi-type I can be described in terms of ellipsoidal momentum distribution functions whose ellipsoidal tensor is built out of the Killing vectors associated with the spatial homogeneity.

We give precise definitions of the weak and strong principles of equivalence and show that the new gravitational theory based on the improved energy‐momentum tensor of Callan, Coleman, and Jackiw [Ann. Phys. (N.Y.) 59, 42 (1970)] satisfies both of these principles. As a consequence of the equality between the 4‐momentum given by the canonical energy‐momentum tensor and the ``momentum'' given by the pseudotensor that is the source of gravitation, the weak principle is also shown to hold in more general Einstein theories. Investigation of the interactions of a scalar field in the new gravitational theory shows that, besides the familiar long range interaction, there exists a new short range gravitational interaction between any scalar field and other matter.

Nuclear magnetic resonance (NMR) has long been a useful technique for the study of solids (1, 2). Within recent years, however, a number of advances have been made both along theoretical and experimental lines which hold promise to expand greatly the utility of nuclear magnetic resonance as an experimental technique for the study of physical and chemical phenomena in solids. Of particular importance are those techniques which are furnishing generally applicable means of measuring chemical shift tensors in solids for the first time. This review has been restricted to a discussion of those developing techniques which, among other possibilities, are allowing high resolution chemical shift tensor measurements in solids. No attempt will be made here for a definitive essay of the theoretical basis of these somewhat complex new techniques, but rather the emphasis will be placed on discussing the nature of the information they allow one to obtain on physical and chemical interactions within solid materials and what the present and future potentials and limitations of the techniques appear to be. Recent efforts in other aspects of solid state NMR have been reviewed (3-12), and the reader is particularly referred to the discussion of sample spinning techniques by Andrew (3), to the summary of theoretical developments in pulsed NMR by Mansfield (4), and to discussions of using NMR for the study of motional effects in solids (5-7) and of surface adsorbed species (8, 9). Additionally, attention is called to the development of more precise techniques for measuring moments of NMR lines by Lowe et al (13) and locating the center of such solid state lines by Kunitomo (14).

It is shown that the canonical tensor and not the Frenkel tensor is the correct tensor to describe the gravitational influence of direct particle fields. This corrects an earlier calculation which had resulted in favour of the Frenkel tensor. The forms of the two tensors are examined in cosmological models with perfect future absorbers and imperfect past absorbers, and it is shown that only the canonical tensor is physically reasonable.

Separable potentials of rank one to five for the nucleon-nucleon $^{3}S_{1}\ensuremath{-}^{3}D_{1}$ channel are constructed using a method introduced by Ernst, Shakin, and Thaler. The potentials all have the same deuteron wave function as the Reid soft-core potential. In addition the phase shifts of the rank-four and -five potentials are quite close to the experimental phase shifts. A survey of previously published separable potentials of rank greater than one is made and it is concluded that most of these potentials have unacceptably poor deuteron properties. A number of errors that were made in the fitting of these potentials to the data are pointed out. A simplification of the Reid soft-core potential for the $^{3}S_{1}\ensuremath{-}^{3}D_{1}$ channel is also presented. The simplified potential consists entirely of Yukawa terms but has experimental properties that generally differ from the original Reid soft-core properties by less than the present experimental error.[NUCLEAR REACTIONS Separable potentials fitted to Reid $^{3}S_{1}\ensuremath{-}^{3}D_{1}$ phase shifts, deuteron wave function. Previous $^{3}S_{1}\ensuremath{-}^{3}D_{1}$ separable potentials surveyed, errors pointed out. Compared to UPE constructions. Simplified (local) Reid tensor force presented.]

With the aim of treating the elastic scattering of neutrons by deuterons at low energies in presence of tensor forces, the resonating-group method is considered, which gives a set of coupled integro-differential equations. The Hulthen wave function is used for thes-state component of the deuteron ground state and it is shown that the coupled integro-differential equations results essentially from the inclusion of the tensor forces.

It is shown that the direct application of Volkoff's method for determining the quadrupole coupling tensor in a case where the magnetic nuclei are situated in crystallographically related, but magnetically nonequivalent, sites does not lead to a unique solution. For the deuteron quadrupole coupling tensor in gypsum two solutions are found, and it is shown that the tensor chosen earlier is not correct.

The nucleus 44 Ti is studied in the framework of the conventional shell model using a pairing-plus-surface-tensor interaction (PSTI). The nucleus 40 Ca is assumed to be an inert core. We include only 1 f 7 2 and 2 p 3 2 single-particle orbits in the model vector space. Low-lying energy levels are calculated and compared with the experimental spectrum. The agreement is good. We also present B (E2) values for transitions between levels of the ground-state band. The theory-experiment comparison suggests that the PSTI provides a reasonably good description of 44 Ti.

The correction of measured integrated intensities for the first-order thermal diffuse scattering (TDS) is considered on the basis of the existing theory of X-ray thermal diffuse scattering for an elastic wave of long wave length. Generalized formula for the TDS correction α is found to be represented by a quadratic form in the Miller indices h, k, l and a tensor Δβ, as α = Δβ11h2 + Δβ22k2 + Δβ33l2 + 2Δβ12hk + 2Δβ23kl + 2Δβ31lh. Δβ is a tensor introduced in this paper which characterizes the anisotropy of the TDS correction. The form of the tensor Δβ is shown to depend only on the crystallographic system. The relation between Δβ and the temperature-parameter tensor is presented.

Long wavelength ac bias record sensitivity is computed utilizing a model in which both longitudinal and vertical record fields are considered. It is assumed that the recorded magnetization is related by an anhysteretic susceptibility tensor to the signal field. At each depth in the tape the signal field is evaluated where the bias field amplitude is equal to the tape coercivity. Neither the effects of demagnetizing fields nor spreads in particle switching fields are included. The calculation yields excellent quantitative agreement with experimental sensitivity and correlates well with the shape of measured sensitivity vs. bias curves. The predictions of this model are compared in detail with those calculated utilizing solely longitudinal fields.

Finite field perturbation theory coupled with configuration interaction techniques have been used to evaluate the components of the polarisability tensor of H2 at and near its equilibrium internuclear separation. The values are fitted to a polynomial which then yields the derivatives of the polarisability with respect to internuclear distance. Our results compare favourably with theoretical values obtained by the use of highly accurate correlated wavefunctions.

ESR spectrum of Cr 3+ doped in TGS was reinvestigated. The principal values and the directions of the d -tensor for the lines which appeared at lower fields, were determined at 60°C in the paraelectric region. Each line was split into doublet in the ferroelectric region. It was ascertained from the angular dependence of the lines in the b -plane that this anomalous splitting originates from a tilting of the principal axis by the movement of the ligand atoms surrounding the Cr 3+ ion.

Using fixed-t dispersion relations together with experimental data on π−p → π0n we determine the real and imaginary parts of the s-channel helicity amplitudes (SHA) of the vector (ϱ) exchange contribution to πN → πN. The results are in good agreement with amplitude analyses. We also make comparison with finite-energy sum-rules (FESR) and discuss certain implications. Furthermore, on the basis of two-component duality and FESR we calculate the tensor (f) contribution. The t-structure of the resulting SHA is compared with independent information on hadronic tensor exchanges.

The problem of labelling the representations ofSO(3) contained in each representation ofU(3) is solved by using for this purpose an operatorZ with integral eigenvalues. A description is given of the decomposition of any tensor representation ofU(3) intoSO(3)-irreducible tensors labelled by these eigenvalues. An implicit definition ofZ is given in terms of theU(3) generators, and the relationship ofZ to the labelling operators (with irrational eigenvalues) proposed by Racah and others is made clear. A possible application to a quark model of the hadrons is described.

An alternate method of renormalizing a quartic self-interacting boson theory has been developed. We find that one can obtain finite renormalized expressions for the perturbation theory contributions to the Green's function without carrying out limiting procedures. As a consequence of the analysis, one is able to renormalize explicitly the field energy-momentum tensor to all orders. There exists a one-parameter family of renormalized tensors. The method will allow for a simple determination of the asymptotic “zero-mass” theory.

The properties of the nuclear tensor of inertia are discussed in connection with the fission process. In particular, the relations between the nuclear inertial parameters and the characteristics of the nuclear intrinsic motion are considered.

The tensor polarization of the deuteron in elastic electron–deuteron scattering depends on the ratio of deuteron quadrupole and monopole form factors. Measurements at [Formula: see text] are sensitive to the deuteron percentage D state; while measurements at [Formula: see text] are sensitive to a diffraction minimum in the monopole form factor.

Vector $i{T}_{11}$ and tensor $Q=(\frac{1}{2\sqrt{2}})({T}_{20}+\sqrt{6}{T}_{22})$ and $R=(\frac{1}{2\sqrt{2}})({T}_{20}\ensuremath{-}\sqrt{6}{T}_{22})$ analyzing powers have been measured in the elastic scattering of 20.4-, 25.2-, and 29.9-MeV polarized deuterons by protons. Comparison is made with Faddeev-equation calculations including spin-dependent nucleon-nucleon interactions. Qualitative agreement is obtained for all the measured quantities only when both the spin-dependent nucleon-nucleon $P$-wave forces and the tensor ($^{3}S_{1}\ensuremath{-}^{3}D_{1}$) force are taken into account.

We derive from the Maxwell tensor the forces and torques applied to a uniaxial anisotropic crystal when an extraordinary wave passes through it, and we show that the momentum to be associated with the wave in the crystal is collinear with the Poynting vector. Next, we present experimental results of torque measurements using, successively, two uniaxial anisotropic crystals placed inside a variable pressure container. We finally compare our results with those obtained by other authors and show that, according to the types of experiments, either Abraham's or Minkowski's momentum density will appear. [Translated by the journal]

The least-squares method yields well-known estimates which will be called /b G/ macr /sup 2/ and /b K/ macr /sup 2/ for the /b G//sup 2/ and /b K//sup 2/ tensors related to the hyperfine tensor /b T//sup 2/ by /b T//sup 2/=/b G/ /sup -1/ /b K//sup 2/ /b G//sup -1/. However, very little is known about how to estimate the eigenvalues of /b G//sup 2 / and /b T//sup 2/, which are the important EPR parameters. The common procedure used in estimating these EPR parameters consists in computing the eigenvalues of /b G/ macr /sup 2/ and /b T/ macr /sup 2/. The statistical characteristics of these eigenvalue estimators are studied by stimulation. An empirical description of the joint distribution of the eigenvalue estimators is generated. The authors show that a good experimental design is necessary to prevent biased and highly correlated eigenvalue estimators.

The R branch of the pure rotational spectrum of SiH4 in the ground vibronic state has been studied in the spectral region 60 to 134 cm−1 using a Michelson interferometer and a 10.6 m cell. Thirteen lines have been observed and identified as forbidden (J → J + 1) transitions for rotational levels from 10 to 22, with the absorption mechanism being provided by the centrifugal distortion dipole moment μD. The two strongest competing mechanisms are discussed and eliminated from consideration: transitions within the vibrational excited state ν4 and the difference band ν2 – ν4. In the data analysis, the frequency shift due to the fourth rank tensor distortion interaction is taken into account by an extension of the earlier work on GeH4 by Ozier and Rosenberg. It has been determined that the rotational constant B0 = (2.8574 ± 0.0027) cm−1 and the scalar distortion constant Ds = (3.48 ± 0.54) × 10−5 cm−1. From estimates of the line intensities, it has been found that [Formula: see text].

Summary An integral equation method is used to derive the electromagnetic response of a three-dimensional heterogeneity in a three-layer medium. The method consists of replacing the heterogeneity with point dipole scattering currents. The kernel of the integral equation is a tensor Green's function which is derived for the three-layer case. It is shown how the use of iteration methods and Simpson's rule integration might significantly reduce computing requirements. The problem of initial estimates is discussed. The method is extended to include the case of a simultaneous conductivity and permeability anomaly.

A review is given of symmetries in the vector spaces of the nuclear shell model, the goodness of nuclear symmetries, and the irreducible tensor formalism as a calculational tool.

Statistical mechanical averages of vectors and tensors characterizing the configurations of polymethylene chains with n = 5–200 bonds have been calculated, these quantities being expressed in the internal reference frame attached to the first two bonds. Specifically, the persistence vector a ≡ 〈r〉 and the averaged center‐of‐mass vector 〈g〉 have been obtained as functions of chain length. They are compared to the corresponding vectors for various model chains. Configurational averages of the Cartesian tensors formed from ρ = r − a have been computed up to and including the tensor of seventh rank. Density distribution functions Wa(ρ), evaluated from these moment tensors using the three‐dimensional Hermite series elaborated in the preceding paper, are approximately cyclindrically symmetric about a principal axis that lies close to the direction of a. The second moment (and, hence, the dispersion) along this axis is smaller than the moments perpendicular thereto. Acentricity and skewness of the distribution W...

A method is described which allows chemical shifts in solids to be detected to a good accuracy by sharply determining the first moment. The analysis is made by using spin-temperature concepts and it is shown that the chemical-shift tensor can be determined if a single tensor contributes to the spectrum, and the average if more than one tensor contributes. Measurements are made in a single crystal of KH 2 PO 4 . The average tensors obtained are axially symmetric about the c axis with σ // =-25.6 ppm and σ ⊥ =-8.3 ppm relative to C 6 H 12 for 1 H(300 K), and with σ // =-1.1 ppm and σ ⊥ =-16.5 ppm relative to 85% H 3 PO 4 for 31 P(77 K). These values are discussed qualitatively. The 19 F chemical shift in CaF 2 is also obtained, and the result is in good agreement with that of pulse train method.

The geomagnetic induction tensor is a means of summarizing the response of the earth at a given observing site to a geomagnetic variation source field. In this paper the characteristics of the tensor elements are examined, both generally and for the special cases of one-dimensional and two-dimensional geologic structure. The first-order model is taken of uniform source fields originating external to a semi-infinite half-space. Graphical ways of presenting the information contained in an induction tensor are explored, including ellipses of rotation, polar diagrams, and diagrams analogous to the Mohr circles of elasticity theory. Criteria to distinguish “two-dimensional” data from “three-dimensional” data are established. The advantages of simultaneously recording “normal” and “anomalous” variations are demonstrated in terms of the extra tensor elements which may then be estimated. The most practical way of presenting information from many stations on a map may be by drawing, for each site, arrows which summarize the response in the vertical field and quadrics which summarize the response in the horizontal field.

Electron paramagnetic resonance has been observed in φ4AsCuCl3 at room temperature. The spectrum is characteristic of a spin one system. Spin Hamiltonian parameters are gx = 2.08, gy = 2.11, gz = 2.32, D = −0.111 cm−1, and E = 0.009 cm−1. The principal axes of the g and D tensor are coincident. The rhombic spin Hamiltonian is a reflection of the distorted tetrahedral environment of the Cu2+ ions. An expression for the angular dependence of the D tensor components with arbitrary field rotation is developed. The usual method of obtaining the principal axes is discussed. The results from magnetic data compare favorably with the predictions based on structural analysis. The spin Hamiltonian can be rationalized on the basis of a chloride bridged binuclear species. Low temperature spectra show profound increase in intensity but little or no change in line positions. The Cu2+ ions are pairwise coupled to give a triplet ground state and a singlet excited state at 36 cm−1.

Angular distribution of the vector and tensor analyzing powers ${A}_{y}$, ${A}_{\mathrm{xz}}$, $\frac{1}{2}({A}_{\mathrm{xx}}\ensuremath{-}{A}_{\mathrm{yy}})$, and ${A}_{\mathrm{zz}}$ have been measured at 11.4 and 14.4 MeV for $d+^{3}\mathrm{H}$ elastic scattering. Mass identification techniques were used in detecting the scattered deuteron and the recoil triton at most angles. Measurements were made for the 33 to 140\ifmmode^\circ\else\textdegree\fi{} center-of-mass angular range. The angular distributions of the various analyzing powers are very similar at both energies and additionally are similar to the $d+^{3}\mathrm{He}$ analyzing-power data of K\"onig et al. at 11.5 MeV.

The quadratic Stark effect in the hyperfine structure of the 2 P ½ ground level of aluminium has been investigated by the method of atomic beams. A frequency shift caused by the application of an electric field has been interpreted in terms of an effective Stark operator, and the associated tensor polarizability α ten ( J = ½, F = 3) has a measured value of (8.13±0.72) × 10 –4 a 3 0 . The case of J = ½ is a special one in that the theoretical value of α ten ( J = ½, F = 3) vanishes in the absence of hyperfine structure effects. The inclusion of the hyperfine structure operator in a calculation of the tensor polarizability has led to a small theoretical value in agreement with experiment.

A study is made of the motion of an extended body in arbitrary gravitational and electromagnetic fields. In a previous paper it was shown how to construct a set of reduced multipole moments of the charge-current vector for such a body. This is now extended to a corresponding treatment of the energy-momentum tensor. It is shown that, taken together, these two sets of moments have the following three properties. First, they provide a full description of the body, in that they determine completely the energy-momentum tensor and charge-current vector from which they are constructed. Secondly, they include the total charge, total momentum vector and total angular momentum (spin) tensor of the body. Thirdly, the only restrictions on the moments, apart from certain symmetry and orthogonality conditions, are the equations of motion for the total momentum and spin, and the conservation of total charge. The time dependence of the higher moments is arbitrary, since the process of reduction used to construct the moments has eliminated those contributions to these moments whose behaviour is determinate. The uniqueness of the chosen set of moments is investigated, leading to the discovery of a set of properties which is sufficient to characterize them uniquely. The equations of motion are first obtained in an exact form. Under certain conditions, the contributions from the moments of sufficiently high order are seen to be negligible. It is then convenient to make themultipole, in which these high order terms are omitted. When this is done, further simplifications can be made to the equations of motion. It is shown that they take an especially simple form if use is made of the extension operator of Veblen & Thomas. This is closely related to repeated covariant differentiation, but is more useful than that for present purposes. By its use, an explicit form is given for the equations of motion to any desired multipole order. It is shown that they agree with the corresponding Newtonian equations in the appropriate limit.

Under the influence of a constant magnetic field, the electric property of a plasma and the magnetic property of a ferrite are anisotropic. In this paper, the general coordinatefree invariant forms of the dielectric tensor of a plasma and the permeability tensor of a ferrite are obtained. The tensors are expressed explicitly as a sum of three tensors: a unit tensor, a symmetric tensor and an antisymmetric tensor, each of which is weighted by different constants. The symmetric and antisymmetric tensors are related to the unit vector of the constant magnetic field. The invariant forms in terms of the sum of the projectors of the tensors are also derived. When a Cartesian coordinate system is introduced, the invariant forms are easily reduced to the commonly used matrix representations. The invariant forms clearly show the effects of the constant magnetic field on the anisotropies of the media. Moreover, they effectuate and simplify the deduction of the general solutions of problems involving wave propagation and excitation in plasma and ferrite and thus facilitate interpretations of the final results.

The transverse component (${\ensuremath{\sigma}}_{T}$) of the tensor electric conductivity of a weakly ionized magnetized plasma has been measured under near-thermal equilibrium conditions. Care has been taken to minimize the effects of $\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ifmmode\times\else\texttimes\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{B}}$ drifts and finite apparatus size. For magnetic fields that are large enough, it is found that the magnetic field dependence of ${\ensuremath{\sigma}}_{T}$ is $\frac{1}{{B}^{2}}$, in agreement with theory. It is also found that the overall density dependence of ${\ensuremath{\sigma}}_{T}$ agrees with standard conductivity theory. At the highest values of the magnetic field used, the data show some deviation from theoretical predictions which include only electron transport. This deviation is attributed to ion transport, and opens up the possibility of measuring transport properties of the ions alone.

As an ectension of the previous work, Schwinger's variational principle is formulated for the quantized fields which correspond to the classical real scalar fields described by the Lagrangian density L c (∂ τ φ, φ) = − 1 2 g ab (φ)(∂ μ φ a )(∂ μ φ b ) − v(φ) . The c-number variations of field operations φ a (x) and space-time coordinatei x are sufficient to give the laws of the quantized fields. The Euler-Lagrange equations, the canonical equations of motion and the canonical commutation relations are derived from this principle. The energy-momentum tensor Tμν, energy-momentum vector Pμ and angular-momentum tensor Mμν are introduced in a natural way. The differential conservation law ∂μTμν = 0 is also derived and it is inferred that Pμ and Mνλ should obey the commutation relations which characterize the inhomogeneous Lorentz group. A quantized version of Noether's theorem, which relates the conservation law with the invariance of the action, is established. All resulting relations are consistent with one another, although there arises a pathological situation in the explicit calculation of ∂μTμν. Further, it is shown that our formulation is covariant under an arbitrary change of field variables. The appropriate choice of the quantal Lagrangian density is essential to ensure internal consistency and covariance. The ambiguities in the quantal form of the Lagrangian density are discussed. A comment on the use of Schwinger's criterion for Lorentz invariance is also given.

The spectral line shape of hyper-Raman light scattering in liquids is calculated by the method of irreducible spherical tensors and the theory of rotational Brownian motion. The line broadening due to rotational molecular motion in the liquid is expressed by way of relaxation times τ and is found to depend on nonlinear molecular parameters ∣B∣2, which are quadratic functions of the derivatives with respect to appropriate normal coordinates of the hyperpolarizability tensor elements b. Tables of ∣B∣2 are adduced for all vibration symmetries of spherical-top and symmetric-top molecules. The tensor elements of scattered light intensity are calculated for linearly and circularly polarized incident light. Investigation of hyper-Raman spectral line broadening is seen as a source of new data on the dynamics of molecules in the liquid state and on their nonlinear optical properties.

The first accurate measurements of the algebraic values of the photoelastic tensor of silicon are reported. The values, determined acousto-optically, are ${p}_{1111}=\ensuremath{-}0.094$, ${p}_{1122}=+0.017$, and ${p}_{1212}=\ensuremath{-}0.051$. The results are in strong disagreement with calculations based on the Phillips-Van Vechten theory of ionicity. It is shown that the widely used assumption of a universal power law describing both the intermaterial and intramaterial variations of the homopolar average energy gap with volume must be reinterpreted or revised.

This paper presents a new formulation, based on Lagrangian coordinates, for the vibration of plates, both finite and infinite, at varying temperatures. The formulation circumvents a possible difficulty in applying perturbation theory to the usual formulae for infinite plates in a purely mechanical way. Knowledge of the thermal expansion tensor of the material is not required to infer the frequency‐temperature behavior for arbitrary orientation of crystal, once experimental results have been established for a few chosen orientations; this contrasts with earlier work. The formulation is also immediately suitable for use in conjuction with the finite element method to obtain the variation of resonant frequency with temperature of finite plates, either with or without mass loading. Results are given for a particular A T‐cut quartz resonator.

The nonlinear properties of a ferromagnet are studied. Many-time retarded Green's functions are used to obtain an expression for the cubic nonlinearity tensor with allowance for spatial dispersion of a uniaxial ferromagnet. The components due to the dipole-dipole interaction of the spins and also due to the anisotropy energy are found. A comparative analysis is made of the different components of the cubic nonlinearity tensor in both the nonresonance case and for various resonances, in particular when ω ∼ ω0, 3ω ∼ 2w0, 2ω ∼ ω0, 3ω ∼ ω0 for the case in tripling of the frequency. Here, ω is the frequency of the incident wave and ω0 is the frequency of uniform precession. It is shown that in the non-resonance case the largest components are those that are nonvanishing when no allowance is made for spatial dispersion; in the resonance cases the largest components are those due to the dipole-dipole interaction of the ferromagnetism spins.