Cartesian Axes Research Articles

Line profiles for a collection of identical crystals: effect of a linear transformation of shape

A linear transformation is defined as a distortion of the unit lengths along three Cartesian axes and of their angles, preserving the same covariant coordinates of the corresponding points. The mathematical relationship between the line profiles of two mutually transformed crystals is proposed. It makes it possible, for example, to get the line profiles for ellipsoids, parallelepipeds, distorted tetrahedra and octahedra from the exact results reported by Langford & Wilson [J. Appl. Cryst. (1978), 11, 102-113] for spheres, cubes, regular tetrahedra and octahedra.

Elasto-optic studies on potassium acid phthalate single crystal

Potassium acid phthalate (KAP) belongs to the stress-optical constants qe and 12 strain-optical mm2(C2v) class of orthothombic system according constants Pu. The complete matrices of qli and PU to Okaya [1]. Betyaev etal . [2] have studied its and also the expressions for the absolute and rela. optical, dielectric, piezoelectric and elastic properative path retardations for different directions of ties. The acousto-optic interactions and hence the stress and directions of observation for an orthorindividual values of nine of the twelve strain optical hombic crystal are given by Narasimhamurty [5]. constants have been determined by Belikova et al. The crystals of KAP were grown in the labora[3] using the Bragg ultrasonic diffraction techtory by the slow evaporation of an aqueous nique. Since the acousto-optic interaction in this solution of its salt at room temperature. Large crystal characterizes it as an interesting material crystals o f 40ram × 30ram x 8ram size were for use in acousto-optic devices, we have studied obtained, by seeding, in a period of 8 to 10 weeks. in detail the elasto-optic behaviour of the single The crystallographic data reported by Okaya [1] crystal of KAP and now present the complete was used to identify the faces and the crystaUoset of absolute values of its stress-optical and graphic axes of the crystal. The assignment of the strain-optical constants qij and Pij respectively, cartesian axes x, y , z to the crystal follows that by These constants have been obtained from the Belyaev et at. [2]. The orientations of the KAP experimental measurements of relative path retarprisms were accurate to within 1 °. dation employing the bi-refringent compensator Observations on the differential stress bimethod of Veerabhadra Rao and Narasimhamurty refringence at mean sodium yellow light are [4] and absolute path retardation employing a presented in Table I and those on the absolute modified form of Rayleigh refractometer, when path retardation measurements are collected in crystals of different orientations were subject to Table II. linear stresses. The values of the principal refractive indices The photo-elastic tensor of orthohombic and etastic constants used in the calculations are crystals is described in Pockels' scheme by its 12 taken from Belyaev etal . [2] and are

Proton and boron-11 nuclear spin relaxation and the molecular tumbling of nido-decaborane in perdeuterotoluene solution. An interesting transition in solute–solvent interaction behaviour

The variations with temperature of the 1H and 11B longitudinal relaxation times, T1, for B10H14 in CD3C6D5 solution show that there is a transition from anisotropic motion at lower temperatures to essentially isotropic tumbling at higher temperatures. The dominant relaxation mechanisms are (11B–1H) and (1H–1H) dipolar for the 1H nuclei and quadrupolar for the 11B nuclei. The activation energy EA for the high-temperature isotropic tumbling is 9.9(5) kJ mol–1, and analysis of the 1H relaxation at lower temperatures shows activation energies ExA≅ 20, EyA≅ 18 and EzA≅ 15 kJ mol–1 about molecular cartesian axes x, y and z, z being the C2 axis of the molecule. Proton nuclear-shielding behaviour permits description of the modes of solute–solvent interaction and shows that ΔH and ΔS for the motional transition are +25 ± 3 kJ mol–1 and +110 ± 20 J mol–1 K–1, respectively. Some of the observations have important wider implications.

Specification of biaxial orientation in amorphous and crystalline polymers

AbstractThe fundamental concepts for specifying orientation in amorphous and crystalline polymers are reviewed. A new set of orientation factors is proposed to represent the second moments of biaxial orientation. The factors are defined both for the chain axis and for three crystallographic axes of an orthorhombic (or pseudo‐orthorhombic) crystal structure. The orientation factors are defined in terms of the angles between the crystallographic axes and Cartesian coordinate reference axes defining the machine, transverse and thickness directions of films. This makes the orientation factors symmetric with respect to the machine and transverse directions unlike the Stein‐Nomura‐Kawai orientation factors which are defined in terms of Euler's angles. A graphical procedure for representing the state of orientation as a point inside an isoceles triangle is described. Methods of measuring the orientation factors are also reviewed. The paper concludes with examples of the application of these concepts to orientation in amorphous polystyrene films fabricated in our laboratories and to crystalline polyethylene samples discussed in the literature.

Modern Methods for Obtaining Anthropometric Data

Modern concepts and methods are summarized for obtaining measurements of human body static form, dynamic characteristics and muscular strength. A philosophy of moderation is advocated in selecting methods which meet two conflicting goals: (1) Permit predictive analysis of biometric trends by continuing selective usage of traditional anthropometer and tape methods; (2) Take advantage of more powerful stereophotogrammetric techniques to develop 3-dimensional, whole-body, simultaneous records which are easily referenced to consistent cartesian or polar axis systems and thus are better suited to engineering workspace design and analysis. A further consideration is the use of mockups which represent or position the human body in specific and typical workspace conditions, appropriate to the purposes for the anthropometric surveys. In all cases, recording formats should be readily compatible with data analysis by computer. Also described are procedures for organization of personnel and equipment for conducting surveys of civilian personnel in the United States. Trades between accuracy or population description and sample size are considered. Key sources for the information contained in this paper are the following references.

Second-order derivatives in a local frame developed in spheroidal and spherical coordinates

Second-order derivatives of a general scalar function of position (F) with respect to the length elements along a family of local Cartesian axes are developed in the spheroidal and spherical coordinate systems. A link between the two kinds of formulations is established when the results in spherical coordinates are confirmed also indirectly, through a transformation from spheroidal coordinates. IfF becomesW (earth's potential) the six distinct second-order derivatives—which include one vertical and two horizontal gradients of gravity—relate the symmetric Marussi tensor to the curvature parameters of the field. The general formulas for the second-order derivatives ofF are specialized to yield the second-order derivatives ofU (standard potential) and ofT (disturbing potential), which allows the latter to be modeled by a suitable set of parameters. The second-order derivatives ofT in which the property ΔT=0 is explicitly incorporated are also given. According to the required precision, the spherical approximation may or may not be desirable; both kinds of results are presented. The derived formulas can be used for modeling of the second-order derivatives ofW orT at the ground level as well as at higher altitudes. They can be further applied in a rotating or a nonrotating field. The development in this paper is based on the tensor approach to theoretical geodesy, introduced by Marussi [1951] and further elaborated by Hotine [1969], which can lead to significantly shorter demonstrations when compared to conventional approaches.

Self-avoiding walks with span limitations. I. The mean square end-to-end distance

Self-avoiding walks on the simple cubic lattice of lengths up to 11 steps were exactly enumerated. The mean square end-to-end distance and its mean square components were recorded as a function of the span separation along one principle Cartesian axis. The molecular weight dependence for narrow spans (thin slabs—up to two lattice spacings) obeys that of a two-dimensional walk as conjectured by Wall et al. As the slab thickness increases, one encounters a dimensionality transition in the behavior of the mean separations which is compared with similar transitions observed for the self-avoiding walk in the presence of one interacting barrier. For thick slabs the behavior is three dimensional.

Pulsed nuclear magnetic resonance studies of the protons of hexamethyldisiloxane adsorbed on aerosil silica

Measurements are reported of the free-induction decays, spin–lattice relaxation and solid- and dipolar-echo responses of the protons in hexamethyldisiloxane adsorbed onto calcined and deuterated aerosil-200 silica. It is shown that the free-induction decay corresponds to a super-Lorentzian lineshape arising from an inhomogeneous distribution of dipolar second moments. A model is proposed in which a hexamethyldisiloxane molecule adsorbed on the silica surface undergoes rapid but anisotropic motions about three Cartesian axes fixed in the surface as well as, possibly, a number of internal motions. As a consequence the intramolecular dipole–dipole couplings between the protons are partitioned into an average part, which is time independent and a function only of the angle between the external field B0 and the normal to the surface at the adsorption site, and a zero-average fluctuating part which dominates the spin—lattice relaxation behaviour. It is proposed that the average part dominates the linewidth for a given site and thus leads to the super-Lorentzian shape for the powder sample. Spin–lattice relaxation measurements are shown to be consistent with this model in that when it is measured by observing the signal at the beginning of the f.i.d. following the 90° pulse in the 180-τ-90 sequence the relaxation is multicomponent, whereas if it is monitored in the tail of the f.i.d. it is single exponential in character and has the maximum value. Similarly, solid-and dipolar-echoes are shown to be formed even from the long-time part of the f.i.d., confirming that it is dominated by secular dipolar interactions. The observation that the dipolar-echoes always have their maximum intensity at t=τ following the 90x-τ-45y-T-45y sequence is also explained in terms of the inhomogeneous nature of the lineshape with respect to the dipolar couplings. The consequences of these observations for the interpretation of nuclear spin relaxation measurements on adsorbed systems is emphasised.

The Period of the Chandler Wobble

Realistic models of the Earth are known to possess a solid anelastic inner core, mantle and crust, and a fluid core and oceans. How might we go about calculating the theoretical free period of the Chandler wobble of such an Earth model? Let xi be a set of Cartesian axes with an origin at the center of mass, and let ωi be the instantaneous angular velocity of rotation of these axes with respect to inertial space. The net angular momentum is then Cijωj + hi, where Cij is the inertia tensor, and hi is the relative angular momentum. Let us affix the axes xi in the mantle and crust by stipulating that the relative angular momentum is that of the core and oceans alone, i.e., hi (mantle and crust) = 0; hi = hi (core and oceans). For an infinitesimal free oscillation of angular frequency σ, we can write ωi = Ω(δi3 + mi eiσt), Cij = A(δilδjl + δi2δj2) + Cδi3δj3 + cij eiσt, and hi = hi eiσt, where Ω is the mean rate of rotation and A and C are the mean equatorial and polar moments of inertia.

Versatile Mirror Mount for Stable Optical Instrumentation

A mirror mount adjustable along three Cartesian axes and about two angular axes is described. The mount can support mirrors up to 8 in. (20.3 cm) in diameter, can be positionally set to high tolerances, and is stable over a period of years. Examples of mounting a variety of mirrors are given.

Lifshitz points in ising systems

Phase diagrams of Ising systems with competing interactions are calculated using (a) the method of Muller-Hartmann and Zittartz to determine the transition temperature via the vanishing of an interface free energy (b) a Migdal-Kadanoff bond-moving scheme and (c) Monte Carlo simulations. It is shown that in two-dimensional Ising systems a uniaxial Lifshitz point can exist at non-zero temperatures, whereas the lower critical dimensiond l for a Lifshitz point in a system with identical competing interactions along each of its cartesian axis isd l ≧2.

A prescription which predicts functionally equivalent residues at given sites in protein sequences

Grantham (1974) has suggested a compact way of representing those properties of amino acid residues which correlate most strongly with evolutionary substitution frequency and functional equivalence. He establishes a Euclidean space in which the Cartesian axes are measures of composition, polarity and volume. The distance between pairs of residues is a measure of their relatedness. We utilize this representation to present a prescription which predicts all functionally equivalent residues expected at a given site in any protein sequence if at least two such residues are known. Cytochrome c is used as an example and a table is given showing new residues which are predicted at each site. Some sites appear to be complete in the light of present data. The prescription establishes a conceptual framework wherein the validity of Grantham's hypothesis can be tested objectively and, if necessary, generalized. The three properties described by Grantham (1974) appear by themselves unable to explain the substitution of alanine for cysteine observed at the heme attachment site (site 22) in cytochrome c from Euglena gracilis and from Crithidia onocopelti.

Linear and quadratic electric-field dependence of the nuclear magnetic shielding in the hydrogen molecule

An electric field distorts the electron distribution in a molecule, thereby causing changes in the magnetic shielding of the nuclei. In standard notation the component σαβ of the nuclear shielding tensor for a molecule in a uniform field (Ex , Ey , Ez ) can be written as the expansion For the hydrogen molecule in a principal set of cartesian axes, there are three non-vanishing components of , of which two are independent, and 21 non-vanishing components of , of which only seven are independent. For the mean shielding, σ=1/3σαα, one linear and two quadratic electric-field coefficients are required; thus where the subscripts label electric-field components which are parallel and perpendicular to the bond axis. Most of the components of and have been calculated by an ab initio method for a proton in the hydrogen molecule and this has permitted a determination of A, B ‖ and B t].

Anisotropic corrections of measured integrated Bragg intensities for thermal diffuse scattering. II

The method of evaluation for the contribution of TDS to Bragg reflexions is given on the basis of the general formalism developed in the previous paper [Harada & Sakata, Acta Cryst. (1974), A30, 77-82]. TDS tensor Δβ is expressed by a matrix product as {\tilde{\sigma}}Tσ, where T is the tensor that characterizes the anisotropy of TDS in reciprocal space and σ is the transformation matrix of scattering vector from Cartesian axes to crystallographic reciprocal axes. All the components of T and σ are listed in a table for the nine groups of elastic constants for practical use. It is, however, found that there are only seven matrix forms of Δβ corresponding to the seven crystallographic systems. Two different approximations proposed previously for Nillson's formalism in the estimation of the scan area of Bragg reflexion are shown to be available also for the general formula. The numerical calculations of the TDS correction for an NaC1 single crystal are made with these approximations and they are compared with the experimental measurements by Renninger and with the results of calculations given with other approximations. No substantial difference is seen among the results of calculations and they are in good agreement with experiments.

The equations of hydrodynamics for a thermally conducting, viscous, compressible fluid in special relativity

view Abstract Citations (5) References (8) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The equations of hydrodynamics for a thermally conducting, viscous, compressible fluid in special relativity. Greenberg, P. J. Abstract In this paper the equations of hydrodynamics governing the behavior of a thermally conducting, viscous, compressible fluid in special relativity are derived. We also derive the expressions (expansion, rotation, and shear) describing the kinematics of the fluid as seen by an observer comoving with an element of the fluid and employing three Fermi-transported, local, Cartesian axes. It is furthermore observed that there is the possibility for waves of viscosity to exist in the fluid, and we determine, in the case of one-dimensional flow, the condition for the existence of these waves. Publication: The Astrophysical Journal Pub Date: February 1975 DOI: 10.1086/153379 Bibcode: 1975ApJ...195..761G Keywords: Astronomical Models; Compressible Fluids; Hydrodynamic Equations; Relativity; Thermal Conductors; Viscous Fluids; Mathematical Models; One Dimensional Flow; Rotating Fluids; Shear Flow; Thermal Conductivity; Wave Propagation; Physics (General) full text sources ADS |

ChemInform Abstract: THE PROTON MAGNETIC RESONANCE IN THE DPPH‐BENZENE COMPLEX

Chemischer InformationsdienstVolume 5, Issue 19 Physical Organic Chemistry ChemInform Abstract: THE PROTON MAGNETIC RESONANCE IN THE DPPH-BENZENE COMPLEX TOSHIO YOSHIOKA, TOSHIO YOSHIOKASearch for more papers by this authorHIROAKI OHYA-NISHIGUCHI, HIROAKI OHYA-NISHIGUCHISearch for more papers by this authorYASUO DEGUCHI, YASUO DEGUCHISearch for more papers by this author TOSHIO YOSHIOKA, TOSHIO YOSHIOKASearch for more papers by this authorHIROAKI OHYA-NISHIGUCHI, HIROAKI OHYA-NISHIGUCHISearch for more papers by this authorYASUO DEGUCHI, YASUO DEGUCHISearch for more papers by this author First published: May 14, 1974 https://doi.org/10.1002/chin.197419084AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Volume5, Issue19May 14, 1974 RelatedInformation

The Proton Magnetic Resonance in the DPPH–Benzene Complex

Abstract The proton magnetic resonance in single crystals of the DPPH–benzene complex has been measured around three Cartesian axes. All the spectra are highly complex, but some lines are clearly identified around all axes. The principal values of their anisotropic hyperfine tensors have been evaluated, in addition to the isotropic hyperfine coupling constants. In the crystalline state, the unpaired electron-spin distributions of one phenyl group is almost twice as large as that of the other phenyl group, as is to be expected from the crystal structure. Besides, the unpaired electron-spin delocalization is larger in the crystalline state than that in solution, particularly on the picryl group and on one phenyl group.

Computer Controlled Pattern Generating System for use with Electron-Beam Writing Instruments

Recent advances in electron sources and electron resists have resulted in a significant increase in potential exposure rate in electron-beam writing instruments. This in turn places stringent requirements on the pattern generating electronics which must be fast enough to avoid their limiting potential system throughout, yet flexible enough to be capable of creating all normally required geometries. A computer-controlled pattern generation system is described which is designed to meet these requirements. In this system, the speed limitation due to the calculation and data-transfer rate of modern minicomputers is avoided by the use of an auxiliary special-purpose logic processor. Patterns are built up from a basic trapezoid shape with two sides parallel to rectangular cartesian coordinate axes. The computer has only to specify the basic parameters defining a trapezoid and the auxiliary processor will then output coordinates representing the end points of lines defining the figure in a raster-scan format. These coordinates, which may be generated at rates up to 1 MHz are passed to a pair of precision ramp generators. The ramp generators, which are based on commercially available 16-bit digital-to-analog converters, then produce the analog deflection signals needed to define the line, at a sweep rate which may be varied under computer control over six orders-of-magnitude. Particular care has been taken to reduce the switching spikes, inherent in digital-to-analog converters, to negligible proportions on the output waveform.

Overestimation of acute angles.

An explanation was sought for overestimation of acute angles, especially when those angles are presented in a horizontal orientation. 22 Ss ordinally ranked 8 45° angles oriented horizontally and vertically in either a position which would be bisected by the Cartesian axes or in which one arm would be on the axis with respect to how likely it was to suggest a right angle seen tilted in depth. Horizontally oriented angles were ranked most likely to produce the effect (Wilcoxon test, t (18) = 0, p < .01). The explanation was applied to studies which show overestimates of acute angles.

MAGNETIC ANOMALIES OF THIN SHEETS AND THEIR INTERPRETATIONS

A method of quantitative interpretation of the vertical magnetic anomaly (ΔZ) caused by a thin sheet infinite in horizontal extent but limited in depth extent is developed. The determination of the position of the point on the profile above the dike plays an important role in the analysis and is done either by a graphic method or with the help of a rapidly convergent iterative process. When the depth extent of the sheet, either vertical or inclined, becomes infinite, the mathematical steps become simpler, and evaluation of different unknown parameters of the causative body is possible without any prior knowledge of the positions of the cartesian axes. The proposed methods are applied both on theoretical models and on actual field cases.