Calcite Single Crystals Research Articles

Dynamic and Kinematic Analysis of Deformed Crinoid Stems in a Quartz Graywacke

In a Devonian crinoidal quartz graywacke from Lindlar, northeast of Cologne, Germany, crinoid stem disks, each composed of a single crystal of calcite, are elliptical in outline and show distinct dimensional preferred orientation. The crinoid fabric of one specimen has been examined in detail. There is a distinct preferred orientation of c- and a-axes of the calcite lattice. Most of the disks are strongly twinned on two of the three {} planes of the lattice. By dynamic analysis of observed double twinning (construction of "compression" and "tension" axes using a modification of the method of Turner [1953]), a statistically homogeneous stress system has been deduced. This is compatible with related folding on a larger scale: The deduced axial tension is parallel to the axis of regional folding. Strain corresponding to the observed twinning accounts for about 50 per cent of the measured mean elongation of the crinoid disks. It is suggested that the disks were initially elongate in outline, that the direction of primary elongation was parallel to an a crystal axis, and that deposition in a current of flowing water induced an initial sedimentary fabric characterized by distinct dimensional and lattice preferred orientation of the disks. This still survives in the deformed rock. The features of crinoid disk orientation and elongation recorded by Kurtman (1960) in the quartz graywackes of the Lindlar district are explained on this basis. /// In einer Quarzgrauwacke mit Krinoidenstielgliedern aus dem Devon von Lindlar (Bergisches Land, nord -östlich Köln) haben die Stielglieder elliptischen Querschnitt und sind deutlich mit ihren Ellipsenachsen geregelt. Jedes Stielglied besteht aus einem einzigen Kalzitkristall. Das Gefüge der Stielglieder wird in einem Handstück genauer untersucht. Die c- und a-Achsen der Kalzitkristalle sind deutlich geregelt und die meisten Stielglieder stark nach zwei der drei {} Ebenen des Kristalls verzwillingt. Durch dynamische Analyse der eingemessenen Verzwilligung (Konstruktion von "Kompressions"- und "Tensions"-Achsen in Abänderung des Verfahrens von Turner [1953]) lässt sich ein statistisch homogenes Stress-System ableiten, welches sich auf die grossmassstäbliche Faltung beziehen lässt: die ermittelte axiale Zerrung ist der Achse der regionalen Faltung parallel. Der durch beobachtete Verzwilligung ermittelte strain entspricht nur etwa 50 Prozent der mittleren Längung der Stielglieder. Es wird angenommen, dass diese ursprünglich einen elliptischen Querschnitt hatten, dass die Längsachse parallel einer der a-Achsen der ursprünglichen Kristallite war und dass die Sedimentation in strömendem Wasser ein primäres Sedimentteilgefüge der Stielglieder erzeugte, das durch deutliche Vorzugsorientierung der Scheiben (nach Korngestalt und Kristallachsenrichtungen) bezeichnet ist. Dieses Teilgefüge ist im deformierten Gestien noch erhalten. Die Orientierungen der Krinoiden-Stengelglieder und deren Längung, die Kurtman (1960) in den Quarzgrauwacken des Gebietes um Lindlar beschrieben hat, werden auf dieser Grundlage erklärt.

Electron spin resonance absorption spectrum of CO2-molecule-ions in single crystal calcite

An electron spin resonance absorption spectrum due to CO2 - molecule-ions has been observed at 3 cm wavelength in single crystals of optical grade calcite. The spectrum exhibits orthorhombic symmetry with both the 12C and 13C isotopically substituted CO2 - molecule-ions having the same g-tensor components given by gzz = 2·00161, gxx = 2·00320 and gyy = 1·99727, where z refers to the carbonate-ion C-O bond direction, x the crystal [111] direction and y the direction perpendicular to these two. The CO2 - molecule-ions are located at carbonate-ion sites with their planes parallel to those of carbonate-ions and with their two-fold symmetry axes parallel to the z direction. The hyperfine structure tensor components associated with the 13C isotope are Azz = 177·28 oe, Axx = 134·62 oe, and Ayy = 131·69 oe.

Paramagnetic Resonance Absorption Spectrum of Trivalent Iron in Single-Crystal Calcite

The electron paramagnetic resonance absorption spectrum of trace amounts of trivalent iron in single crystals of natural calcite has been observed at 9.6 kMc/sec. The spectrum is symmetric about the crystal [111] direction which, together with the doublet character of each spectral component, suggests that the trivalent iron ions occupy sites closely approximating those of the calcium ions. At 77\ifmmode^\circ\else\textdegree\fi{}K, the best-fit parameters to the spin Hamiltonian are found to be ${g}_{\mathrm{II}}={g}_{\ensuremath{\perp}}=2.0030$, ${{B}_{2}}^{0}=343.23$ Oe, ${{B}_{4}}^{0}=\ensuremath{-}0.69102$ Oe, and $|{{B}_{4}}^{3}|=15.6$ Oe. The signs of the zero-field splitting parameters, determined by comparing relative line intensity ratios at 1.5 and at 77\ifmmode^\circ\else\textdegree\fi{}K, are such that the \ifmmode\pm\else\textpm\fi{}\textonehalf{} doublet spin state lies lowest. From this, it is concluded that the sign of the cubic-field splitting parameter is positive. The spectrum is found to exhibit a strong linewidth anisotropy. From the dependence of this broadening on the polar angle $\ensuremath{\theta}$, it is concluded that the sites occupied by this ion have a mosaic structure whose width is given by $\ensuremath{\delta}\ensuremath{\theta}=0.04$ deg. An x-ray diffraction analysis of the calcite single crystal yields a mosaic angle of $\ensuremath{\delta}\ensuremath{\theta}=0.011$ deg.

Ultrasonic Internal Conical Refraction in Rocksalt and Calcite

Evidence for the existence of ultrasonic internal conical refraction in rocksalt and calcite single crystals was obtained by observing the behavior of shear waves along one of the axes of threefold symmetry in each crystal. A method of selective absorption was used in conjunction with pulse-echo ultrasonic attenuation measurements. Absorbing material was pasted at various spots on the bottom of the specimen opposite the transducer to map the energy density in the wave. Positions of maximum flux coincided with predictions when the shear wave was not perturbed by misorientation-induced ultrasonic double refraction.

Secondary changes of length with pressure in experimentally deformed rocks

After plastic deformation at high confining pressure, marble and limestone show changes of length during release of the pressure that differ markedly from the normal elastic changes with pressure in undeformed specimens; after compression, the specimens tend to elongate much more than undeformed specimens, while after extension, the length changes tend to be less than in undeformed specimens. These departures from normal elastic behaviour, called here secondary length changes, are a function of the changes of pressure, not of time. During a subsequent pressure cycle, they are only partly reversible. They are accompanied by a decrease in density. The absence of such secondary length changes in polycrystalline copper and in single crystals of calcite leads to the hypothesis that they have their origin in intergranular stresses, which give rise to grain distortion and the opening of internal fractures. The chief sources of intergranular stresses in calcite aggregates are the anisotropy of yield stress of the grains, leading to residual stresses after the deformation, and the anisotropy of elastic linear compressibility of the grains, leading to further internal stresses being superposed on the former as the pressure falls. Similar secondary length changes have also been observed in granite, sandstone and serpentinite after deformation in compression, although the magnitudes of the effects are somewhat less than in the calcite rocks, especially in the case of the sandstone. These effects can probably also be attributed to intergranular stresses, especially in the granite where intergranular stresses can arise from the differences of properties between minerals. It is not clear whether such effects are to be expected in nature but they could be of importance in facilitating phenomena involving diffusion.

Petrofabric Analysis of Experimentally Deformed Calcite-Cemented Sandstones

Cylinders of sand crystals, composed of single crystals of calcite that poikilitically enclose detrital grains, and calcite-cemented sandstones from the Tensleep (Pennsylvanian, Wyoming) and Supai (Permian, Nevada) formations were experimentally deformed dry at confining pressures of 1-5 kilobars and temperatures of 150°-300° C. Thin sections of the undeformed and deformed specimens were studied microscopically to gain a better understanding of the behavior of sandstones in simulated tectonic environments. The calcite and the detrital grains (quartz, feldspar, and others), which have radically different physical and mechanical properties, are shown statistically to have deformed with respect to the principal stresses across the boundaries of the whole specimens rather than with regard to local stress concentrations at grain contacts. The deformation mechanisms of calcite and quartz are the same in the sandstones as in monomineralic aggregates, such as marbles and quartz sands. Statistically, twin lamellae are developed in those calcite grains that are favorably oriented for twin gliding with respect to the load axes. The resolved shear-stress coefficient for these twin planes averages 0.27. Compression and extension axes deduced from the best developed set of twin lamellae in each calcite grain yield derived positions for the principal stress axes that are in excellent agreement with those known from the experiments. In addition, the number of lamellae per millimeter increases with increased strain of the specimens. Quartz, feldspar, rock fragments, and garnet grains comprise the bulk of the detrital material. These deform primarily by fracturing. The microfractures in the grains are nearly planar features which, because of their geometric relationship to the known principal stress directions, are recognized as extension and shear fractures. They develop independently of thegrain mineralogy and, in quartz grains, greatly overshadow a slight tendency for fractures to parallel r{} and z{}. The degree of fracturing in a specimen, expressed as a fracture index, tends to increase with increased strain of the specimen. In sand crystals loaded unfavorably for twin gliding in the calcite crystal, extension fracturing in the detrital grains causes local reorientation of the stresses and produces twin lamellae in the adjacent calcite. Twin lamellae in calcite and fractures in detrital grains are shown to be criteria for simulated tectonism. The development of lamellae and microfractures is directly related to the orientations of the principal stresses in heterogeneous, sedimentary rocks at the time of deformation.

Elastic constants of Solenhofen limestone and their dependence upon density and saturation

An ultrasonic pulse technique at 3 Mc/s was used to determine the longitudinal (P) and shear (S) elastic wave velocities in 24 samples of Solenhofen limestone at atmospheric pressure and 25°C, with and without water saturation. The densities of the air-dried samples range from 2.48 to 2.66 g/cm3. Comparison of the results with the single crystal constants of calcite and with the high-pressure data of Hughes and Cross is made. The conclusions are: (1) Both P and S velocities in dry Solenhofen limestone are linear functions of the density at atmospheric pressure and 25°C. The velocity-density slopes for the dry samples are 5.11 and 1.625 km/sec per g/cm3 for the P and S velocities, respectively. (2) Inhomogeneity of Solenhofen limestone is slight, resulting in less than 3 per cent deviation in the P and S velocities from the least-squares curves. (3) Extrapolation of both the dry and the saturated P and S velocities to the limiting density (density of calcite) shows that, as the porosity decreases, the behavior of the limestone can be interpreted as an aggregate of randomly oriented single crystals of calcite. (4) Application of hydrostatic pressure to dry Solenhofen results in the same limiting P velocity as is obtained from the compressibility and density of the poly crystalline components at 1 atmosphere pressure. (5) Water saturation of the limestones decreases both the P and S velocities. The saturated velocities are linear functions of the saturated densities. There seems to be a small decrease in the matrix rigidity of the limestone with saturation.

Suprageneric Classification of Rhizopodea: ABSTRACT

A proposed suprageneric classification of the class Rhizopodea is given, with particular revision to the order Foraminiferida. The subclass Lobosia contains the orders Amoebida, Arcellinida, and Mycetozoida. The subclass Filosia includes the orders Aconchulinida and Gromida, and the subclass Granuloreticulosia includes the orders Athalamida, Monothalamida, Foraminiferida, Xenophyophorida, and Proteomyxida. The order Foraminiferida includes 7 superfamilies based on wall composition and structure, and method of test growth: Lagynacea (gelatinous to chitinous tests) Astrorhizacea (non-sepate, agglutinated test), Lituolacea (septate and agglutinated), Parathuramminacea (non-septate, with wall of calcareous granules in calcareous cement), Endothyracea (septate, with granular or fibrous calcareous wall, generally with two distinct layers), Fusulinacea (basically three distinct wall layers), Miliolacea (porcellaneous). The so-called calcareous perforate foraminifera represent ten superfamilies, six of which have radially built walls: the Nodosariacea (with basically radiate aperture), Buliminacea (high spired test and commonly with internal toothplate), Asterigerinacea (enrolled, no-canalicula e, walls and septa single layered), Rotaliacea (canaliculate, enrolled), Globigerinacea (planktonic), Orbitoidacea (with double walls and primarily formed double septa). The last four superfamilies include the Cassidulinacea (walls of perforate granular calcite), Carterinacea (walls of calcareous secreted spicules), Robertinacea (wall of aragonite and chambers internally subdivided) and Spirillinacea (wall optically acts as a single crystal of calcite). A complete synonymy of all suprageneric categories has been compiled, in order to determine the correct family group names to be utilized, on a priority basis. End_of_Article - Last_Page 133------------

Electron Spin Resonance in Neutron-Irradiated Calcite

Electron spin resonance spectra were stadied for pure single calcite crystals irradiated with fast neutrons in which extremely narrow lines (width approximately 0.1 gauss) with thermal relaxation times at liquid helium temperatures of several seconds were observed. Crystals irradiated with doses of 1017 , 1018, and 1019 fast neutrons became brownish in color, and the main features of the spectrum of the sample irradiated with 1017 neutrons were studied at 8700 Mc and 77 K; whereas H0 was rotated in a plane normal to the c axis. The implications of the observations are discussed.

Exsolution of Dolomite from Calcite

Calcites from a variety of marbles have been investigated by single-crystal X-ray techniques. Most of the specimens were relatively high-grade metamorphic carbonates, from amphibolite and granulite facies rocks. The typically clouded calcite of these rocks commonly contains dolomite, revealed by the X-ray photographs but not always observed under the microscope. This finely disseminated dolomite is in the same crystallographic orientation as the host calcite and is almost certainly an exsolution product. An oriented intergrowth of dolomite in calcite was produced in the laboratory. The original material was an echinoid test made up of spongy but single-crystal calcite with approximately 10 per cent $MgCO_3$ in metastable solid solution. This amount of Mg is stable in calcite only at temperatures well above 600° C; therefore exsolution of some of the Mg as dolomite took place when the sample was heated under $CO_2$ pressure at 500° C. The host magnesian calcite retained its identity as a single crystal, an...

Crystallography of Echinoid Calcite

Each element of an echinoderm skeleton is a single crystal of calcite. The universal stage method is used to determine orientations of crystals in coronal plates of fossil and Recent regular echinoids. Stereographic diagrams are presented which relate crystallographic orientation to the symmetry of the echinoid skeleton. Most species exhibit one of two preferred orientations and show little variation within a single individual. In one of the preferred orientations, c-axes are perpendicular to the test surface; in the other, they are tangent to the test surface and are aligned longitudinally. In the present sample, crystallographic orientation shows no variation within single genera, and most echinoid families studied have but one orientation. The two preferred orientations appear to have evolved independently several times during echinoid history. Preferred orientation, as an adaptation to the environment, may be related to the organisms' sensitivity to light.

Studies of Individual Dislocations in Crystals by X-Ray Diffraction Microradiography

The distribution of imperfections within the interior of crystals has been studied using ``projection topographs'' which are x-ray diffraction images showing a projection of a slice of crystal and the imperfections in it. Individual dislocations have been observed in single crystals of diamond, silicon, germanium, lithium fluoride, sodium chloride, silver chloride, magnesium oxide, calcite, quartz, and aluminum. From the variation of dislocation contrast with the orientation of the x-ray reflecting plane the direction of Burgers vector can be found. Dislocations can be seen with good contrast when the product of linear absorption coefficient μ and slice thickness t is of the order of unity or less. If μt≫1 the contrast is reversed through the Borrmann effect. Stereo pairs of projection topographs can be prepared from the pair of reflections hkl and h̄k̄l̄.

The solubility of calcite in carbon dioxide solutions

The solubility of calcite in water is reported for temperatures between 100 and 300 degrees , at partial pressures of carbon dioxide ranging from 1 - 40 atmospheres. Values for the solubility product and the free energy of solution of calcite are derived. From a study of the kinetics of solution of a single calcite crystal at these temperatures, the Arrhenius energy of activation was found to be very low. The rate determining step is suggested to be a diffusion or desorption process.

INTERNAL FRICTION OF FINE‐GRAINED LIMESTONES AT ULTRASONIC FREQUENCIES

Ultrasonic pulse measurements in the 3 to 10 mc/sec frequency range were made of shear and dilatational absorption and velocity for three samples of fine grained, homogeneous, and well compacted limestones at atmospheric conditions. The complex moduli of elasticity were calculated for two limestone specimens of different grain size and porosity. A “hysteresis” or “solid‐friction” type of loss for waves of dilatation was found in all three limestone specimens. Shear measurements made in one of the limestones also show a “hysteresis” type of loss. Comparison of the dilatational loss in a single crystal of calcite to the dilatational loss in the limestones indicates that the internal friction for the limestone specimens probably occurs at the grain boundaries.

EFFECTS OF GAMMA RADIATION ON THE EXPERIMENTAL DEFORMATION OF CALCITE AND CERTAIN ROCKS

Calcite single crystals, Yule marble, and certain other rocks were deformed dry at room temperature under constant confining pressures of 2000 or 2750 bars at a strain rate of 1 per cent per minute. Uniaxial compression and extension experiments were made; for the marble, parallel and normal to c v fabric maximum; for the single crystals, parallel to c v and normal to m {1010}. Yule marble exposed to 17 megaroentgens of gamma radiation before deformation turned from white to bright blue when deformed. Intensity of color varied with orientation and amount of strain. Irradiated single crystals changed from amber to ultramarine blue when the greatest principal stress paralleled c v . No color change occurred when the least principal stress paralleled c v . Irradiation resulted in small reductions in yield stress of most of the rocks, but not of the single crystals. Petrographic studies of the single crystals revealed that established principal deformation mechanisms, twin gliding on e {0112} and translation gliding on r {1011}, were still operative and confirmed that blue color was associated with translation gliding only. In Yule marble blue grains had deformed by translation, clear grains by twinning. Thermoluminescence studies indicated that compression along c v yielded a new peak in the glow curves (at 280°), not present in undeformed irradiated calcite. Extension along c v increased the magnitude of the glow curve at about 240° but gave rise to no new peaks.

DEFORMATION OF YULE MARBLE. PART VII: DEVELOPMENT OF ORIENTED FABRICS AT 300°C–500°C

Yule marble has been deformed at 400° and 500° C and a confining pressure of 5000 atmospheres. Its strength, relative to that at room temperature, is 47 per cent at 400° C and 41 per cent at 500° C, measured in all cases at 3 per cent strain. Textures of the deformed rock are similar to those of marble comparably deformed at 300° C. Sander's method of “axial-distribution analysis” has been used to test homogeneity of fabric in the original marble and in a specimen shortened 19 per cent at 300° C. In both cases aggregates of a few tens of grains show local very strong preferred orientation of the lattice. In this respect there is no effect that can be correlated with deformation. Petrofabric analysis of specimens shortened 40 per cent or elongated 90–120 per cent demonstrates strong preferred orientation of the c axis at 10°–30° to the axis of compression or at 60°–80° to the axis of extension. The symmetry of the deformed fabrics corresponds with the symmetry of strain, which is determined by the orientation of the axis of extension or compression in relation to the preferred orientation pattern of the original fabric. Behavior of grains is predicted on the basis of a hypothesis of approximately homogeneous deformation, involving twin gliding on ![Graphic][1] and translation gliding on ![Graphic][2] . Behavior of individual grains is also reconstructed from evidence of observed internal rotation of early e lamellae, interpreted in the light of previous observations on single crystals of calcite. The same data are used to compute shortening or elongation of individual grains parallel to the axis of applied force. The behavior and strain so deduced and the details of observed preferred-orientation patterns agree with prediction. It should prove possible to compute strain in geologically deformed rocks on the basis of petrographically measured effects of internal rotation. However, at present this can be attempted only for those few minerals, such as calcite and dolomite, whose glide systems have been ascertained experimentally. Possible analogy between a natural marble fabric and a fabric resulting from experimental compression is noted. [1]: /embed/inline-graphic-1.gif [2]: /embed/inline-graphic-2.gif

EXPERIMENTAL DEFORMATION OF CALCITE CRYSTALS

This paper reports the results of experimental plastic deformation of cylinders cut from single crystals of clear calcite. A wide range of crystallographic orientation in relation to compression or extension of cylinders is involved. Most experiments were conducted at 20°C and 5000 or 10,000 atmospheres confining pressure, or at 300°C and 5000 atmospheres. Temperatures of 150°C and 400°C were employed in a few additional cases. Shortening or extension of the whole cylinder ranges from 2 to 20 per cent; but in some extension experiments necking of the cylinder has locally increased the strain by a factor of 3 or 4. Stress-strain curves for typical experiments are given. Where the orientation permits, the dominant mechanism of deformation at all temperatures is twin gliding on ![Graphic][1] . Cylinders so oriented that twin gliding cannot occur deform plastically by some alternative mechanism. At 20°C calcite is many times stronger when oriented unfavorably for ![Graphic][2] twin gliding than when favorably oriented; but with rising temperature this difference in strength rapidly diminishes. Analysis of stress-strain data for variously oriented crystals at 300°C points to translation gliding on ![Graphic][3] as the alternative mechanism to twin gliding on ![Graphic][4] . However, no satisfactory correlation of stress-strain data for 20°C could be established on the basis of this or any other simple glide system. An independent approach to the problem is based on analyses of rotational effects observed microscopically in thin sections of the deformed material. Deformed sectors ( e.g ., kink bands) in the cylinder are found to be externally rotated about an axis parallel to the glide plane and normal to the glide line of the active system. At the same time, early-formed lamellae (such as ![Graphic][5] twin lamellae) become internally rotated within the deformed crystal, the axis of rotation being the intersection of the glide plane and the rotated lamella. The senses of internal and external rotation in a given sector of the crystal are mutually opposed, and for a given glide plane each can be deduced for a given stress system. Analysis of directions and amounts of internal and external rotation in many instances leads to unique identification of the active glide system. The glide systems so identified include (1) twin gliding on ![Graphic][6] , parallel to the edge ![Graphic][7] (2) translation gliding on ![Graphic][8] parallel to the edge ![Graphic][9] , effective at all temperatures; (3) translation gliding on ![Graphic][10] , parallel to the edge ![Graphic][11] , effective at low temperatures. Translation gliding on ![Graphic][12] in the sense opposite to that of twin gliding is discarded as a possible mechanism of deformation; there is likewise no evidence of gliding on {0001}. Visible effects of deformation (lamellae, partings, deformation bands, kink bands, etc.) for individual experiments embracing the complete range of orientation are described in detail and illustrated by photographs, line drawings, and projections. The criteria by which various kinds of internally rotated lamellae may be recognized are summarized (Table 6), and the possible applications of our conclusions in interpreting the fabric of an experimentally deformed multicrystalline aggregate—Yule marble—are discussed. [1]: /embed/inline-graphic-1.gif [2]: /embed/inline-graphic-2.gif [3]: /embed/inline-graphic-3.gif [4]: /embed/inline-graphic-4.gif [5]: /embed/inline-graphic-5.gif [6]: /embed/inline-graphic-6.gif [7]: /embed/inline-graphic-7.gif [8]: /embed/inline-graphic-8.gif [9]: /embed/inline-graphic-9.gif [10]: /embed/inline-graphic-10.gif [11]: /embed/inline-graphic-11.gif [12]: /embed/inline-graphic-12.gif

An Unusual Stalactite from Saltville, Virginia

ABSTRACT The stalactite described and illustrated shows an unusual development of a single crystal of calcite expressed in external as well as internal form.

Measurement of the Refraction of X-Rays in a Prism by Means of the Double X-Ray Spectrometer

In the double spectrometer used, x-radiation was reflected from a crystal surface $A$ through a slit $S$ on to a second crystal surface $B$, and thence to the ionization chamber. Surfaces $A$ and $B$ were obtained by splitting a single crystal of calcite. The curve obtained for Mo $\mathrm{K}\ensuremath{\alpha}$ by rocking $B$ is a triangle with base about 8 seconds wide. The effect of interposing a prism of aluminum (angle 166\ifmmode^\circ\else\textdegree\fi{}) between $S$ and $B$ was to shift the curve 5.62\ensuremath{''} on the average, giving $1\ensuremath{-}\ensuremath{\mu}=1.68\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$. The value calculated from the Lorentz dispersion formula is 1.77\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$. For Cu $\mathrm{K}\ensuremath{\alpha}$ the shift was 5.53\ensuremath{''} for a 116\ifmmode^\circ\else\textdegree\fi{} prism, giving $1\ensuremath{-}\ensuremath{\mu}=8.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ as compared with the calculated value 8.36\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$. The accuracy of this method is thus within 5 percent.