The properties of anisotropic conical failure surfaces in relation to the Mohr–Coulomb criterion

An earlier publication¹ considered the properties of circular conical failure surfaces whose axes coincide with the space diagonal in principal stress space. The present work uses a similar approach to analyse conical surfaces that are offset from the space diagonal. It is shown that cones fitted to the Mohr–Coulomb surface in triaxial compression contain a potential singularity. The occurrence and location of the singularity depends on the Mohr–Coulomb friction angle to which the surface is fitted in triaxial extension. It is shown that for a cone fitted to the same friction angle in both triaxial extension and compression, singular conditions occur when that angle reaches \documentclass{article}\pagestyle{empty}\begin{document}$ \sin ^{ - 1} \left({\sqrt 7 - 2} \right)\left({ = 40.22^\circ } \right) $\end{document}. Even cones fitted to smaller friction angles give significant overestimations of material strength for certain stress paths.     

Study on the action of the active earth pressure by variational limit equilibrium method

Within the framework of limiting equilibrium approach, the problem of active earth pressure on rigid retaining wall is formulated in terms of the calculus of variations by means of Lagrange multipliers. It is transcribed as the functional of extreme‐value problem by two undetermined function arguments, and is further transformed into determining the minimax solution of restrained functions incorporating the geometrical relations of the problem. The function of (fmincon) in the optimization toolbox of MATLAB 6.1 can be used to find the minimax solution. Computation results show there exist two kinds of modes of failure sliding along plane surface and rotating around log‐spiral cylinder surface when the soil behind the walls reaches the critical active state. The magnitude of active earth pressure in the case of translational mode is less than that in the case of rotational mode. The location of action point of earth pressure in the case of translational mode is at or below $\frac{1}{3}$ height of the wall, and in the case of rotational mode, is above $\frac{1}{3}$ height of the wall. Preliminary study indicates a pair of numbers by two theoretical modes can be regarded as an interval estimation of active pressure. Copyright © 2009 John Wiley & Sons, Ltd.     

基于摩擦与变形耗能的滚石切向恢复系数影响因素

切向恢复系数是滚石碰撞回弹的重要控制参数,目前的理论公式不能完全反映其作用机制,这是滚石动力学研究的一个难点问题.为此,根据滚石不同的回弹状态,提出基于入射角度变化的切向力模型;进一步,以切向接触理论和动能定理为基础,考虑碰撞过程中切向的摩擦耗能与变形耗能,推导了切向恢复系数的理论公式;最后研究入射速度、入射角、被撞击物体的变形模量对切向恢复系数的影响.结果表明:滚动回弹的切向恢复系数主要受切向变形量的影响;滑动回弹时,入射速度对切向恢复系数的影响参数为\begin{document}$ {v}^{\frac{1}{20}} $\end{document},切向恢复系数随着其增加而缓慢减少;入射角度对切向恢复系数的影响参数为$ \frac{\mathrm{c}\mathrm{o}{\mathrm{s}}^{\frac{1}{20}}{\beta }_{i}}{\mathrm{t}\mathrm{a}\mathrm{n}{\beta }_{i}} $,切向恢复系数随其增加而增大;被撞击物体的变形模量对切向恢复系数的影响参数为$ {E}_{2}^{-\frac{5}{8}} $,切向恢复系数随其增加而增加.基于摩擦与变形耗能的切向恢复系数计算公式为滚石的碰撞回弹过程提供了新的计算模型.      

Wave propagation in an inhomogeneous cross‐anisotropic medium

Analytical solutions for wave velocities and wave vectors are yielded for a continuously inhomogeneous cross‐anisotropic medium, in which Young's moduli (E, E′) and shear modulus (G′) varied exponentially as depth increased. However, for the rest moduli in cross‐anisotropic materials, ν and ν′ remained constant regardless of depth. We assume that cross‐anisotropy planes are parallel to the horizontal surface. The generalized Hooke's law, strain–displacement relationships, and equilibrium equations are integrated to constitute governing equations. In these equations, displacement components are fundamental variables and, hence, the solutions of three quasi‐wave velocities, V_P, V_SV, and V_SH, and the wave vectors, $\mathop{\mathop{l}\limits^{\rightharpoonup}}\nolimits_{P}$ $\mathop{\mathop{l}\limits^{\rightharpoonup}}\nolimits_{\mathit{SV}}$, and $\mathop{\mathop{l}\limits^{\rightharpoonup}}\nolimits_{{\mathit{SH}}}$, can be generated for the inhomogeneous cross‐anisotropic media. The proposed solutions and those obtained by Daley and Hron, and Levin correlate well with each other when the inhomogeneity parameter, k, is 0. Additionally, parametric study results indicate that the magnitudes and directions of wave velocity are markedly affected by (1) the inhomogeneous parameter, k; (2) the type and degree of geomaterial anisotropy (E/E′, G′/E′, and ν/ν′); and (3) the phase angle, θ. Consequently, one must consider the influence of inhomogeneous characteristic when investigating the behaviors of wave propagation in a cross‐anisotropic medium. Copyright © 2009 John Wiley & Sons, Ltd.     

Cation stacking faults in magnesium germanate spinel

Widely extended, cation stacking faults in experimentally deformed Mg₂GeO₄ spinel have been studied using transmission electron microscopy (TEM). The faults lie on {110} planes. The displacement vector is of the form \(\frac{1}{4}\left\langle {1\bar 10} \right\rangle \) and is normal to the fault plane. The partial dislocations which bound the stacking fault have colinear Burgers vectors of the form \(\frac{1}{4}\left\langle {1\bar 10} \right\rangle \) which are normal to the fault plane.     

Parameter identification method for determination of elastic modulus of rock based on adjoint equation and blasting wave measurements

The purpose of this paper is to present a parameter identification method to determine the force of a blast and the elastic modulus of the ground using the measurements of a dynamic elastic wave, the adjoint equation method of optimal control theory, and the finite element method. Before the excavation of rocky ground, it is important to estimate the ground properties. In this paper, the elastic modulus is determined as the performance function is minimized using a technique based on the first‐order adjoint method. The performance function is a square sum of the discrepancies between the computed and the observed values of the velocities. After the determination of the magnitude of the blasting force, we can determine the elastic modulus of the rock. As the basic equation to calculate the velocities of dynamic elastic body, elastic equilibrium equations with linear viscosity are employed. The adjoint equation method has been utilized in order to calculate the gradient of the performance function with respect to the parameters. The gradient of the performance function is calculated using the first‐order adjoint equation. The weighted gradient method is applied for minimization. In order to solve the state equations in space and time, the finite element method and the Newmark $\frac{1}{4}$ method are used. In this paper, we tested the practical application of our proposed method for determination of the elastic modulus of rock at the Ikawa tunnel located in the Tokushima prefecture, Japan. Copyright © 2009 John Wiley & Sons, Ltd.     

A seismological overview of the April 25, 2015 M<Subscript>w</Subscript>7.8 Nepal earthquake

This paper aims to determine the damage distribution and to analyze the available strong motion records of the April 25, 2015 Nepal earthquake and its eight aftershocks. For this purpose, an earthquake investigation team was dispatched to Nepal from May 6 to 11, 2015 to evaluate the damages of the epicentral region and the four affected cities containing Kathmandu, Bhaktapur, Gorkha, and Pokhara. Based on the observations from the damages to the built environment, an iso-intensity map is prepared on the EMS-98 intensity scale in which the maximum intensity in the epicentral region is estimated to be about VIII. However, based on the geological and geotechnical evidences such as landslide volumes and ground fissures, the maximum intensity can be inferred about IX or X on the International Union for Quaternary Research (INQUA) intensity scale. In addition, the available strong motion data of the 2015 Nepal mainshock and its eight large aftershocks recorded at the KATNP accelerometric station in Kathmandu were processed and analyzed. In order to investigate the probable site effects, the Fourier amplitude spectra (FAS) of the horizontal north-south (N) and east-west (E) components and the average of them (H _avg) were divided to the FAS of the vertical (Z) component and thus, the \( \raisebox{1ex}{$ N$}\!\left/ \!\raisebox{-1ex}{$ Z$}\right. \), \( \raisebox{1ex}{$ E$}\!\left/ \!\raisebox{-1ex}{$ Z$}\right. \), \( \raisebox{1ex}{${H}_{\mathrm{avg}}$}\!\left/ \!\raisebox{-1ex}{$ Z$}\right. \) spectral ratios were calculated. Based on these horizontal to vertical spectral ratios, a low-frequency peak at about 0.2–0.3 Hz (3.5–5-s period) is observed clearly in all the records. Accordingly, the repeated results might imply site amplification due to the thick alluvial deposits and the high groundwater level at the KATNP accelerometric station within the Kathmandu basin. It should be noted that all the horizontal to vertical spectral ratios of the aftershocks show a high peak at around 1.5–3 Hz, which is missed in the horizontal to vertical spectral ratio of the mainshock. On the other hand, considering the low angle of the causative fault plane, a near-source directivity effect on the fault normal direction (here, the vertical component) of the April 25, 2015 mainshock rupture may exist. Therefore, vertical to horizontal spectral ratios (\( \raisebox{1ex}{$ Z$}\!\left/ \!\raisebox{-1ex}{$ N$}\right. \) and \( \raisebox{1ex}{$ Z$}\!\left/ \!\raisebox{-1ex}{$ E$}\right. \)) were also calculated to find the vertical peak more clearly. The figures confirmed a peak at the frequency of 1.5–3 Hz in the mainshock spectra which is not repeated on the aftershock spectra and thus can probably be attributed as the pulse of directivity effect toward Kathmandu. This inferred directivity pulse can be also well distinguished on the velocity and displacement time histories of the mainshock.     

Water diffusion in Mount Changbai peralkaline rhyolitic melt

Diffusion couple experiments with wet half (up to 4.6 wt%) and dry half were carried out at 789–1,516 K and 0.47–1.42 GPa to investigate water diffusion in a peralkaline rhyolitic melt with major oxide concentrations matching Mount Changbai rhyolite. Combining data from this work and a related study, total water diffusivity in peralkaline rhyolitic melt can be expressed as:

$ D_{{{\text{H}}_{ 2} {\text{O}}_{\text{t}} }} = D_{{{\text{H}}_{ 2} {\text{O}}_{\text{m}} }} \left( {1 - \frac{0.5 - X}{{\sqrt {[4\exp (3110/T - 1.876) - 1](X - X^{2} ) + 0.25} }}} \right), $

$ {\text{with}}\;D_{{{\text{H}}_{ 2} {\text{O}}_{\text{m}} }} = \exp \left[ { - 1 2. 7 8 9- \frac{13939}{T} - 1229.6\frac{P}{T} + ( - 27.867 + \frac{60559}{T})X} \right], $

where D is in m² s^?1, T is the temperature in K, P is the pressure in GPa, and X is the mole fraction of water and calculated as X = (C/18.015)/(C/18.015 + (100 ? C)/33.14), where C is water content in wt%. We recommend this equation in modeling bubble growth and volcanic eruption dynamics in peralkaline rhyolitic eruptions, such as the ~1,000-ad eruption of Mount Changbai in North East China. Water diffusivities in peralkaline and metaluminous rhyolitic melts are comparable within a factor of 2, in contrast with the 1.0–2.6 orders of magnitude difference in viscosities. The decoupling of diffusivity of neutral molecular species from melt viscosity, i.e., the deviation from the inversely proportional relationship predicted by the Stokes–Einstein equation, might be attributed to the small size of H₂O molecules. With distinct viscosities but similar diffusivity, bubble growth controlled by diffusion in peralkaline and metaluminous rhyolitic melts follows similar parabolic curves. However, at low confining pressure or low water content, viscosity plays a larger role and bubble growth rate in peralkaline rhyolitic melt is much faster than that in metaluminous rhyolite.

     

On Bagnold's sediment transport equation in tidal marine environments and the practical definition of bedload

Bagnold's sediment transport equation has proved to be important in studying tidal marine environments. This paper discusses three problems concerning Bagnold's transport equation and its practical application:

• 1 Bagnold's suspended-load transport equation and the total-load transport equation with are incorrect from the viewpoint of energy conservation. In these equations the energy loss due to bedload transport has been counted twice. The correct form should be for suspended-load transport and for total-load transport with
• 2 The commonly used Bagnold's transport coefficient K varies as a non-linear function of the dimensionless excess shear stress, which can be represented best by the power law , where the coefficient A and exponent B depend on sediment grain size D. The empirical values of A and B for fine to medium grained sands are determined using Guy et al.'s (1966) flume-experiment data.
• 3 The sediment transport rates predicted from this equation are compared with bedform migration measurements in the flume and the field. This comparison shows that the sediment transport rates measured from bedform migrations are higher than the predicted bedload transport rates, but comparable to the calculated total-load (bedload plus intermittent suspended-load) transport rates. This indicates that bedform migration involves both bedload and intermittent suspended-load transport. As a logical conclusion, bedform migration data should be compared with Bagnold's total-load transport equation rather than with his bedload transport equation. In this respect the term ‘bed material’ might be more appropriate than the term ‘bedload’ for estimating sediment transport rate from bedform migration data.

The sediment transport rates predicted from this modified Bagnold transport equation are in good agreement with field measurements of bedform migration rates in four individual tidal marine environments, which cover a wide range of sediment grain size, flow velocity and bedform conditions (ranging from small ripples, megaripples to sandwaves).     

Analysis of dislocations in some naturally deformed plagioclase feldspars

Dislocations in intermediate plagioclase feldspars, which were deformed under granulite facies conditions, have been analysed. The study reveals extensive ductile deformation by intracrystalline slip and by twinning. Six out of the seven possible Burgers vectors were identified: \(b = \left[ {001} \right],\tfrac{1}{2}\left[ {110} \right],\tfrac{1}{2}\left[ {1\bar 10} \right],\left[ {101} \right],\tfrac{1}{2}\left[ {112} \right]and\tfrac{1}{2}\left[ {1\bar 12} \right]\) . Most, perhaps all, dislocations are dissociated by up to 200 Å. The microstructure is dominated by [001] screw dislocations, most of which appear to be dissociated in (010). The dominant slip system appears to be (010) [001]. Large grain-to-grain variations in the density of free dislocations indicate that the plastic strain in individual grains depended upon the Schmid factor for (010) [001]. The microstructure suggests that the rate-controlling step for high-temperature creep of plagioclase is cross-slip of extended [001] screw dislocations. The rheological contrast between feldspar and quartz is partly due to a difference in stacking fault energy.     

Kinetic rate laws as derived from order parameter theory II: Interpretation of experimental data by laplace-transformation,the relaxation spectrum,and kinetic gradient coupling between two order parameters

A unifying theory of kinetic rate laws, based on order parameter theory, is presented. The time evolution of the average order parameter is described by $$\langle Q\rangle \propto \smallint P(x)e^{^{^{^{^{^{^{ - xt} } } } } } } dx = L(P)$$ where t is the time, x is the effective inverse susceptibility, and L indicates the Laplace transformation. The probability function P(x) can be determined from experimental data by inverse Laplace transformation. Five models are presented:

1. Polynomial distributions of P(x) lead to Taylor expansions of 〈Q〉 as $$\langle Q\rangle = \frac{{\rho _1 }}{t} + \frac{{\rho _2 }}{{t^2 }} + ...$$
2. Gaussian distributions (e.g. due to defects) lead to a rate law $$\langle Q\rangle = e^{ - x_0 t} e^{^{^{^{^{\frac{1}{2}\Gamma t^2 } } } } } erfc\left( {\sqrt {\frac{\Gamma }{2}} t} \right)$$ where x ₀ is the most probable inverse time constant, Γ is the Gaussian line width and erfc is the complement error integral.
3. Maxwell distributions of P are equivalent to the rate law 〈Q〉∝e^?k√t .
4. Pseudo spin glasses possess a logarithmic rate law 〈Q〉∝lnt.
5. Power laws with P(x)=x ^a lead to a rate law: ln〈Q〉=-(α + 1) ln t.

The power spectra of Q are shown for Gaussian distributions and pseudo spin glasses. The mechanism of kinetic gradient coupling between two order parameters is evaluated.     

Thermal expansion and spontaneous strain associated with the normal-incommensurate phase transition in melilites

The linear thermal expansions of åkermanite (Ca₂MgSi₂O₇) and hardystonite (Ca₂ZnSi₂O₇) have been measured across the normal-incommensurate phase transition for both materials. Least-squares fitting of the high temperature (normal phase) data yields expressions linear in T for the coefficients of instantaneous linear thermal expansion, $$\alpha _1 = \frac{1}{l}\frac{{dl}}{{dT}}$$ for åkermanite: $$\begin{gathered} \alpha _{[100]} = 6.901(2) \times 10^{ - 6} + 1.834(2) \times 10^{ - 8} T \hfill \\ \alpha _{[100]} = - 2.856(1) \times 10^{ - 6} + 11.280(1) \times 10^{ - 8} T \hfill \\ \end{gathered} $$ for hardystonite: $$\begin{gathered} \alpha _{[100]} = 15.562(5) \times 10^{ - 6} - 1.478(3) \times 10^{ - 8} T \hfill \\ \alpha _{[100]} = - 11.115(5) \times 10^{ - 6} + 11.326(3) \times 10^{ - 8} T \hfill \\ \end{gathered} $$ Although there is considerable strain for temperatures within 10° C of the phase transition, suggestive of a high-order phase transition, there appears to be a finite ΔV of transition, and the phase transition is classed as “weakly first order”.     

Observations on the extended Matsuoka–Nakai failure criterion

The equivalent Mohr–Coulomb (M‐C) friction angle ? (J. Geotech. Eng. 1990; 116 (6):986–999) of the extended Matsuoka–Nakai (E‐M‐N) criterion has been examined under all possible stress paths. It is shown that ? depends only on the ratio of cohesion to confining stress c^′/σ and the frictional angle ?, where ? is the friction angle measured in triaxial compression (or extension) to which the E‐M‐N surface is fitted. It is also shown that ? is independent of c^′, when σ=0 and of σ when c^′=0, with the former representing an upper bound and the latter a lower bound of ? for any particular stress path. The closest point projection method has also been implemented successfully with the E‐M‐N criterion, and plane strain and axisymmetric element tests performed to verify some theoretical predictions relating to failure and post‐yielding behavior. Finally, a bearing capacity problem was analyzed using both E‐M‐N and M‐C, highlighting the conservative nature of M‐C for different friction angles. Copyright © 2009 John Wiley & Sons, Ltd.     

Theoretical calculation of equilibrium Mg isotope fractionation between silicate melt and its vapor

Isotope fractionation during the evaporation of silicate melt and condensation of vapor has been widely used to explain various isotope signals observed in lunar soils, cosmic spherules, calcium–aluminum-rich inclusions, and bulk compositions of planetary materials. During evaporation and condensation, the equilibrium isotope fractionation factor (α) between high-temperature silicate melt and vapor is a fundamental parameter that can constrain the melt’s isotopic compositions. However, equilibrium α is difficult to calibrate experimentally. Here we used Mg as an example and calculated equilibrium Mg isotope fractionation in MgSiO₃ and Mg₂SiO₄ melt–vapor systems based on first-principles molecular dynamics and the high-temperature approximation of the Bigeleisen–Mayer equation. We found that, at 2500 K, δ²⁵Mg values in the MgSiO₃ and Mg₂SiO₄ melts were 0.141?±?0.004 and 0.143?±?0.003‰ more positive than in their respective vapors. The corresponding δ²⁶Mg values were 0.270?±?0.008 and 0.274?±?0.006‰ more positive than in vapors, respectively. The general \(\alpha - T\) equations describing the equilibrium Mg α in MgSiO₃ and Mg₂SiO₄ melt–vapor systems were: \(\alpha_{{{\text{Mg}}\left( {\text{l}} \right) - {\text{Mg}}\left( {\text{g}} \right)}} = 1 + \frac{{5.264 \times 10^{5} }}{{T^{2} }}\left( {\frac{1}{m} - \frac{1}{{m^{\prime}}}} \right)\) and \(\alpha_{{{\text{Mg}}\left( {\text{l}} \right) - {\text{Mg}}\left( {\text{g}} \right)}} = 1 + \frac{{5.340 \times 10^{5} }}{{T^{2} }}\left( {\frac{1}{m} - \frac{1}{{m^{\prime}}}} \right)\), respectively, where m is the mass of light isotope ²⁴Mg and m′ is the mass of the heavier isotope, ²⁵Mg or ²⁶Mg. These results offer a necessary parameter for mechanistic understanding of Mg isotope fractionation during evaporation and condensation that commonly occurs during the early stages of planetary formation and evolution.     

Über p-Veatehit,eine neue Veatehit-Varietät aus dem Zechsteinsalz

Zusammenfassung Im Älteren Steinsalz von Reyershausen bei Göttingen wurde eine neue Veatchit-Varietät gefunden mita ₀ = 6,721 Å,b ₀ = 20,81 Å,c ₀ = 6,64₇ Å, = 119° 4; Raumgruppe oderP2₁,Z = 4[(Sr, Ca) O · 3 B₂O₃ · 2 H₂O]. (010) ist die Ebene der vollkommenen Spalt-barkeit. Die Polymorphie der Veatehit-Minerale wird geometrisch durch geringfügige Deformationen der rhombischen Raumgruppe (bzw.A2₁ a m) erklärt.Der neue Vertreter wirdp-Veatehit (mit einfach-primitivem Raumgitter) genannt im Unterschied zum Original-Veatehit, der in die Raumgruppe gehört und dessen Symametrieebene senkrecht auf der vollkommenen Spaltebene steht.     

Thermodynamic modelling of non-convergent ordering in orthopyroxenes: a comparison of classical and Landau approaches

The excess Gibbs free energy due to non-convergent ordering is described by a Landau expansion in which configurational and non-configurational entropy contributions are separated:

A thermodynamic formulation of hydrous cordierite

A thermodynamic formulation of hydrous Mg-cordierite (Mg₂Al₄Si₅O₁₈·nH₂O) has been obtained by application of calorimetric and X-ray diffraction data for hydrous cordierite to the results of hydrothermal syntheses. The data include measurements of the molar heat capacity and enthalpy of hydration and the molar volume. The synthesis data are consistent with a thermodynamic formulation in which H₂O mixes ideally on a single crystallographic site in hydrous cordierite. The standard molar Gibbs free energy of hydration is-9.5±1.0 kJ/mol (an average of 61 syntheses). The standard molar entropy of hydration derived from this value is-108±3 J/mol-K. An equation providing the H₂O content of cordierite as a function of temperature and fugacity of H₂O is as follows (n moles of H₂O per formula unit, n<1): $$\begin{gathered}n = {{f_{{\text{ H}}_{\text{2}} O}^{\text{V}} } \mathord{\left/{\vphantom {{f_{{\text{ H}}_{\text{2}} O}^{\text{V}} } {\left( {f_{{\text{ H}}_{\text{2}} O}^{\text{V}} + {\text{exp}}\left[ { - {\text{3}}{\text{.8389}} - 5025.2\left( {\frac{1}{T} - \frac{1}{{298.15}}} \right)} \right.} \right.}}} \right.\kern-\nulldelimiterspace} {\left( {f_{{\text{ H}}_{\text{2}} O}^{\text{V}} + {\text{exp}}\left[ { - {\text{3}}{\text{.8389}} - 5025.2\left( {\frac{1}{T} - \frac{1}{{298.15}}} \right)} \right.} \right.}} \hfill \\{\text{ }}\left. {\left. { - {\text{ln}}\left( {\frac{T}{{{\text{298}}{\text{.15}}}}} \right) - \left( {\frac{{298.15}}{T} - 1} \right)} \right]} \right) \hfill \\\end{gathered}$$ Application of this formulation to the breakdown reaction of Mg-cordierite to an assemblage of pyrope-sillimanite-quartz±H₂O shows that cordierite is stabilized by 3 to 3.5 kbar under H₂O-saturated conditions. The thermodynamic properties of H₂O in cordierite are similar to those of liquid water, with a standard molar enthalpy and Gibbs free energy of hydration that are the same (within experimental uncertainty) as the enthalpy and Gibbs free energy of vaporization. By contrast, most zeolites have Gibbs free energies of hydration two to four times more negative than the corresponding value for the vaporization of water.     

On the energy gap for Fe-poor oxide and silicate minerals

A relationship between the energy gap (E _G) and the density (ρ) over mean atomic weight (〈A〉) ratio for Fe-poor oxide and silicate minerals is derived from simple properties of their free atom-components. Theoretical considerations are based on the Lorentz electron theory of solids. The eigenfrequency ν ₀ of elementary electron oscillators, in energy units h ν ₀, is identified with the energy gap of a solid. The numerical relation is of the form $$(\langle U_0 \rangle ^2 - E_G^2 )\frac{{\langle A\rangle }}{\rho } = \frac{4}{3}\pi \hbar ^2 \frac{{e^2 }}{m}N = 276.79 eV^2 cm^3 /mol$$ where 〈U ₀〉 is the average first ionization potential (per free atom), ? is crossed Planck's constant, e is the electron charge, m is the electron rest mass, and N is Avogadro's number. For several geophysically interesting oxide and silicate minerals which are in general composed of four different elements (O, Si, Mg and Al), we obtain from laboratory data that the mean value of $$\left\langle {[\langle U_0 \rangle ^2 - (E_G^{lab} )^2 ]\frac{{\langle A\rangle }}{\rho }} \right\rangle \approx 248.2 \pm 20.9eV^2 cm^3 /mol.$$ .     

The concept of geochemical potential field exemplified by ore formation field

Geochemical potential field is defined as the scope within the earth’s space where a given component in a certain phase of a certain material system is acted upon by a diffusion force, depending on its spatial coordinatesX, Y andZ. The three coordinates follow the relations: $$NF_{ix} = - \frac{{\partial \mu }}{{\partial x}}, NF_{iy} = - \frac{{\partial \mu }}{{\partial y}}, NF_{iz} = - \frac{{\partial \mu }}{{\partial z}}$$ The characteristics of such a field can be summarized as: (1) The summation of geochemical potentials related to the coordinatesX, Y, Z, or pseudo-velocity head, pseudo-pressure head and pseudo-potential head of a certain component in the earth is a constant as given by $$\mu _x + \mu _y + \mu _z = c$$ or $$\mu _{x2} + \mu _{y2} + \mu _{z2} = \mu _{x1} + \mu _{y1} + \mu _{z1} $$ Derived from these relations is the principle of geochemical potential conservation. The following relations have the same physical significance: $$\mu _k + \mu _u + \mu _p = c$$ or $$\mu _{k2} + \mu _{u2} + \mu _{p2} = \mu _{k1} + \mu _{u1} + \mu _{p1} $$ (2) Geochemical potential field is a vector field quantified by geochemical field intensity which is defined as the diffusion force applied to one molecular volume (or one atomic volume) of a certain component moving from its higher concentration phase to lower concentration phase. The geochemical potential field intensity is given by $$\begin{gathered} E = - grad\mu \hfill \\ E = \frac{{RT}}{x}i + \frac{{RT}}{y}j + \frac{{RT}}{z}K \hfill \\ \end{gathered} $$ The present theory has been inferred to interpret the mechanism of formation of some tungsten ore deposits in China.     

Fluid–rock interaction at a carbonatite-gneiss contact, Alnö, Sweden

We evaluate balanced metasomatic reactions and model coupled reactive and isotopic transport at a carbonatite-gneiss contact at Alnö, Sweden. We interpret structurally channelled fluid flow along the carbonatite-gneiss contact at ～640°C. This caused (1) metasomatism of the gneiss, by the reaction: ${\hbox{biotite} + \hbox{quartz} + \hbox{oligoclase} + \hbox{K}_{2} \hbox{O} +\,\hbox{Na}_{2}\hbox{O} \pm \hbox{CaO} \pm \hbox{MgO} \pm \hbox{FeO} = \hbox{albite} + \hbox{K-feldspar} + \hbox{arfvedsonite} + \hbox{aegirene-}\hbox{augite} + \hbox{H}_{2} \hbox{O} + \hbox{SiO}_{2}}We evaluate balanced metasomatic reactions and model coupled reactive and isotopic transport at a carbonatite-gneiss contact at Aln?, Sweden. We interpret structurally channelled fluid flow along the carbonatite-gneiss contact at ∼640°C. This caused (1) metasomatism of the gneiss, by the reaction: , (2) metasomatism of carbonatite by the reaction: calcite + SiO₂ = wollastonite + CO₂, and (3) isotopic homogenization of the metasomatised region. We suggest that reactive weakening caused the metasomatised region to widen and that the metasomatic reactions are chemically (and possibly mechanically) coupled. Spatial separation of reaction and isotope fronts in the carbonatite conforms to a chromatographic model which assumes local calcite–fluid equilibrium, yields a timescale of 10²–10⁴ years for fluid–rock interaction and confirms that chemical transport towards the carbonatite interior was mainly by diffusion. We conclude that most silicate phases present in the studied carbonatite were acquired by corrosion and assimilation of ijolite, as a reactive by-product of this process and by metasomatism. The carbonatite was thus a relatively pure calcite–H₂O−CO₂–salt melt or fluid.