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Multipeak solutions for a semilinear Neumann problem. (English) Zbl 0866.35039
The paper is concerned with the semilinear Neumann problem: $\varepsilon^2 \Delta u-u+f(u) =0,\;u>0, \text{ in } \Omega,\quad {\partial u\over \partial\nu} =0\text{ on } \partial \Omega,$ where $$\Omega$$ is a bounded domain in $$\mathbb{R}^N$$, $$\nu$$ is the outer normal to $$\partial\Omega$$ and $$\varepsilon$$ is a positive constant. In addition to suitable conditions on $$f(t)$$, typically satisfied by the function $$f(t)= t^p-at^q$$ if $$a\geq 0$$ and $$1<q<p <(N+2)/(N-2)$$, the domain $$\Omega$$ is assumed to satisfy the condition that there exist $$k$$ disjoint patches $$\Lambda_1$$, $$\Lambda_2, \dots, \Lambda_k$$ on $$\partial \Omega$$ such that $$\max_{P\in \Lambda_i} H(P)> \max_{P\in\partial \Lambda_i} H(P)$$, where $$H(P)$$ denotes the mean curvature of $$\partial \Omega$$ at $$P$$. Under these conditions, the author proves the existence of a classical solution $$u_\varepsilon$$ which has exactly $$k$$ local maxima, precisely one on each $$\Lambda_i (i=1,2, \dots,k)$$, and then analyses the asymptotic behavior as $$\varepsilon \downarrow 0$$.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 35B25 Singular perturbations in context of PDEs
##### Keywords:
local maxima of a solution; asymptotic behavior
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##### References:
 [1] S. Alama and Y. Y. Li, On “multibump” bound states for certain semilinear elliptic equations , Indiana Univ. Math. J. 41 (1992), no. 4, 983-1026. · Zbl 0796.35043 · doi:10.1512/iumj.1992.41.41052 [2] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state , Arch. Rational Mech. Anal. 82 (1983), no. 4, 313-345. · Zbl 0533.35029 [3] H. Brezis and T. Kato, Remarks on the Schrödinger operator with singular complex potentials , J. Math. Pures Appl. (9) 58 (1979), no. 2, 137-151. · Zbl 0408.35025 [4] C.-C. Chen and C.-S. Lin, Uniqueness of the ground state solutions of $$\Delta u+f(u)=0$$ in $$\mathbf R^ n,\;n\geq 3$$ , Comm. Partial Differential Equations 16 (1991), no. 8-9, 1549-1572. · Zbl 0753.35034 · doi:10.1080/03605309108820811 [5] V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials , J. Amer. Math. Soc. 4 (1991), no. 4, 693-727. JSTOR: · Zbl 0744.34045 · doi:10.2307/2939286 · links.jstor.org [6] V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $$\mathbf R^ n$$ , Comm. Pure Appl. Math. 45 (1992), no. 10, 1217-1269. · Zbl 0785.35029 · doi:10.1002/cpa.3160451002 [7] B. Gidas, W.-M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $$\mathbf R\spn$$ , Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York, 1981, pp. 369-402. · Zbl 0469.35052 [8] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order , 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001 [9] C. Gui, Existence of multibump solutions for nonlinear Schrödinger equations via variational method , to appear in Comm. Partial Differential Equations. · Zbl 0857.35116 · doi:10.1080/03605309608821208 [10] M. K. Kwong and L. Zhang, Uniqueness of the positive solution of $$\Delta u+f(u)=0$$ in an annulus , Differential Integral Equations 4 (1991), no. 3, 583-599. · Zbl 0724.34023 [11] Y. Y. Li, On $$-\Delta u=K(x)u^ 5$$ in $$\mathbf R^ 3$$ , Comm. Pure Appl. Math. 46 (1993), no. 3, 303-340. · Zbl 0799.35068 · doi:10.1002/cpa.3160460302 [12] Y. Y. Li, Prescribing scalar curvature on $$S^ 3,\;S^ 4$$ and related problems , J. Funct. Anal. 118 (1993), no. 1, 43-118. · Zbl 0790.53040 · doi:10.1006/jfan.1993.1138 [13] Y. Y. Li, Prescribing scalar curvature on $$S^ n$$ and related problems. I , J. Differential Equations 120 (1995), no. 2, 319-410. · Zbl 0827.53039 · doi:10.1006/jdeq.1995.1115 [14] C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system , J. Differential Equations 72 (1988), no. 1, 1-27. · Zbl 0676.35030 · doi:10.1016/0022-0396(88)90147-7 [15] 1 P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I , Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109-145. · Zbl 0541.49009 · numdam:AIHPC_1984__1_2_109_0 · eudml:78069 [16] 2 P. L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II , Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 4, 223-283. · Zbl 0704.49004 · numdam:AIHPC_1984__1_4_223_0 · eudml:78074 [17] W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem , Comm. Pure Appl. Math. 44 (1991), no. 7, 819-851. · Zbl 0754.35042 · doi:10.1002/cpa.3160440705 [18] W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem , Duke Math. J. 70 (1993), no. 2, 247-281. · Zbl 0796.35056 · doi:10.1215/S0012-7094-93-07004-4 [19] E. Sere, Existence of infinitely many homoclinic orbits in Hamiltonian systems , Math. Z. 209 (1992), no. 1, 27-42. · Zbl 0725.58017 · doi:10.1007/BF02570817 · eudml:174347
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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Talk:RLC circuit

copy + paste

Hello,

this seems to be only a copy from Wikipedia's article w:RLC circuit. Everyone can copy + paste. But not everyone can make a new learning experiment now with this and try to rephrase this in our own words or use Wikipedia's article to generate information or a better learning experience on another level ?

Please also see: Wikiversity:What Wikiversity is not

"A duplication of other Wikimedia projects. While Wikiversity complements other Wikimedia projects, it will not simply duplicate their content. So, if you want to read about a topic, you may be better off visiting, say, Wikipedia or Wikibooks, but if you want to learn about this topic, you can do so at Wikiversity. Learning materials will be created and used on Wikiversity, but materials on other projects may also be used as learning materials themselves or even places to consolidate this learning, i.e. writing an article, manual etc based on what you've learned. There may be some overlapping, but each project will maintain its own focus."

Also worthwile reading: Wikiversity:What is Wikiversity? How to Cite from Wikipedia. ----Erkan Yilmaz uses the Wikiversity:Chat (try) 09:20, 19 July 2008 (UTC)

Yes it is copy paste, but that is not a bad way to start.

I think this needs to be simplified and adapted to OpenStax University Physics. See https://cnx.org/contents/eg-XcBxE@8.87:JOs6racw@3/153-RLC-Series-Circuits-with-A

This would make a good student project in such a course. What we really need is a derivation, based on complex numbers, of this formula:
▭ RLC series circuit ${\displaystyle V_{0}=I_{0}Z}$ where ${\displaystyle Z={\sqrt {R^{2}+\left(X_{L}-X_{C}\right)^{2}}}}$ and ${\displaystyle \phi =\tan ^{-1}{\frac {X_{L}-X_{C}}{R}}}$

format: plain


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Article

Talc-chlorite-amphibole felses of the KTB pilot hole, Oberpfalz, Bavaria: Protolith characteristics and phase relationships

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To read the full-text of this research, you can request a copy directly from the authors.

Abstract

The metagabbro-amphibolite sequences in the KTB pilot hole contain intercalations of talc-chlorite-amphibole felses (or ''hosbachites''), which show transitional contacts to the adjacent metagabbros. The hosbachites are characterized by relics of a primary igneous texture and still contain igneous minerals like clinopyroxene, biotite and pseudomorphs after olivine, while brown Ca-amphibole was presumably formed in a late-magmatic stage. The geological, textural, mineralogical and geochemical evidence indicates that the hosbachites were derived from ultramafic cumulates, differentiated from a basaltic magma, either in the inner parts of dolerite sills or in small gabbro intrusions. A pervasive metamorphic overprint under medium-pressure, amphibolite-facies conditions which was accompanied by penetrative deformation led to assemblages with green Ca-amphibole +/- anthophyllite +/- cummingtonite +/- tremolite/actinolite + clinochlore + talc + olivine + ilmenite +/- Cr-bearing spinel + sulfides. Phase relationships are consistent with a prograde P-T path leading to the formation of anthophyllite from olivine + talc at peak metamorphic temperatures of 640-700-degrees-C, at assumed pressures of 8-10 kbar, similar to those derived from mineral assemblages in the adjacent metabasites and metasediments. High-pressure relics locally present in coronitic metagabbros and retrograded eclogites of the KTB pilot hole were not recognized in the hosbachites. A retrograde overprint under greenschist-facies conditions led to the total replacement of igneous or metamorphic olivine by aggregates of antigorite + magnetite, chloritization of biotite and the formation of late tremolite/actinolite.

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... In this paper, we present a transmission electron microscope (TEM) and electron microprobe (EMP) study of the assemblage Ca-amphibole-cummingtonite-anthophyllitegedrite-chlorite-talc encountered in a metamorphosed ultramafic rock of the pilot hole of the Kontinentales Tiefbohr-Programm (KTB; German Continental Deep Drilling Program), northeastern Bavaria, Germany. Earlier studies of this rock were aimed at its protolith characteristics and phase relationships (Matthes et al., 1995) and the stability field of the mineral assemblage observed (Schmädicke & Okrusch, 1997). In the present study, the dominant points are the chemical composition of coexisting amphiboles, their exsolution phenomena and mutual orientation relationships, and those with associated chlorite and talc within the same ultramafic rock. ...
... Patzak, 1991Patzak, , 1996Patzak, et al., 1991;Schalkwijk, 1991;O'Brien et al., 1992) are repeatedly intercalated with layers and lenses of ultramafic talc-chlorite-amphibole rock. Contact relationships, relic textures, relics of igneous minerals and bulk rock compositions indicate that these ultramafic layers were derived from mafic cumulates, differentiated from a basaltic magma, either in the lower parts of dolerite sills or small gabbro intrusions (Matthes et al., 1995). The investigated sample, derived at a depth of 1382.36 m, is a massive ultramafic rock (Fig. 1). ...
... These features are interpreted as relics of late-igneous amphibole, originally derived from magmatic Ti-augite. The porphyroclasts are spatially related with fine, colourless aggregates of talc, opaque phases and serpentine interpreted as pseudomorphs after olivine (Matthes et al., 1995). The pseudomorphs mostly occur in mutual contact with the porphyroclasts, in many cases as inclusions. ...
Article
Full-text available
&#112Lenses of ultramafic rocks intercalated within a metagabbroamphibolite sequence were encountered in the KTB pilot hole. A pervasive metamorphic overprint formed the dominating assemblage calcic amphiboleorthoamphibolechloritetalc. In this study the phase relations of ultramafic rocks are investigated in order to (a) constrain the stability field of this assemblage in general, and (b) define the equilibrium pressuretemperature (PT) conditions of this assemblage in the ultramafic rocks from the KTB borehole. For that purpose, phase equilibria were calculated in the model systems CaOMgOAl2O3SiO2-H2O (CMASH) and CaOMgOFeOAl2O3SiO2-H2O (CMFASH). Thereby, the continuous compositional change of solid solutions with pressure and temperature was modeled, including the Tschermaks substitution and the MgFe-1 exchange. Based on these results, petrogenetic grids were constructed, revealing that calcic amphiboleorthoamphibolechloritetalc assemblages cover a stability field of <650770&#117C/1M 14&#114kbar (CMASH) and <550650&#117C/1&#7714&#114kbar (CMFASH), respectively. This explains the widespread occurrence of the considered assemblage. Based on the bulk rock composition of the KTB samples, a special PT diagram was constructed, limiting the stability field of the calcic amphiboleorthoamphibolechloritetalc assemblage. At 580&#117C the stability field extends from 6 to 14&#114kbar pressure, and shrinks to 1011&#114kbar at 630&#117C. Conventional estimates using the mineral compositions of the KTB samples yield a temperature around 630&#117C, at which the calculated stability field of calcic amphiboleorthoamphibolechloritetalc extends from 10 to 11&#114kbar.
... In this paper, we present a transmission electron microscope (TEM) and electron microprobe (EMP) study of the assemblage Ca-amphibole-cummingtonite-anthophyllitegedrite-chlorite-talc encountered in a metamorphosed ultramafic rock of the pilot hole of the Kontinentales Tiefbohr-Programm (KTB; German Continental Deep Drilling Program), northeastern Bavaria, Germany. Earlier studies of this rock were aimed at its protolith characteristics and phase relationships (Matthes et al., 1995) and the stability field of the mineral assemblage observed (Schmädicke & Okrusch, 1997). In the present study, the dominant points are the chemical composition of coexisting amphiboles, their exsolution phenomena and mutual orientation relationships, and those with associated chlorite and talc within the same ultramafic rock. ...
... Patzak, 1991Patzak, , 1996Patzak, et al., 1991;Schalkwijk, 1991;O'Brien et al., 1992) are repeatedly intercalated with layers and lenses of ultramafic talc-chlorite-amphibole rock. Contact relationships, relic textures, relics of igneous minerals and bulk rock compositions indicate that these ultramafic layers were derived from mafic cumulates, differentiated from a basaltic magma, either in the lower parts of dolerite sills or small gabbro intrusions (Matthes et al., 1995). The investigated sample, derived at a depth of 1382.36 m, is a massive ultramafic rock (Fig. 1). ...
... These features are interpreted as relics of late-igneous amphibole, originally derived from magmatic Ti-augite. The porphyroclasts are spatially related with fine, colourless aggregates of talc, opaque phases and serpentine interpreted as pseudomorphs after olivine (Matthes et al., 1995). The pseudomorphs mostly occur in mutual contact with the porphyroclasts, in many cases as inclusions. ...
Article
Full-text available
A metamorphosed ultramafic rock, penetrated by the KTB pilot hole at a depth of 1382.36 m, contains the assemblage Caamphibole - anthophyllite - chlorite - talc, formed at about 630degreesC/10 kbar under conditions of the high-P amphibolite facies. EMP analyses of the Ca-amphibole yielded magnesiohastingsite (to pargasite) compositions with (Al+Fe3++Cr+Ti)([6]) of 1.0-1.2 p.f.u. and (Fe2++Mn)/(Mg+Fe2++Mn) ratios of 0.10-0.16, while coexisting anthophyllite has (Al+Fe3++Cr+Ti)([6]) less than or equal to 0.16 and (Fe2++Mn)/(Mg+Fe2++Mn) ratios of 0.21-0.29. Chlorite is clinochlore with Al-[4]/(Al-[4]+Si) of 0.27-0.30 and Fe2+/(Fe2++Mg) of 0.12-0.14. TEM investigations revealed, for the first time, a complex lamellar intergrowth in the sequence talc --> anthophyllite anthophyllite/cummingtonite intercalations --> anthophyllite --> Ca-amphibole. This lamellar intergrowth is about 15 mum in width, but individual cummingtonite lamellae intercalated with dominant anthophyllite are only 1 mum wide. All the amphiboles share the directions b* and a*; anthophyllite and the monoclinic amphiboles are intergrown along (100); anthophyllite and talc are intergrown with (100)(Ath)//(001)(Tlc). In other areas anthophyllite and chlorite are intergrown with a*(Ath)//c*(Chl), and b*(Ath)//b*(Chl) and Ca-amphibole exsolves lamellae of cummingtonite//(100). With one possible exception, cummingtonite has the space group C2/m. Anthophyllite displays chain multiplicity faults //(010), anthophyllite and cummingtonite chain arrangement faults //(100). Microstructures suggest that anthophyllite was formed at the expense of cummingtonite that is interpreted as a high-T precursor phase testifying to a possible earlier, granulite-facies metamorphic stage. During the retrograde P-T path, on cooling presumably below 500-550degreesC, anthophyllite exsolves platelets or lamellae of gedrite with {hk0} composition planes. Orientations close to {230} and {110} have been recorded, so far not described in the literature. Alteration products of biotite are composed of clinochlore and a sheet silicate intermediate in composition between chlorite and biotite, i.e., with partly high K contents, but with chlorite metric.
... From petrological studies on talc-chlorite-amphibole felses or hoesbachites, it was inferred that these rocks are the result of complex metasomatic alteration of mafic to ultramafic cumulates under amphibolite-to greenschist-facies conditions (Matthes and Schubert 1967;Schubert 1969;Okrusch et al. 1995). Cumulates form by gravimetric differentiation of early crystals that sink to the bottom of a magma chamber (Hunter 1996). ...
Article
Full-text available
The production of lathe-turned tripod vessels made from softstone is one of the major features of the so-called Rasikajy population that inhabited northern Madagascar from the ca. 8th to the late 15th century a.d. The raw material for the vessels was quarried in the hinterland, and over 30 quarries have recently been visited, documented and sampled. The quarry of Bobalila is the first to ever be excavated, and a large sample suite was taken for petrological analysis. The results reveal significant mineralogical and chemical variation that is almost as large as the variation between all other quarries in northern Madagascar. The underlying processes could affect other softstones and should be considered in provenance attempts. Nonetheless, the petrographic study has permitted us to understand and characterize the type of material that was sought-after by Rasikajy workers, which can now be easily distinguished from other softstone vessels in the Indian Ocean trade network.
... Minor (~10%) primary late-magmatic hornblende is present in some dykes and these can be identified as hornblende dolerites. Hornblende grains exhibit light green, brownish green and brown pleochroism; presumably formed in a late-magmatic stage (e.g., Matthes et al., 1995). Rare small grains of quartz are also identified in a few thin sections. ...
Article
A precise U–Pb baddeleyite age (2251 ± 4 Ma) has been obtained for an ENE-trending dyke exposed near the Chhura region of the Bastar craton, and additional dykes of similar trend are also reported from other parts of the northern Bastar craton such as in the Dongergarh and Pakhanjore areas. We propose a new widespread ENE-trending ca. 2.25 Ga Chhura mafic dyke swarm in the Bastar craton, which represents a continuation of the NE- to ENE-trending ca. 2.26 Ga Kaptipada swarm in the Singhbhum craton. The adjoining Dharwar craton also hosts ca. 2.26–2.25 Ga magmatism, the N- to NNE-trending Ippaguda-Dhiburahalli dyke swarm. A geochemical match between the ca. 2.25 Ga Chhura, the ca. 2.26–2.25 Ga Ippaguda-Dhiburahalli, and most samples of the ca. 2.26 Ga Kaptipada swarms corroborates a genetic link. Together these mafic dykes in Bastar, Singhbhum and Dharwar cratons, define the ca. 2.26–2.25 Ga Ippaguda (-Dhiburahalli-Chhura-Kaptipada) LIP. The relative post-2.25 Ga rotations between the three cratons would allow two alternative locations for the plume centre either at the eastern end of the Kaptipada swarm in the Singhbhum craton or at the south end of the Ippaguda-Dhiburahalli swarm in the Dharwar craton. Alternatively, convergence of the Churra, Kaptipada and Ippaguda-Dhiburahalli sub-swarms could suggest a plume centre location in the Godavari rift, between the Bastar and Dharwar cratons. Magmatic events of approximately this 2.26–2.25 Ga age are known from other parts of the globe including East Antarctica, and Kaapvaal and Zimbabwe cratons.
... Few samples contain hornblende (Hbl; Fig. 3e), that shows light green, brownish green and brown pleochroism; presumably formed in a late-magmatic stage (e.g. Otten, 1984;Matthes et al., 1995). As remnants of the original mineralogy Cr 20 120 140 80 260 230 640 200 110 50 90 250 Ni 50 100 120 100 230 130 510 110 100 80 100 100 Sc 27 39 37 38 24 38 20 34 32 33 38 44 V 260 296 319 384 148 306 188 304 298 342 372 283 Rb 71 27 25 23 16 57 34 7 21 23 30 10 Ba 608 163 191 231 107 169 214 61 307 107 230 85 Sr 179 111 140 158 297 140 264 118 287 158 147 113 Nb 15 8 4 4 3 3 ...
Article
Full-text available
A vast tract of ENE–WSW to NE–SW trending mafic dyke swarm transects Archaean basement rocks within the eastern Dharwar craton. Petrographic data reveal their dolerite/olivine dolerite or gab-bro/olivine gabbro composition. Geochemical characteristics, particularly HFSEs, indicate that not all these dykes are co-genetic but are probably derived from more than one magma batch and different crystallization trends. In most samples the La N /Lu N ratio is at ∼2, whereas others have a La N /Lu N ratio >2 and show higher concentrations of high-field strength elements (HFSEs) than the former group. As a consequence , we assume that the ENE–WSW to NE–SE trending mafic dykes of the eastern Dharwar craton do not represent one single magmatic event but were emplaced in two different episodes; one of them dated at about 2.37 Ga and another probably at about 1.89 Ga. Trace element modelling also supports this inference: older mafic dykes are derived from a melt generated through ∼25% melting of a depleted mantle, whereas the younger set of dykes shows its derivation through a lower degree of melting (∼15%) of a comparatively enriched mantle source.
... Hornblende grains exhibit light green, brownish green and brown pleochroism; presumably formed in a late-magmatic stage (e.g. Otten, 1984;Matthes et al., 1995), are present in all the thin sections. These petrographic observations suggest low-temperature late hydrothermal alteration. ...
Article
Full-text available
Four distinct Proterozoic mafic dyke swarms indentified in northern Bastar craton.•They are emplaced in a stable continental rift setting.•Paleoproterozoic dykes derived from melts generated within spinel stability field.•Mesoproterozoic dykes derived from melts generated within garnet stability field.•These dykes may be related to the assembly and break-up of Columbia supercontinent.
... Few samples contain hornblende (Hbl; Fig. 3e), that shows light green, brownish green and brown pleochroism; presumably formed in a late-magmatic stage (e.g. Otten, 1984;Matthes et al., 1995). As remnants of the original mineralogy Cr 20 120 140 80 260 230 640 200 110 50 90 250 Ni 50 100 120 100 230 130 510 110 100 80 100 100 Sc 27 39 37 38 24 38 20 34 32 33 38 44 V 260 296 319 384 148 306 188 304 298 342 372 283 Rb 71 27 25 23 16 57 34 7 21 23 30 10 Ba 608 163 191 231 107 169 214 61 307 107 230 85 Sr 179 111 140 158 297 140 264 118 287 158 147 113 Nb 15 8 4 4 3 3 ...
Article
A vast tract of ENE–WSW to NE–SW trending mafic dyke swarm transects Archaean basement rocks within the eastern Dharwar craton. Petrographic data reveal their dolerite/olivine dolerite or gabbro/olivine gabbro composition. Geochemical characteristics, particularly HFSEs, indicate that not all these dykes are co-genetic but are probably derived from more than one magma batch and different crystallization trends. In most samples the LaN/LuN ratio is at ∼2, whereas others have a LaN/LuN ratio >2 and show higher concentrations of high-field strength elements (HFSEs) than the former group. As a consequence, we assume that the ENE–WSW to NE–SE trending mafic dykes of the eastern Dharwar craton do not represent one single magmatic event but were emplaced in two different episodes; one of them dated at about 2.37 Ga and another probably at about 1.89 Ga. Trace element modelling also supports this inference: older mafic dykes are derived from a melt generated through ∼25% melting of a depleted mantle, whereas the younger set of dykes shows its derivation through a lower degree of melting (∼15%) of a comparatively enriched mantle source.
... These are dusted with tiny platelets of hemo-ilmenite exsolved from a former, Ti rich amphibole which, in turn, was presumably derived from an igneous Ti augite by late-magmatic replacement. It is assumed that this process took place under subsolidus conditions and was caused by water influx during cooling, as described, for the Artfjallet gabbro and dolerites, central Sweden (Otten, 19841, the Frankenstein gabbro, Odenwald, Germany (Kreher, 19941, and the metagabbros and related meta-ultramafics in the KTB pilot hole, Oberpfalz, Germany (O'Brien et al., 1992; Matthes et al., 1995). Such an amphibole forming reaction can be formulated as:Fig. ...
Article
In the Proterozoic Schist Belt of Nigeria, lenticular bodies of metabasites and meta-ultramafics are frequently intercalated within staurolite bearing metapelitic schists. Such a metamorphosed mafic-ultramafic complex is particularly well exposed in the Mokuro riverbed between the towns of Ife and Ilesha. These outcrops display contact relationships with the surrounding metasediments, as well as between the individual mafic and ultramafic rock types. The most common mafic rocks are indistinctly layered amphibolites, accompanied by apatite rich amphibolites and massive amphibolites, in part rich in ilmenite and pyrrhotite. Among the generally massive ultramafic rocks, nearly monomineralic amphibole rocks predominate, while chlorite-amphibole, talc-chlorite-amphibole and talc bearing olivine-chlorite-amphibole rocks occur in subordinate amounts.
Chapter
Full-text available
The metamorphic sequences constituting the MGCR frequently contain intercalations of metabasites. Judging from structural and textural relics, from rarely preserved magmatic minerals and from geochemical constraints, most of these metabasites are derived from igneous protoliths, especially from mafic volcanites, including volcanoclastites, in places also from gabbros.
Article
Full-text available
Metaserpentinite bodies in the metamorphic rocks of northern Veporic were emplaced by means of low-temperature protrusion of completely serpentized plagioclase harzburgites. No relics of primary harzburgites were preserved in metalserpentinites and the newly formed mineral assemblages differed in two massifs. Phase equilibria and mineral compositions of metaserpentinites depended on their bulk compositions.
Article
A thermodynamic model has been developed and calibrated for monoclinic and orthorhombic amphiboles compositionally contained within the Ca-Mg-Fe2+ amphibole quadrilateral. The model incorporates the energetic consequences of cation ordering of Fe2+ and Mg over nonequivalent sites in the crystal structure, and accounts for the temperature, pressure, and compositional dependence of the orthorhombic-monoclinic phase transition. Calibration is based on previously published work on the thermodynamic properties of Fe2+-Mg amphiboles and tremolite, and experimental cation-ordering data, along with solvus width and tieline orientations of natural coexisting amphiboles in the quadrilateral. Among derived parameters is the enthalpy of formation of end-member ferroactinolite (-10534.966 kJ/mol). A calculated FeMg-1 isopotential solvus at mid-composition in the quadrangle agrees well with a revised calibration of experimental data from Cameron (1975). The solvus is not strongly asymmetric and narrows with increasing Fe/Mg ratio and temperature. Phase relations along the Mg and Fe sides of the quadrilateral are compared and contrasted. Cummingtonite-anthophyllite phase relations are shown in the T-XFe plane, with and without coexisting calcic amphibole. The breakdown reactions of quadrilateral amphiboles to assemblages of two pyroxenes, olivine, quartz, and H2O are depicted as functions of temperature, pressure, composition, H2O activity, and oxygen fugacity. Some highlights of their relevance to metamorphic and igneous systems are discussed.
Article
As laid out in Chapters V.B.1 and V.B.2, the parautochthonous Thuringian fades of the Saxothuringian Belt has been overthrust from the SE by nappes, which have survived in the core of the Vogtland Synform. These are the tectonic klippen of Münchberg, Wildenfels, and Frankenberg (in order from W to E; see Fig. 1). The klippen represent, in fact, erosional remnants of one or several nappe piles, which were originally more extensive, and possibly laterally continuous with each other. The tectonic sequence in each of these klippen is characterized by inversion of the stratigraphy and of the metamorphic grades. The lower nappes comprise (in order from bottom to top): proximal Early Carboniferous flysch, Silurian/Devonian bedded chert, and a volcanosedimentary sequence of largely Ordovician age. These very-low-grade, fossiliferous sequences are overlain by a number of metamorphic thrust sheets. This chapter describes the lithology, tectonic sequence, and internal structure of the metamorphic nappes in the klippen, and of their probable equivalents at the northwestern margin of the Moldanubian Region, from which they are derived. We set out from the Münchberg klippe, which is the largest, most differentiated and best studied example of its kind, and then briefly discuss the closely related units. Details on the metamorphic evolution are available in the contribution by Blümel (Chap. V.C.2)
Article
Experiments were performed to quantitatively confirm the relationship between open dry microcracks and P wave velocity on crystalline rocks. Equations derived by Hudson [1980] (first-order corrections only) describe the experimentally obtained relationship. Microcrack analysis consists of chemical contact imagery to make the cracks visible, image analysis, and stereologic data processing to obtain three-dimensional crack characteristics from data obtained in three orthogonal plane images. Pressure-dependent measured P wave velocities are used to estimate intrinsic P wave velocities (velocities characteristic for the uncracked rock) by extrapolating the linear part of the pressure-velocity curve from high pressure to low pressure. Therefore the impact of open microcracks on P wave velocity is given by normalized P wave velocities, which are the ratio between measured and intrinsic velocities at a hydrostatic pressure of 1 MPa. Three-dimensional crack characteristics and normalized P wave velocities, are obtained in three orthogonal directions. Velocities in all other directions were estimated by interpolation assuming an orientation distribution that can be described by a triaxial ellipsoid. In this way it is possible to study the relationship between open dry cracks and P wave velocity experimentally and to compare the results with the theoretically predicted relationship using one single model. In a case study, pressure-dependent measured P wave velocities have been used to infer the distribution of open microcracks in 84 drill cores from the Continental Deep Drilling Project of the Federal Republic of Germany (Kontinentales Tiefbohrprogramm der Bundesrepublik Deutschland, or KTB) covering the depth range between 621-8080 m. The cores were studied after stress relaxation (1-7 years after drilling). It was not possible to obtain information about the in situ stress field owing to the anisotropic nature of the rock fabric.
Article
: and . John C. Schumacher Dept. of Earth Sciences, University of Bristol, Bristol, BS8
Chapter
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ABSTRACT Volcanic arc basalts are all characterized by a selective enrichment in incompatible elements of low ionic potential, a feature thought to be due to the input of aqueous ﬂuids from subducted oceanic crust into their mantle source regions. Island are basalts are additionally characterized by low abundances [for a given degree of fractional crystallization) of incompatible elements of high ionic potential, as feature for which high degrees ot'melting, stability of rninor residual oxide phases, and remelting of depleted mantle are all possible explanations. Calc-alkaline basalts and shoshonites are additionally characterised by enrichment of Th, P and the light REE in addition to elements of low ionic potential, a feature for which one popular explanation is th contamination of their mantle source regions by a melt derived from subducted sediments. By careful selection of variables, discrimination diagrams can be drawn which highlight these various characteristics and therefore enable volcanic arc basalts to he recognized in cases where geological evidence is ambiguous. Plots of Y against Cr, K[Yb, Ce/Yb, or Th/Yb against Ta/Yb, and Ce/Sr against Cr are all particularly successful and can be modelled in terms of vectors representing different petrogenctic processes. An additional plot of Ti/Y against Nb/Y is useful for identifying 'anomalous' volcanic arc settings such as Grenada and parts of the Aleutian arc. Intermediate and acid rocks from volcanic are settings can also be recognized using a simple plot of Ti against Zr. The lavas from the Oman ophiolite complex provide a good test of the application of these techniques. The results indicate that the complex was made up of back-arc oceanic crust intruded by the products of volcanic arc magmatism.
Article
Full-text available
Clinochlore, which is, within the limits of error, the thermally most stable member of the Mg-chlorites, breaks down at $$P_{{\text{H}}_{\text{2}} {\text{O}}}$$ = P tot to the assemblage enstatite+forsterite+spinel+H2O along a univariant curve located at 11 kb, 838 ° C; 15kb, 862 ° C; and 18 kb, 880 ° C (±1 kb ±10 ° C). At water pressures above that of an invariant point at 20.3 kb and 894 ° C involving the phases clinochlore, enstatite, forsterite, spinel, pyrope, and hydrous vapor, clinochlore disintegrates to pyrope+forsterite+spinel+H2O. The resulting univariant curve has a steep, negative dP/dT slope of −930 bar/ °C at least up to 35 kb. Thus, given the proper chemical environment, Mg-chlorites have the potential of appearing as stable phases within the earth's upper mantle to maximum depths between about 60 and 100 km depending on the prevailing undisturbed geotherm, and to still greater depths in subduction zones. However, unequivocal criteria for mantle derived Mg-chlorites are difficult to find in ultrabasic rocks.
Book
In October 1986 the German Minister for Research and Technology (Bundesminister fUr Forschung und Technologie), Dr. H. Riesenhuber, officially announced that the super-deep borehole of the Continental Deep Drilling Program of the Federal Republic of Germany (KTB) would be drilled in the Oberpfalz area of Northern Bavaria. The site selection was based on a recommendation from the Deutsche Forschungsgemeinschaft (DFG) made after an evaluation by the Project Management of the technical and financial risks involved. This decision was preceded by a conference held from September 19 to 21, 1986 in Seeheim/Odenwald at which the results of the site studies in the Oberpfalz and the Schwarzwald were presented and thoroughly debated. The models and scientific targets resulting from these investigations formed the basis for a vote by the DFG Senate Commission for Geoscientific Interdisciplinary Research which was taken immediately after the conference. After evaluation of all scientific and technical aspects, the members of the commission voted almost unanimously for the Oberpfalz site. It was, ho",'ever, strongly emphasized that both locations had a wealth of attractive research objectives and that despite clear-cut differences in some major aspects scientifically the two could be regarded as more or less equivalent. Both'locations would be excellent sites for research drilling and would certainly cor.
Chapter
By the time the transition zone between the Saxothuringian and the Moldanubian units in northeastern Bavaria was selected as a possible location for the German continental deep-drilling project (KTB), great differences with respect to the interpretation of the geological evolution of the area were obvious, differences which had important consequences for the geochronological evolution of this area. Stettner (1975,1979) assumed that the high-grade Moldanubian gneisses were metamorphosed during Cadomian time, whereas the Saxothuringian sediments of Early Paleozoic age were supposed to have been subjected to an Early Hercynian reworking. In contrast, Schreyer (1966) proposed that the Saxothuringian and the Moldanubian units had suffered the same low-pressure metamorphism, which would mean that at least the last metamorphic overprinting of both units should be of Hercynian age. This was confirmed by structural studies, which showed a continuous structural development from the Moldanubian into the Saxothuringian (Stein 1987). The structural model of the transition zone between Moldanubian and Saxothuringian is also in agreement with the petrological studies of Wagener-Lohse and Blümel (1984) and Blümel (this Vol.).
Chapter
The Variscan basement at the Continental Deep Drilling Site (KTB) Oberpfalz on the western margin of the Bohemian Massif is composed of three polyphase-deformed structural units: the Saxothuringian, the Moldanubian and MP to HP metamorphic nappe complexes of the Zone of Erbendorf-Vohenstrauß (ZEV) and the Münchberg Massif (MM). The boundary between the Saxothuringian und Moldanubian, which is interpreted as a cryptic suture, is represented by the northwestern rim of the HT-mylonite belt of the Zone of Tirschreuth-Mähring (ZTM). This forms part of a formerly active continental margin, whereas the Saxothuringian terrane represents a formerly passive continental margin. Deformation of the Moldanubian active continental margin already began during the oceanic subduction stage and was therefore a longer-lasting, more penetrative deformational event than in the Saxothuringian, where the whole Variscan deformation is related only to the collisional stage. The klippen of the ZEV and MM form part of a previously coherent nappe complex which was originally connected with the western part of the Bohemian (Zone of Tepla-Taus, ZTT) and now covers the Saxothuringian-Moldanubian suture. The Erbendorf Line near Erbendorf is a nappe boundary. Its interpretation as suture zone can no longer be upheld. The Variscan crustal structure evolved during multiphase deformation in which two main events can be distinguished. The older events up to D3 are characterized by NW-directed tectonic transport. This is revealed by the seismic crustal structure which is characterized by dominantly SE-dipping reflectors in all NW-SE-directed reflection seismic profiles. Based on reflection seismic data, D2-backfolding and backthrusting of the Saxothuringian seems to be restricted to higher crustal levels and is interpreted as having resulted from crustal wedging and ramp tectonics. In the roof of a crustal wedge foreland-dipping backthrusts and antivergent folds develop, whereas foreland-directed overthrusting continues inside and below the wedge. The younger event is characterized by NW-trending and SW-vergent D4-folding and thrusting and merges into late-orogenic dextral wrench faulting and extensional tectonics. The ZEV nappe forms a D4-syncline, in which the rocks have been preserved from erosion. The D4-deformation, which increases in intensity to the south, documents a reorientation of the regional stress field at the southwestern rim of the Bohemian Massif. This is interpreted on the basis of geophysical and geological data as resulting from a late-collisional indentation process. Variscan crustal convergence was probably much stronger than previously assumed. Initial; oceanic subduction zones have been completely overridden by continental crust. Therey, the complexly deformed cryptic sutures seen at the present outcrop level of the crystalline basement have lost their contact woth the previous subduction zone. During collision, terranes previously separated by oceanic crust were thus thrust over one another, so that suture zones widely separated before collision are now closely neighboring or oevrridden by other terranes. The latter is presumably the case at the northwestern margin of the Bohemian Massif, where the Bohemian terrane overthrust the Saxothuringian-Moldanubian suture zone.
Article
Eclogitic relics testifying to one or more early events of high-pressure metamorphism are present in different tectonic units of the NE Bavarian crystalline basement forming the north-western margin of the Bohemian Massif. Eclogites and their amphibolized derivatives are most frequent in the Munchberg nappe complex where they form constituents of the uppermost crystalline nappe, the so-called Hangendserie. -from Authors
Article
New experimental data on the upper thermal stability of chlorite in the system H2O-MgO-Al2O3-SiO2 are reported here and are combined with other experimental data, molar volumes of the solid phases, heat capacities, H2O fugacities, and activity expressions for cordierite, chlorite and orthopyroxene to obtain an array of univariant curves about an invariant point at 720 + or - 10oC and 2.75 + or - 0.3 kbar where chlorite, cordierite, forsterite, orthopyroxene, spinel and water are in equilibrium.-J.A.Z.
Chapter
The crustal structure and the geotectonic development of the Variscan Belt of Europe have been controversial for al long time. Early in this century, F.E.Suess (1912), Kossmat (1927) and others had published thrust-tectonic concepts for the Variscan Belt, which are nothing less but anticipations of modern structural concepts stimulated by plate tectonics. The target area of the present KTB project served, in this early phase of research, as an outstanding example of horizontally directed tectonism. Since this time, the Variscan basement of NE Bavaria has occupied a key role in the ensuing geotectonic debate.
Article
The phase relations of three basalts, the Picture Gorge tholeiite, the 1921 Kilauea olivine tholeiite, and the 1801 Hualalai alkali basalt, were studied at 5 kb water pressure, 680-1000°C, at the oxygen fugacities of the quartz-fayalite-magnetite (QFM) and hematite-magnetite (HM) buffers.In the range 680-850 °C, the crystalline assemblage on the QFM buffer is dominantly hornblende+ plagioclase, ± ilmenite, magnetite, sphene, fayalitic olivine, and phlogopitic mica. From 875 to 1000 °C the crystalline assemblage is hornblende+ olivine± augite+ ilmenite± magnetite. A melt phase is present from 700 to 1000 °C; a vapor phase was present in all charges.The hornblendes formed on the QFM buffer range in composition from common green hornblendes at low temperatures to kaersutitic hornblendes at 1000 °C. A1(IV) and Ti increase temperature. AI(VI) passes through a maximum near 825 °C, decreasing both above and below this temperature. AI(IV) is proportional to the sum A1(VI)+2Ti. There is a positive linear correlation of approximately 3: 1 between AI(IV) and the number of cations in the A-site. The most likely explanation for this correlation at present is that the substitution of AI(VI) or Ti+4for a divalent cation creates local charge imbalances in the amphibole structure which can be compensated only by further A-site substitution. There also appears to be a correlation between the a-cell dimension of hornblende and the A-site occupancy. Above a thresh hold value of approxmately 0.5 cations in A, a increases as A-site occupancy increases.Phase relations on the hematite-magnetite buffer are considerably simpler. The hornblendes show relatively little change in composition as temperature increases, and in the tholelitic compositions break down at or below 970 °C 35-60 °C above the first appearance of augite±olivine. The melting of hornblende is incongruent in all cases. The Fe-Ti oxides are pseudo-brookite and titanohematite; at 1000 °C these oxides make up 10 per cent by weight of the assemblage and contain most of the Tio2 and FeO in the charge.The patterns of hornblende variation observed in this study compare closely with those reported in a wide range of experimental and field data. The appearance of high-TiO2 kaersutitic hornblendes in the tholeities at 1000° C, PH2O= 5 kb on the QFM buffer implies that the restricted occurence of kaersutite in nature (where it is associated only with mafic to intermediate alkalic rocks) is controlled by volatile content (H2O,F2)rather than by differences in condensed bulk composition.
Article
The pure magnesium orthorhombic amphibole, anthophyllite, has been synthesized and its upper and lower stability limits have been established by reversible hydrothermal experiments. The synthetic mineral has refractive indices n x=1·587±0·001, n y=1·602±0·005, n z=1·613±0·001, and unit-cell dimensions of a 0=18·61±0·02 Å, b 0=18·01±0·06 Å, c 0=5·24±0·01 Å. The mineral is stable over a narrow temperature range in the presence of the phase H 2O. At a P H2O of 1,000 bars the upper stability limit is 745°±10° C and the lower stability limit is 667°±8° C.Rate studies indicate that anthophyllite can nucleate at temperatures above its upper stability limit by disintegration of talc sheets into strips of double chains. The activation energy for this process is 150±30 kcal mol -1.Application of the data to rocks of the Balmat area, New York, suggests that the equilibrium pressure of water during the metamorphism was significantly less than the total pressure. The data indicate that monomineralic zones of anthophyllite in ultramafic rocks are due to the presence of a steep gradient in the activity of H 2O, or a steep gradient in temperature, or both, across the zones.
Article
The paper comments on the recommendations on which the IUGS Subcommission on the Systematics of Igneous Rocks agreed at Montreal, August 1972. Plutonic rocks are classified and named according to mineral contents. For nomenclature are considered: Q = quartz, A = alkali feldspar (incl. albite), P = plagioclase, F = feldspathoids, M = mafic and related minerals. Rocks with M less than 90 are named according to their positions in the QAPF diagram, the light-colored constituents being calculated to the sum 100. The following are treated: granitoids and related rocks, ultramafic and gabbroic rocks, charnockitic rocks, feldspathoidal rocks. A color index is used to distinguish the leuco- and mela-types of each rock group in comparison with normal types.
Article
The Variscan belt of Europe resulted from the collision of Africa with Baltica and intervening microplates. In early Paleozoic time, Baltica was probably separated from the areas farther south by a North-German/Polish ocean, whose trace is now under younger cover in the northern part of continental Europe. To the south followed an assembly of microplates with African affinities: a Cadomian (Pan-African) orogen (550 to 600 Ma); traces of the Late Ordovician glaciation (South Pole in the Sahara); and, correspondingly, relatively high paleolatitudes. Some of these units also contain fragments of much older crust (up to 2.5 Ga). These microplates were separated from Africa during an important rifting episode in Cambro-Ordovician time. Rifting is reflected in several belts with spilite-keratophyre volcanism, thick sedimentary sequences, and a thermal event at deeper crustal levels. Small yet widespread occurrences of MOR-type basalts (now mostly eclogites) suggest that rifting attained at least the narrow-ocean stage. To the south of the North- German/Polish ocean, in Ordovician time, the following elements can be recognized (Fig. 1): mid-European microplate; Saxothuringian rift/narrow ocean; Tepla/Barrandian microplate; Barrandian/Moldanubian basin, which is possibly continued westward in the Southern Vosges/Black Forest basin; Vindelician microplate; Massif Central/ Western Alps narrow ocean; and southern Europe, possibly separated by an additional basin from Gondwanaland. Closure of all these basins was accompanied by northward drift of the African assembly, during Late Ordovician through Carboniferous time. Opening of the Rhenohercynian basin within the mid-European plate, during Devonian time, contrasts with the general trend of convergence. The Rhenohercynian basin possibly opened when the mid-European plate overrode the spreading-center of the pre-Caledonian ocean farther north, or alternatively, due to back-arc spreading related to northward subduction of oceanic crust farther south. Subduction zones developed in each of these basins. They exhibit a bilateral symmetry, with the Moldanubian block acting as a median zone of structural parting. Subduction of oceanic crust started in Late Ordovician-Early Devonian time (depending on the region), and was largely terminated by Late Devonian or Early Carboniferous time; convergence continued during Late Carboniferous time and involved the subduction of continental crust. Crustal shortening has brought about alpinotype nappe-thrusting, commonly resulting in an inversion of early Variscan, pressure-dominated metamorphic facies. The tectonic assemblages were welded together in Carboniferous time by extensive lowpressure metamorphism and widespread post-tectonic granites.
Article
New experimental data on the upper thermal stability of chlorite in the system H'O- MgO-AlrOr-SiO, are reported here and are combined with other experimental data, molar volumes of the solid phases, heat capacities, HrO fugacities, and activity expressions for cordierite, chlorite, and orthopyroxene to obtain an array of univariant curves about an invariant point at 720 + l0"C and2.75 + 0.3 kbar where chlorite, cordierite, forsterite, orthopyroxene, spinel, and water are in equilibrium. A crystallochemical study of chlorite synthesized from three different bulk compositions at 3 and 14 kbar and over the tem- perature range of 650-850"C suggests that Mg-chlorite attains a composition of (Mg.rAl'.r) (Sir 8Alr ,)Oro(OH)8 at its upper thermal stability at all pressures. Thermochemical data for clinochlore were derived from the experimental data on the dehydration of chlorite above 3.5 kbar, yielding values of -8220.452 + 27 kJ/mol and -2186.2 + 33 J/(K'mol) for
Article
The univariant reactions (l) Ta + Fo : 5En * Stm and (2)Ta:3En * Qtz * Stm have been experimentally reversed at P",o : Ptot"r. Smooth curves drawn between the brackets pass through the coordinates 0.5 kbar, 617'C; I kbar,638.C; 2kbar,662"C;3 kbar, 679.C;4 kbar, 696oC for reaction (1); and 0.5 kbar,660'C; I kbar, 697"C;2kbar,738. C, for reaction (2). The positions of the curves are probably within 5"C of the stated values. Intersections of reactions (l ) and (2) with the reversed reactions (3) 9Ta * 4Fo : 5Ant * 4Stm and (4) Ant : 7En * Qtz * Stm, respectively, generate two invariant points. The reaction (7) Ant : Ta * 4En is common to both invariant points and together with reactions (3) and (4) serves to bound the stability field ofanthophyllite. Reactions (2) and (4) intersect at Ps,6: 5 kbar, and their positions are consistent with the P-T diagram proposed by Greenwood. However, the slope of reaction (1) is less positive than the slope ofreaction (3), and this is inconsistent with Greenwood's proposed P-T diagram.lf reaction (l ) is accurately positioned, the conflict can be resolved in one of two ways. Reaction (3) can be rotated until its slope is less positive than the slope of reaction (l), or the P-T diagram can be inverted. The first alternative is most plausible. If Greenwood's proposed P-T diagram is correct, the data presented here limit the stability of pure Mg-anthophyllite with respect to the assemblage Ta * En to a maximum pressure of about 5 kbar Pr,o.
Article
We have examined experimental data for reactions among anthophyllite, enstatite, forster- ite, quartz, talc, and water, using linear programming methods, in order to examine the range of internally consistent thermodynamic data sets that is permitted by the experiments. Phase diagrams calculated using the derived thermodynamic parameters may have all topologies previously proposed to describe the anthophyllite stabitty field as well as others. Depending upon the experimental data analyzed, phase diagrams in close agreement with calorimetrically measured thermodynamic parameters may have the topology originally pro- posed by Greenwood (1963) or the topology he sketched (Greenwood, l97l) in revising his original ideas. The experimental data of Chernosky (1976) and Greenwood (1963) appear not to be consistent with the available calorimetric data for the above minerals within their stated uncertainties. This discrepancy is most easily explained if there exists a major difference be- tween the properties of talcs used in the experiments and that used for calorimetric measure- ments or if the calorimetric measurements are incorrect. A similar di-fference in the properties of anthophyllite may also be indicated.
Article
The upper stability limits for intermediate magnesium-irbn chlorites of the clinochlore- daphnite solid solution series were investigated at 2.07 kbar water pressure equal to total pres- sure at oxygen fugacities defined by the nickel-nickel oxide buffer. Five distinct high temperature breakdown assemblages of chlorite have been found: (i) cordieriteo * olivine"" * spinel"" from ChlrooDa" to Chl.uDaru, (ii) cordierite"" * olivine"" * hercyniteo from Chl.uDasu to Chl.rDaur, (iii) cordierite"" * olivine"" * magnetite"" from approximately ChlnrDar. to Chl.uDauu, (iv) cordierite"" * orthoamphibole"" * magnetite"" from ChlnuDauu to ChlrrDa6, and (v) cordierite"" + quartz * magnetite"" from ChlrrDa* to ChloDaroo. The upper limit of chlorite stability varies over the temperature range 720" to 535oC between the clinochlore and daphnite end members respectively. Unit cell parameters for synthetic chlorites of the clinochlore-daphnite solid solution series have been computed and expressed as functions of composition. Electron microprobe analyses of synthetic cordierite indicate a relatively uniform variation in composition with up to 11.9 wt percent iron as FeO in solid solution. The composition of the orthoamphibole phase has been calculated to be intermediate between anthophyllite and gedrite, with a small amount of iron substitution. The results are applicable in the interpretation of some contact metamorphic assemblages derived from ultramafic rocks, and are used to explain the origin of cordierite-anthophyllite bearing rocks found in chlorite alteration pipes below the base metal deposits of the Noranda area.
Article
The stability of amphiboles on the join MgrSi.Orr(OH)r-FezSisorr(OH), has been hydro- thermally investigated at 2 kbar fluid pressure as a function of oxygen fugacity and temper- ature. Atf;, defined by the MH buffer, the maximum extent of solution of Feend-member in amphibole is l4 and 22mole percent at725" and 630'C respectively; amphibole is unstable below 630'C, being replaced by the assemblage talc + quartz * magnetite + hematite. AtI), defined by the NNO buffer the extent of solid solution expands to 54,62, and 65 mole percent Fe end-member at 7250, 625', and 600"C, respectively. Results obtained in this study have been combined with previously published data to produce a I-X section of the upper thermal stability of amphibole at2kbar andl,, defined by the FMQ buffer. Temperatures for the reaction: amphibole + pyroxene + quartz + vapor decrease from -765'C for the pure Mg end-member to -710'C for 62 mole percent Fe end- member. The breakdown reaction: amphibole - olivine + quartz + vapor, was observed for the more iron-rich amphiboles, and takes place at -675"C for amphibole of 73 mole percent Fe end-member. Comparison of the experimental results to selected natural cases shows good agreement in maximum iron contents for the appropriate oxygen fugacity range. Estimates of temperatures of crystallization of metamorphic and igneous Fe-Mg amphiboles are also consistent with pnor reports.
Article
Brown hornblende occurs in minor amounts in the Artfjllet gabbro and dolerites, except in quartz-dolerites where a pale green hornblende occurs. In the gabbro, brown hornblende is mostly Ti-bearing pargasite or kaersutite. It occurs along veins of orthopyroxene, as rims around and blebs in pyroxenes, with orthopyroxene in coronas between olivine and plagioclase and in coronas between ilmenite and plagioclase. In the olivine-dolerites and orthopyroxene-dolerites brown hornblende is ferroan titanian pargasite or ferroan kaersutite. The pale green hornblende in the quartz-dolerites is a magnesio-hornblende. The hornblendes in the dolerites are interstitial or granular, in some dolerites occurring as coarse oikocrysts. It is proposed that under certain conditions the Ti content of hornblende can be used as a thermometer, derived from experimental data of Helz (1973). Microstructures, compositions and formation temperatures (< 1,040="" c)="" show="" that="" the="" brown="" hornblende="" in="" the="" gabbro="" is="" not="" magmatic,="" but="" of="" subsolidus="" origin.="" probably="" it="" formed="" as="" a="" result="" of="" the="" introduction="" of="" water="" into="" the="" gabbro="" during="" a="" deformation="" event="" that="" occurred="" early="" in="" the="" cooling="" history="" of="" the="" gabbro.="" least-squares="" modelling="" of="" hornblende="" formation="" indicates="" that="" all="" magmatic="" minerals="" must="" have="" participated="" in="" the="" reaction="" and="" that="" the="" reaction="" probably="" was="" not="" isochemical.="" microstructures,="" compositions="" and="" formation="" temperatures="" (1,030-965="" c)="" of="" brown="" hornblende="" in="" the="" dolerites="" are="" consistent="" with="" late-stage="" crystallization="" from="" the="" magma.="" for="" the="" pale="" green="" hornblende="" in="" the="" quartz-dolerites="" a="" magmatic="" origin="" is="" likely,="" but="" cannot="" be="">
Article
Complex reaction textures in coronitic metagabbros and retrograded eclogites of the KTB pilot and an adjacent drilling provide evidence for a multistage metamorphic history in the Variscan basement of the NW Bohemian Massif. The eclogites show complete metamorphic recrystallization leaving no textural or mineral relics of their igneous precursors. In contrast, textural relics of the igneous protolith are still preserved in the metagabbros where the metamorphic overprint under high pressure conditions achieved only partial replacement of the initial assemblage plagioclase + augite + amphibole (+olivine or orthopyroxene?) + ilmenite to form the eclogite facies assemblage garnet + omphacite + kyanite + zoisite + quartz+rutile. The garnets in the metagabbros occur in the typical necklace fashion at the borders between the original plagioclase and mafic phase domains. In the same rocks, omphacite formed by a topotactic reaction mechanism replacing igneous augite as well as in smaller grains at the margins of the texturally igneous clinopyroxene where it occurs without fixed orientation with respect to the relict phase. Both eclogites and metagabbros show a partial breakdown under high pressure granulite (transitional to high pressure amphibolite) facies conditions during which omphacite broke down to vermicular symplectites of diopside + plagioclase. A later pervasive medium pressure metamorphism under amphibolite facies conditions led to the development of assemblages dominated by hornblende + plagioclase+titanite: phases prevailing in the overwhelming majority of the surrounding metabasites. Subsequent vein-associated retrogression produced minerals typical of the greenschist to zeolite facies. All metamorphic stages may be represented in a single thin section but although the overall reaction sequence is apparent, the obvious disequilibrium in the rocks makes the use of conventional geothermobarometry difficult. However, calculations made by assuming an approach to domainal equilibrium show that both the eclogite facies and early breakdown occurred above 10 kb. As the metamorphic unit hosting these particular metabasites is generally characterized by pressures below 10 kb these results have important implications for understanding the tectonometamorphic evolution of the region. The relationship between the studied rocks and other units in the NW Bohemian Massif exhibiting a multistage metamorphic evolution is discussed and possible tectonic models evaluated.
Article
The Trinity peridotite was emplaced over metabasalts and metasedimentary rocks of the central metamorphic belt along the Devonian Trinity thrust zone. Three metamorphic events can be recognized in the Trinity peridotite: (1) antigorite (δD= −63 to −65%.) formation related to regional underthrusting of the central metamorphic belt; (2) contact metamorphism associated with Mesozoic dioritic plutons; and (3) late-stage formation of lizardite ± brucite and chrysotile (δD= −127 to −175%.) due to infiltration of meteoric waters. Abundant relict phases indicate incomplete reactions and strongly suggest that the availability of H2O was a controlling factor during serpentinization. Antigorite (event 1) formed as a result of infiltration into the Trinity peridotite of mixed H2O-CO2 fluids derived from the underlying central metamorphic belt. Foliation defined by magnetite veins and shear zones within antigorite serpentinites are subparallel to the Trinity thrust. The assemblage Fo + Atg + Chl + Mag ± Tr ± Carb reflects partial hydration of peridotite at 425–570° C. Talc-rich serpentinite formed along the thrust as a result of the infiltration of silica-bearing fluids. Metasomatic mass-balance calculations based on silica solubilities and the extent of antigorite serpentinization suggest that 80–175 volumes of fluid have passed through a given volume of original peridotite at the Trinity thrust. The Trinity thrust probably represents a Devonian subduction zone. Thermodynamic calculations suggest that hydration reactions account for ∼30–35% of the total heat released by the cooling Trinity peridotite. By analogy, similar hydration reactions are to be expected in the overlying mantle wedge of a subduction zone which act to retard cooling of the hanging wall, just as dehydration reactions delay heating of the downgoing slab. Metasomatic zones formed in peridotite at the Trinity thrust may reflect similar metasomatic processes to those proposed to occur in the mantle wedge above a subducting slab.
Article
Ultrabasic metamorphic rocks with typical mineral assemblage of cummingtonitic hornblende+Mg-chlorite+talc (described byMatthes u.Okrusch, 1965, and called “hösbachit”) were found for the first time in Odenwald/Germany. Three steps of metamorphic development can be described: the primary magmatic ultrabasic rock consisted of (1) orthopyroxene (bronzite), clinopyroxene (diallage), poikilitic olivine, and some hornblende. This is indicated by textural relicts, structural and geochemical investigations. A regional metamorphic process under conditions of the staurolite-almandine-subfacies ofWinkler’s andTurner andVerhoogen’s almandine-amphibolite facies transformed this assemblage to (2): cummingtonitic hornblende I+Mg-chlorite I (clinochlore)+talc. The third step followed under more diaphthoritic conditions and brought mineral assemblage (3): tremolitic hornblende II+Mg-chlorite II (pennine)+talc.


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# Circles

In the circle with a radius 7.5 cm are constructed two parallel chord whose lengths are 9 cm and 12 cm. Calculate the distance of these chords (if there are two possible solutions write both).

Correct result:

x =  10.5 cm
y =  1.5 cm

#### Solution:

$r^2 = 7.5^2 = x_1^2 + (9/2)^2 \ \\ r^2 = 7.5^2 = x_2^2 + (12/2)^2 \ \\ x_1 = 6 \ cm \ \\ x_2 = 4.5 \ cm \ \\ \ \\ x = x_1+x_2 = 10.5 \ \text{cm}$
$y = |x_1-x_2| = 1.5 \ \text{cm}$

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# Thread: Differential equation solver

1. ## Differential equation solver

Hello,
How to solve $4*x^2*y" +12*x*y' +3*y=0, y(4)=\frac18, y'(4)=\frac{-3}{64}$ with the help of Droid48 calculator application? It involves stiff initial value problem as well as vector-value differential equation.

2. ## Re: Differential equation solver

Originally Posted by Vinod
Hello,
How to solve $4*x^2*y" +12*x*y' +3*y=0, y(4)=\frac18, y'(4)=\frac{-3}{64}$ with the help of Droid48 calculator application? It involves stiff initial value problem as well as vector-value differential equation.
Try for a solution of the form $y=x^r$ and see what values of $r$ work. You shouldn't need a calculator for this.

3. ## Re: Differential equation solver

The Wronskian is going to be $Ce^{-\displaystyle \int \dfrac{3}{x}dx} = Ce^{-3\ln x} = e^{\ln x^{-3k}} = x^{-3k}$ for some constant $k$. So, as Walagaster says, the solution will be $x^r$ for some $r$.

4. ## Re: Differential equation solver

Originally Posted by SlipEternal
The Wronskian is going to be $Ce^{-\displaystyle \int \dfrac{3}{x}dx} = Ce^{-3\ln x} = e^{\ln x^{-3k}} = x^{-3k}$ for some constant $k$. So, as Walagaster says, the solution will be $x^r$ for some $r$.
Hello,
How to compute wronskian? How it can be utilised to solve differential equations? Would you give me some more information of Wronskian. As far as i know it is a determinant that is related to linear independence of a set of differential equations.

5. ## Re: Differential equation solver

Originally Posted by SlipEternal
The Wronskian is going to be $Ce^{-\displaystyle \int \dfrac{3}{x}dx} = Ce^{-3\ln x} = e^{\ln x^{-3k}} = x^{-3k}$ for some constant $k$. So, as Walagaster says, the solution will be $x^r$ for some $r$.
Hello, So applying initial conditions $y=x^{-\frac32}$

6. ## Re: Differential equation solver

Originally Posted by Vinod
Hello, So applying initial conditions $y=x^{-\frac32}$
So, after 6 months and no . or ? after your quote, are you asking us or telling us? Do you have a question for us? Or are you hoping someone will check your work (which you didn't show) for you?

7. ## Re: Differential equation solver

$$4x^2y''+12xy'+3y =0 \quad y(4)=\tfrac18, \, y'(4) = -\tfrac3{64}$$
\begin{align}
&\text{Let} & x &= e^t \\ &&\implies \tfrac{\mathrm dx}{\mathrm dt} &= e^t = x \\[8pt]
&\text{and} & y(t) &= y\big(x(t)\big) \\
&& \tfrac{\mathrm dy}{\mathrm dt} &= \tfrac{\mathrm dy}{\mathrm dx} \tfrac{\mathrm dx}{\mathrm dt} = x\tfrac{\mathrm dy}{\mathrm dx} &\implies \dot y &= xy' \\
&& \tfrac{\mathrm d^2y}{\mathrm dt^2} &= \tfrac{\mathrm dy}{\mathrm dx} \tfrac{\mathrm dx}{\mathrm dt} + x\tfrac{\mathrm d^2y}{\mathrm dx^2} \tfrac{\mathrm dx}{\mathrm dt} \\ &&&= x^2 \tfrac{\mathrm d^2y}{\mathrm dx^2} + x\tfrac{\mathrm dy}{\mathrm dx} &\implies \ddot y &= x^2y'' + xy' \end{align}
And so
\begin{align} 4x^2y''+12xy'+3y &= 4(x^2y''+ xy') + 8xy'+3y \\ &= 4\ddot y + 8\dot y + 3y = 0 \end{align}
The characteristic equation \begin{align} 4r^2 + 8r + 3 &= (2r)^2 + 4(2r) + 3 \\ &= (2r+1)(2r+3) = 0 \end{align}
Has roots $$r_1 = -\tfrac12 \quad \text{and} \quad r_2 = -\tfrac32$$
so \begin{align}y(t) &= c_1 e^{-\frac12 t} + c_2 e^{-\frac32 t} \\ &= c_1 (e^{t})^{-\frac12} + c_2 (e^t)^{-\frac32} \\ y(x) &= c_1 x^{-\frac12} + c_2x^{-\frac32 } \\ &= \frac{c_1}{\sqrt x} + \frac{c_2}{\sqrt{x^3}}\end{align}
And plug in the initial conditions to find the constants.

8. ## Re: Differential equation solver

Originally Posted by Archie
$$4x^2y''+12xy'+3y =0 \quad y(4)=\tfrac18, \, y'(4) = -\tfrac3{64}$$
\begin{align}
&\text{Let} & x &= e^t \\ &&\implies \tfrac{\mathrm dx}{\mathrm dt} &= e^t = x \\[8pt]
&\text{and} & y(t) &= y\big(x(t)\big) \\
&& \tfrac{\mathrm dy}{\mathrm dt} &= \tfrac{\mathrm dy}{\mathrm dx} \tfrac{\mathrm dx}{\mathrm dt} = x\tfrac{\mathrm dy}{\mathrm dx} &\implies \dot y &= xy' \\
&& \tfrac{\mathrm d^2y}{\mathrm dt^2} &= \tfrac{\mathrm dy}{\mathrm dx} \tfrac{\mathrm dx}{\mathrm dt} + x\tfrac{\mathrm d^2y}{\mathrm dx^2} \tfrac{\mathrm dx}{\mathrm dt} \\ &&&= x^2 \tfrac{\mathrm d^2y}{\mathrm dx^2} + x\tfrac{\mathrm dy}{\mathrm dx} &\implies \ddot y &= x^2y'' + xy' \end{align}
And so
\begin{align} 4x^2y''+12xy'+3y &= 4(x^2y''+ xy') + 8xy'+3y \\ &= 4\ddot y + 8\dot y + 3y = 0 \end{align}
The characteristic equation \begin{align} 4r^2 + 8r + 3 &= (2r)^2 + 4(2r) + 3 \\ &= (2r+1)(2r+3) = 0 \end{align}
Has roots $$r_1 = -\tfrac12 \quad \text{and} \quad r_2 = -\tfrac32$$
so \begin{align}y(t) &= c_1 e^{-\frac12 t} + c_2 e^{-\frac32 t} \\ &= c_1 (e^{t})^{-\frac12} + c_2 (e^t)^{-\frac32} \\ y(x) &= c_1 x^{-\frac12} + c_2x^{-\frac32 } \\ &= \frac{c_1}{\sqrt x} + \frac{c_2}{\sqrt{x^3}}\end{align}
And plug in the initial conditions to find the constants.
Hello,
Your method of solving is different than mentioned in my PDF.

9. ## Re: Differential equation solver

Originally Posted by Vinod
Hello,
Your method of solving is different than mentioned in my PDF.
Typically there is more than one way to solve a differential equation, though Archie's method is "standard." Can you show us what your pdf 's method looks like? Once we know that we can walk you through any difficulties you are having with it.

-Dan

10. ## Re: Differential equation solver

Originally Posted by Vinod
Hello,
Your method of solving is different than mentioned in my PDF.
\begin{align}
&\text{Let} & t &= \ln x \\ &&\implies \tfrac{\mathrm dt}{\mathrm dx}&= \tfrac1x \\[8pt]
&\text{and} & y(x) &= y\big(t(x)\big) \\
&& \tfrac{\mathrm dy}{\mathrm dx} &= \tfrac{\mathrm dy}{\mathrm dt} \tfrac{\mathrm dt}{\mathrm dx} = \tfrac1x\dot y &\implies xy' &= \dot y \\
&& \tfrac{\mathrm d^2y}{\mathrm dx^2} &= -\tfrac1{x^2}\tfrac{\mathrm dy}{\mathrm dt} + \tfrac1{x}(\tfrac{\mathrm d^2y}{\mathrm dt^2}\tfrac{\mathrm dt}{\mathrm dx}) \\ &&&= \tfrac1{x^2}\ddot y - \tfrac1{x^2}\dot y &\implies
x^2y'' &= \ddot y - \dot y\end{align}
And so
\begin{align} 4x^2y''+12xy'+3y &= 4(\ddot y - \dot y) + 12\dot y + 3y \\ &= 4\ddot y + 8\dot y + 3y = 0 \end{align}
And continue as before.

11. ## Re: Differential equation solver

The substitutions $x=e^t$ and $t = \ln x$ are the same, one going one way and one the other. Both are significantly more complicated than just trying $y = x^r$ in the first place, which is the method I would regard as "standard" for an Euler equation.

12. ## Re: Differential equation solver

Originally Posted by Walagaster
The substitutions $x=e^t$ and $t = \ln x$ are the same, one going one way and one the other. Both are significantly more complicated than just trying $y = x^r$ in the first place, which is the method I would regard as "standard" for an Euler equation.
They are the same, yes. But for me, each is better in parts than the other. They are more complicated than trying $y=x^r$, but that doesn't really help you if the roots of the characteristic equation are not real - unless you were taught the derivation of the real solution (see below). In the case that you have a repeated root, you also need to use the method of reduction of order to get the second solution.

The simplest approach is probably to ask Wolfram Alpha or look for a formula on Wikipedia, but I wouldn't recommend either of those as they tell you nothing about what's actually going on in the equation.

With solutions $$y_1 = x^{u+iv} \qquad y_2 = x^{u-iv}$$
We have \begin{align}
y &= c_1 x^{u+iv} + c_2 x^{u-iv} \\
&= c_1 x^u x^{iv} + c_2 x^u x^{-iv} \\
&= x^u(c_1 x^{iv} + c_2 x^{-iv}) \\
&= x^u(c_1 e^{iv \ln x} + c_2 e^{-iv \ln x}) \\
&= \tfrac12x^u\big((c_1 + c_2) (e^{iv \ln x} + e^{-iv \ln x}) + (c_1 - c_2)(e^{iv \ln x} - e^{-iv \ln x})\big) \\
&= x^u\big(\tfrac{c_1 + c_2}2 \cos{(v \ln x)} + \tfrac{c_1 - c_2}2(\sin{(v \ln x)} \big) \\
&= x^u\big(c_3 \cos{(v \ln x)} + c_4(\sin{(v \ln x)} \big) \end{align}
Not tremendously difficult, but not often taught in my experience.

13. ## Re: Differential equation solver

Originally Posted by Archie
They are the same, yes. But for me, each is better in parts than the other. They are more complicated than trying $y=x^r$, but that doesn't really help you if the roots of the characteristic equation are not real - unless you were taught the derivation of the real solution (see below). In the case that you have a repeated root, you also need to use the method of reduction of order to get the second solution.

The simplest approach is probably to ask Wolfram Alpha or look for a formula on Wikipedia, but I wouldn't recommend either of those as they tell you nothing about what's actually going on in the equation.

With solutions $$y_1 = x^{u+iv} \qquad y_2 = x^{u-iv}$$
We have \begin{align}
y &= c_1 x^{u+iv} + c_2 x^{u-iv} \\
&= c_1 x^u x^{iv} + c_2 x^u x^{-iv} \\
&= x^u(c_1 x^{iv} + c_2 x^{-iv}) \\
&= x^u(c_1 e^{iv \ln x} + c_2 e^{-iv \ln x}) \\
&= \tfrac12x^u\big((c_1 + c_2) (e^{iv \ln x} + e^{-iv \ln x}) + (c_1 - c_2)(e^{iv \ln x} - e^{-iv \ln x})\big) \\
&= x^u\big(\tfrac{c_1 + c_2}2 \cos{(v \ln x)} + \tfrac{c_1 - c_2}2(\sin{(v \ln x)} \big) \\
&= x^u\big(c_3 \cos{(v \ln x)} + c_4(\sin{(v \ln x)} \big) \end{align}
Not tremendously difficult, but not often taught in my experience.
Hello,

But $\sin{\theta}=\frac{1}{2i}(e^{i\theta}-e^{-i\theta})$? and your secondlast step does not contain i in the denominator. It is only$\frac{c_1-c_2}{2}$. How is that?

14. ## Re: Differential equation solver

Oops! I thought it came out a bit easy! I need to sort it out.

Essentially, the real solutions are when $c_1$ and $c_2$ are conjugate pairs, and in this case $c_1 - c_2$ is purely imaginary, so the $i$ cancels. There's also an odd factor of $\tfrac12$ lying around, but that doesn't matter so much.

15. ## Re: Differential equation solver

Something like

With solutions $$\DeclareMathOperator{\im}{Im} \DeclareMathOperator{\re}{Re} y_1 = x^{u+iv} \qquad y_2 = x^{u-iv}$$
We have \begin{align}
y &= c_1 x^{u+iv} + c_2 x^{u-iv} \\
&= c_1 x^u x^{iv} + c_2 x^u x^{-iv} \\
&= x^u(c_1 x^{iv} + c_2 x^{-iv}) \\
&= x^u(c_1 e^{iv \ln x} + c_2 e^{-iv \ln x}) \\
&= x^u\big((c_1 + c_2) \tfrac12(e^{iv \ln x} + e^{-iv \ln x}) + (c_1 - c_2)\tfrac{i}{2i}(e^{iv \ln x} - e^{-iv \ln x})\big) \\
&= x^u\big((c_1 + c_2)\cos{(v \ln x)} + (c_1 - c_2)i\sin{(v \ln x)} \big)\end{align}
For this expression to be real, we require that $\im{(c_1+c_2)} = 0$ and $\re{(c_1-c_2)} = 0$. This means that we must choose $c_1 = \overline{c_2}$. This also leads to the two coefficients being independent, so we may write.
$$y = x^u\big(c_3\cos{(v \ln x)} + c_4\sin{(v \ln x)} \big) \qquad (c_3,c_4) \in \mathbb R^2$$