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Yury L. Voytekhovsky, Alena A. Zakharova
Pétrographie structures and Hardy - Weinberg equilibrium
as one of the agreements than as a principle" [20, p.20]. By the way, a completely modern classification of rocks is "used below".
The attitude towards the situation is gradually changing: "No matter how diverse the associations of rocks mapped in different regions may seem, there is a confidence that, with a systematic approach, they, like chemical elements in the Periodic table, can be naturally classified, paving the way for the unification of legends to the geological maps of the new generation" [16, p.5]. The question is what is meant by a systematic approach and which mathematical apparatus to use.
Methodology. If a systematic approach to the description of petrographic structures means the use of as many parameters as possible, then this already takes place and is rather a weakness than a strength. The reason is clear - very different morphological (euhedral, xenomorphic, etc.), large-scale (even- and uneven-grained; fine-, medium-, coarse-grained, etc.) and genetic (blastasy, etc.) characteristics of minerals (elements) that compose the rock (rock system), cannot be linked within a coherent theory [14, p.110-121, 283-294]. By the principles of systems theory, it was previously proposed to shift the focus from the morphometric characteristics of minerals to the statistics of their contact relationships in the description of the rocks organization [3, 5, 7].
The organization of «-mineral rock is proposed to be expressed through the algebraic relation:
n
Z pijmmj = [m ot2 .. m„] h j=1
revealing the fundamental role of the symmetric matrix [PiJ] of probabilities (frequencies) piJ- of different intergranular contacts mimJ- of the minerals mi and mj. At the same time, it defines in space (m1, ..., mn) non-degenerate central quadratic surfaces (n-dimensional ellipsoids and hyperboloids) -structural indicatrixes of rocks.
The mathematical expression of the petrographic structure is the canonical diagonal form [Dii] of the matrix [PiJ], which determines the type of structure with the signs of the coefficients dii. The real symmetric matrix is always reducible to the diagonal form by the non-degenerate transformation [Dii] = [Qj] [Pj] [Qj] _1, corresponding to the space rotation of the structural indicatrix (m1, ..., mn) and reduction to the principal axes [9, 19]. The nomenclature of the petrographic structure Sm means that among the coefficients dii there are exactly m positive ones.
So, a continuous change in the probabilitiespij of various intergranular contacts (rock organization) does not contradict a sharp change in the type of structural indicatrix (petrographic structure). A complete classification of petrographic structures, with which their nomenclature is strictly connected, is based on indicatrixes. It seems that the proposed methodology follows a systematic approach and at least partially resolves the doubts of E.S.Fedorov and A.Harker. The theory was used to differentiate the gabbronorites monotonic section of the Fedorovo-Panskii intrusion on the Kola Peninsula [4, 6, 8]. We present new applications of the theory with necessary additions.
Granites of the Salma pluton, Karelia. An important geological problem is the development of methods for granitoid intrusions mapping [16]. Let us consider the granites of the Salma pluton in terms of description and comparison of petrographic structures [2]. They consist of five mineral phases (Fig. 1): quartz (yellow), plagioclase (blue), K-Na feldspar (red), biotite (dirty green), and accessory minerals (purple). Do petrographic structures differ in two thin sections?
In both cases, the matrices [PiJ] are reduced to a diagonal form corresponding to the structure S52. Structural indicatrix is a space three-cavity hyperboloid (m1, ..., m5). Indexes correspond to rows and columns of matrices and mean: 1 - quartz, 2 - plagioclase, 3 - K-Na feldspar, 4 - biotite, 5 - accessory minerals. Thus, both samples belong to the same structural type, although their organization, fixed by the whole set of intergranular contacts probabilities pj, is different (typification of the structure does not require conversion of the numbers of intergranular contacts to probabilities):
Pll P12 Pin mi
P21 P22 P2n m2 = 1
P„1 Pn2 . Pnn .mn.
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Journal of Mining Institute. 2020. Vol. 242. P. 133-138 • Geology
Yury L. Voytekhovsky, Alena A. Zakharova
Pétrographie structures and Hardy - Weinberg equilibrium
a
tm
Fig. 1. Medium-grained piterlite (a) and porphyry microcline-albite granite (b), no porphyry phenocrysts in the groundmass. The vertical size is 1 cm
b
164 87.5 106.5 7 10.5" +
87.5 35 49.5 2.5 3.5 +
106,5 49.5 46 1.5 3.5 -
7 2.5 1.5 1 3.5 -
10.5 3.5 3.5 3.5 2 —
J5 '
148 98 182 21.5 4.5" +
98 53 102.5 15 2 +
182 102.5 105 17.5 2.5 —
21.5 15 17.5 5 2.5 —
4.5 2 2.5 2.5 0 —
Analysis of thin sections can be continued since accessory minerals and biotite make up a very small share in granite. We apply the method of accessory minerals subtraction [7], removing them from consideration one by one, that is, removing the corresponding rows and columns from the matrices [Pj]. In both cases, we obtain the type of structure S42, the indicatrix is a space two-cavity hyperboloid (m1, ..., m4):
164 87.5 106.5 7 " +
87.5 35 49.5 2.5 +
106.5 49.5 46 1.5 —
7 2.5 1.5 1 —
'4 '
148 98 182 21.5" +
98 53 102.5 15 +
182 102.5 105 17.5 —
21.5 15 17.5 5 —
S:
Yury L. Voytekhovsky, Alena A. Zakharova
Pétrographie structures and Hardy - Weinberg equilibrium
But without biotite we get structures S\ and S42 cavity hyperboloids (mi, ..., ^4):
" 164 87.5 106.5 10.5
87.5 35 49.5 3.5
106.5 49.5 46 3.5
10.5 3.5 3.5 2
Indicatrixes are space three-cavity and two-
+
+
^ S
'4 '
148 98 98 53
182 102.5 4.5 2
182 4.5
102.5 2
105 2.5
2.5 0
+
S1.
Along with the previous result, this means that in one thin section (Fig.1, a) accessory minerals and biotite form statistically equivalent intergrowths, in another (Fig.1, b) they are different. Without accessory minerals and biotite, the main (carcass-forming) part of the studied granites has a structure S], indicatrix is a space two-cavity hyperboloid (m1, m2, m3):
" 164 87.5 106.5 87.5 35 49.5 106.5 49.5 46
+
^ S1;
148 98 182 98 53 102.5 182 102.5 105
+
^ S1
Thus, the suggested methodology for petrographic structures standardization according to the statistics of intergranular contacts makes it possible to identify even subtle features of rock organization. This allows it to be used for granitoid intrusions mapping. Routine algebraic operations with arbitrarily large matrices are computerized.
Granites of the Akzhailau massif, Kazakhstan. Massive textures are very common in rocks of predominantly magmatic origin. They mean the random spatial distribution of rock-forming minerals, in contrast to layered, banded, or spotted textures. But is the random spatial distribution always consistent with the perfect mixing of minerals? To test the hypothesis, granite from the Akzhailau massif was taken (Fig.2).
Hardy - Weinberg equilibrium is accepted as a mathematical model corresponding to the ideal mixing of minerals [10; 11, p. 126-128]. In relation to our task, this means the following. If p and
Fig.2. The typified physiography of granite of the Akzhailau 4 are the frequencies (probabilities) of minerals A massif [1]. Gray - quartz, white - feldspars, and B (conditionally pA + qB = 1), then the equi-
mica andamphiboles are not sh°wn. librium frequencies of their contacts AA, AB (the
Polished chip sample, 15 x 10 cm A v
Yury L. Voytekhovsky, Alena A. Zakharova
Petrographic structures and Hardy - Weinberg equilibrium
same as BA) and BB in the rock can be calculated by the formula
(pA + qB)2 = p2 AA + 2pq AB + q2 BB = 1.
Estimating deviations from equilibrium frequencies is a statistical routine. The Hardy-Weinberg formula is obviously generalized by the number of terms on polymineralic rocks
(P1A1 + ... + pnAn)2 = X p,j AAj = 1, where i, j = 1, ..., n.
It is theoretically and practically important that it is generalized in degrees to:
• ternary
(P1A1 + ... + pnAn)3 = XPijkAAA = 1, where i,J, k = 1, ..., n;
• quarternary intergranular contacts
(P1A1 + ... + pnAn)4 = X Pjki AiAjAkAi = 1, where i, J, k, l = 1, n.
The study of petrographic structures and textures should be done in 3D. Analysis in 2D (in thin section or polished samples) is justified only by the fact that their classification and nomenclature were formed from observations in 2D. Today it is technically impossible to obtain statistics of quaternary contacts of mineral grains in rock. The joints of four grains do not fall into the petrographic section. But the statistics of their ternary (triple) contacts in n-mineral rocks can be found in thin sections.
Ternary contacts. The use of quaternary and even ternary intergranular contacts in practice is interesting because the resulting classification of petrographic structures is much more extensive than in the case of binary contacts. It should be expected that in this case, through the statistics of intergranular contacts, more and more subtle details of the rocks organization are revealed and typed.
For Akzhailau granite, the probabilities of quartz and feldspar grains in thin sections (208 and 428, respectively) are p1 = 0.327 and p2 = 0.673. The calculated probabilities of ternary contacts corresponding to the Hardy-Weinberg equilibrium are p111 = 0.035, p112 = 0.216, p122 = 0.444, p222 = = 0.305. Of the total number of 1116 ternary contacts, this is: 39, 241, 496, and 340. The actual number of contacts is calculated in the section: 17, 211, 490, and 398. The value of the non-parametric chi-square criterion of 26.02 significantly exceeds the threshold value of 11.3 for a confidence probability of p = 0.99 and the number of degrees of freedom df = 3. Thus, the hypothesis of the correspondence of the studied granite massive texture to the Hardy - Weinberg equilibrium is rejected. The reason is clearly lower frequencies p111, p112 and overestimated p222.
The peculiarity of the situation is that even if the calculated probabilities correspond to theoretical ones (i.e., the real texture to the Hardy - Weinberg equilibrium), the conclusion about the correspondence is made with a certain probability. The calculated probabilities always differ from the equilibrium ones. Hardy - Weinberg equilibria determine classification boundaries in a variety of statistically nonequilibrium situations (in our case, rock structures). To typify them as structural indicatrices, by analogy with [3, 5, 7], we use the Newtonian classification of plane cubic curves [17, p. 44-53]. Unfortunately, for n > 2, the mathematical theory for petrography is not adapted [13]. The equation of the desired curve
X pijk AAjAk = 1, where i, J, k = 1, n
for the case n = 2 (bimineral rock) is as follows (the coefficients of the equation are the probabilities of ternary contacts calculated in the thin section)
0,015 m13 + 0,189 m12 m2 + 0,439 m1 m22 + 0,357 m23 = 1 Journal of Mining Institute. 2020. Vol. 242. P. 133-138 • Geology
Yury L. Voytekhovsky, Alena A. Zakharova
Petrographic structures and Hardy - Weinberg equilibrium
Fig.3. The third order structural indicatrix of granite of Akzhailau massif
and defines a curve of two hyperbolic and one rectilinear branches (Fig. 3). This is one of two dozen possible structural indicatrices [17], whereas in the description based on binary intergranular contacts at n = 2, only two structures are possible. The study of structural indicatrices of the third order at least for n = 3 (trimineral rocks) by computer simulation methods is a promising development of the theory.
Conclusions. Thus, the proposed methodology allows us to construct a complete classification and nomenclature of petrographic structures according to the statistics of binary (for polymineral) and ternary (for bimineral rocks) intergranular contacts.
A continuous change in the organization (probabilities of intergranular contacts) of a rock does not contradict a dramatic change in the structure of the rocks, located at the classification boundaries corresponding to Hardy - Weinberg equilibria.
Correspondence of petrographic structures to one or another state (for example, Hardy -Weinberg equilibria) is checked by statistical criteria and reflects their probabilistic nature. Rock is the realization of a spatially distributed random function of the initial field of concentrations, conditions and mechanisms of crystallization.
The study of petrographic structures and textures should be performed in 3D. Analysis in 2D is justified only by the fact that their classification and nomenclature were also formed from observations in two-dimensional space. But this situation reveals the problem of stereo reconstruction of any parameters.
We gratefully acknowledge D.A.Petrov, Candidate of Geological andMineralogical Sciences (Department of Mineralogy, Crystallography and Petrography, Saint Petersburg Mining University) for the results discussion.
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Journal of Mining Institute. 2020. Vol. 242. P. 133-138 • Geology
Yury L. Voytekhovsky, Alena A. Zakharova
Petrographic structures and Hardy - Weinberg equilibrium
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Authors: Yury L. Voytekhovsky, Doctor of Geological and Mineralogical Sciences, Professor, Voytekhovskiy_ [email protected] (Saint Petersburg Mining University, Saint Petersburg, Russia), Alena A. Zakharova, Postgraduate Student, [email protected] (Saint Petersburg Mining University, Saint Petersburg, Russia). The paper was received on 29 July, 2018. The paper was accepted for publication on 15 November, 2019.
Idylle Plachini Loufouandi Matondo, Mikhail A. Ivanov
Composition and probable ore body of columbite...
UDC 553.078
Composition and probable ore body of columbite from alluvial deposits of Mayoko region (Republic of the Congo)
Idylle Plachini LOUFOUANDI MATONDO, Mikhail A. IVANOV »
Saint Petersburg Mining University, Saint Petersburg, Russia
The article presents the results of optical, electron microscopic and electron microprobe studies of columbite-group minerals, collected during heavy mineral concentrate sampling of alluvial deposits in the Mayoko region (Republic of the Congo). The aim of the study is revealing tantalum niobates ore body in this region. We found that these minerals in loose deposits are represented by two grain-size groups: less than 1.6 mm (fine fraction) and 1.6-15 mm (coarse fraction). The grains of both fractions belong mainly to columbite-(Fe), less often to columbite-(Mn), tantalite-(Mn) and tantalite-(Fe), contain impurities of Sc, Ti, and W. The crystals have micro-scaled zoning (zones varies slightly in the Ta/Nb ratio values) and contains a lot of mineral inclusions and veins represented by zircon, pyrochlore group minerals and others. Columbite-(Fe) and columbite-(Mn) are characterized by an increased content of Ta2O5 up to the transition to tantalite-(Fe) and tantalite-(Mn). This allows us to exclude the formation of subalka-line rare-metal granites, their metasomatites (albitites and greisenes) and carbonatites, from the list of possible colum-bite-tantalite ore bodies in the Mayoko region. Thus, beryl and complex spodumene rare-metal pegmatites of the mixed petrogenetic family LCT-NYF (according to P. Cherni) should be considered as a probable root source. The results of the research should be taken into account when developing the methodology for prospecting in this area.
Keywords: columbite group minerals; columbite-(Fe) - ferrocolumbite; columbite-(Mn) - manganocolumbite; tantalite-(Mn) - manganotantalite and tantalite-(Fe) - ferrotantalite; alluvial placers; indigenous source of minerals; rare metal pegmatites; Mayoko area; Republic of the Congo
How to cite this article: Loufouandi Matondo I.P., Ivanov M.A. Composition and probable ore body of columbite from alluvial deposits of Mayoko region (Republic of the Congo). Journal of Mining Institute. 2020. Vol. 242, p. 139-149. DOI: 10.31897/PMI.2020.2.139
Introduction. In 1945, during the gold placers development, columbite grains along with gold were found in the heavy fraction of alluvial deposits of modern river valleys in the Mayoko region, located in the south-west of the Republic of the Congo . Since then, no data on the ore body of this tantalum-niobium-bearing mineral have appeared in the published materials of geological organizations of the Republic of the Congo, as well as in the scientific articles. This find has stand out for a long time in connection with the problem of the rare metal deposits exploration facing the geological service of the Republic of the Congo. Major niobium and tantalum deposits within the African continent are known in neighboring territories with a similar geological structure - in the Democratic Republic of the Congo, Rwanda and other countries. Therefore, the study results presented in this article correspond to the quite urgent task of determining the nature of the probable ore body of this very valuable mineral raw material and taking these data into account when choosing the optimal exploration methodology in this area.
The studies are based on the results of grain-size classification, analysis of columbite chemical composition and its grains heterogeneity in ten stream sediment samples, collected during heavy mineral concentrate sampling and geological traversing of the area.
Location and geology of Mayoko area. Mayoko District is located in the Niari department in the rainforest area covering the Shayu granitoid massif. This massif is a part of the Congo Craton and can be traced in the territories of both the Republic of the Congo and neighboring Gabon. The geology of this massif is poorly studied. Route exploration of Cosson, Devigne, Donnot, Boineau,
* Columbite in this article refers to the following mineral species of columbite and euxenite groups: columbite-(Fe) - ferrocolumbite, columbite-(Mn) - manganocolumbite, tantalite- (Mn) - manganotantalite, and tantalite-(Fe) - ferrotanthalite (in according to the modern minerals classification adopted by the International Mineralogical Association).
