HOSHINO Mihoko

NISHIMURA Koshi

SANEMATSU Kenzo

TAKAGI Tetsuichi

WATANABE Yasusi

English

NISHIMURA, Koshi

WATANABE, Yasusi

SANEMATSU, Kenzo

HOSHINO, Mihoko

TAKAGI, Tetsuichi

東洋大学学術情報リポジトリ

A Norm Calculation Method Using Square Matrix:Implications for the Total Volume Estimationof Microscale Ore Minerals著者 NISHIMURA Koshi, WATANABE Yasusi, SANEMATSUKenzo, HOSHINO Mihoko, TAKAGI Tetsuichi著者別名 西村 光史, 渡辺 寧, 実松 健造, 星野 美保子, 高木 哲一journal orpublication titleJournal of Toyo University. Natural sciencevolume 57page range 63-72year 2013-03URL http://id.nii.ac.jp/1060/00005993/Creative Commons : 表示 - 非営利 - 改変禁止http://creativecommons.org/licenses/by-nc-nd/3.0/deed.jaJournal ofToyo Universiザ，Natural Science， No.57 : 63-72 (2013) 63 A N orm Calculation Method U sing Square Matrix: Implications for the Total Volume Estimation of Microscale Ore Minerals Koshi NISHlMURA1ヘYasushiW ATANABE2， Kenzo SANEMATSU2 Mihoko HOSHIN02， Tetsuichi T AKAGI2 Abstract An approach using square matrix is employed to calculate mineral volume abundance from whole-rock composition. Mineral species and those chemical compositions are treated as known parameters. The volume abundances of the minerals can be determined simultaneously from whole-rock element data without assignment of whole-rock FeOIFez03 ratio. This approach has been tested for some holocrystalline rocks and a good agreement with point-counting results has been obtained. In principle， the present method can be applied to any type of rock. Ore minerals in ore are sometimes too small for point-counting analysis. In such a case， the present norm calculation method becomes a useful tool to estimate the mineral volume abundance. Keywords: norm calculation， CIPW， iron-oxidation ratio， square matrix， ore mineral， modal analysis 1. Introduction Total volume estimation of the ore mineral in the ore provides crucial information to evaluate mineral-resource potentia1. One of the most popular methods of mineral volume analysis is the point-counting under optical microscope. However， this method is not available for the rocks with cryptocrystalline minerals or with microscale minerals whose size is great1y less than the thickness of the thin section (~35μm). Economic and sub-economic deposits often contain the microscale minerals that are enriched in useful elements such as REEs. In such cases， the volume of ore minerals can be briefiy estimated using data of whole rock and mineral chemistry. Numerous norm calculation methods have been used over a cen知町 toestimate abundances of standard minerals from the whole-rock chemical data of igneous rocks. 1 Natural Science Laboratory， Toyo University， 5-28-20 Hakusan， Bun勾ro-ku，Tokyo 112-8606， Japan 2 Institute for Geo-Resources and Environrnent， Geological Survey of Japan， AIST No. 7， 1-1-1 Higashi， Tsukuba， Ibaraki 305-8567， Japan 64 Koshi NISHTh瓜JRAetal. The idea of the norm calculation was first introduced into petrology by W. Cross， J.P. Iddings， L.V. Pirsson， and H. S. Washington (CIPW norm; Cross et al.， 1902) and there have been a number of modifications and refinements (e.g. Washington， 1917; Johannsen， 1931; Kelsey， 1965; Cox et al.， 1979; Ragland， 1989; Verma et al.， 2002). The norm calculation schemes using square matrix have also been developed (e.g. Nishimura & Yanagi 2000; Pruseth 2009). These calculation schemes need to assign whole rock FeOIFe203 ratio. Unfortunately， however， there is no standardized method for apportionment the oxidation ratio to the rocks whose Fe203 and FeO contents were not separately determined (Verma etα1.， 2002). The results of the conventional norm calculations va可 greatlydepending on the adjustment method of Fe-oxidation ratio (e.g. Le Maitre 1976; Middlemost 1989). In addition， most conventional models were not designed for the minerals those are rare in igneous rocks. We develop a simple norm calculation method that is applicable not only to igneous rocks but also to any other rocks without assignment ofthe oxidation ratio. Advances in micro-analytical technique， such as EPMA and LA-ICトMS，now allow the accurate chemical analysis for almost al rock-forming minerals even if a rock consists of microscale minerals. Our model treats mineral species and those chemical compositions as known parameters. In this case， mineral contents can be calculated企omwhole同rockelement data using square matrix without assignment of whole-rock FeOIFe203 ratio. Once the mineral compositions and proportions are fixed， the whole-rock iron-oxidation ratio is inevitably determined. This method can be applied to the ore minerals whose size is too small for point-counting analysis. Table 1 Mineral parameters used in the calculation Min巴ral Formula Molecular weight Specific gravityネ日uorite CaF2 78.08 3.18 Hematite Fe203 159.70 5.25 Magnetite FeO・Fe203 231.55 5.2 Quartz Si02 60.09 2.65 Goethite FeO・OH 88.86 4.3 Phlogopite K2Mg6[Si6A1202o](OH)4 834.59 3 Annite K2Fe6[Si6A1202o](OH)4 1023.83 3 Fluorapatite CaS(P04)3F 504.31 3.23 Monazite-(Ce) CeP04 235.07 5.15 Celestine SrS04 183.69 3.96 *Deer et al. (1992) Norm Calculation Using Square Ma甘lX 65 2. Method Norm calculation begins with conversion of the analysis in weight percent to molecular form. The molecular proportion of each constituent is determined by dividing by the appropriate molecular weight. Conventional norm calculation methods recast the molecular proportion of each oxide into a series of normative minerals and simultaneously determine the normative mineral assemblages. However， most of the conventional methods were designed for igneous rocks and could not be directly applicable to the other types of rock such as hydrothermally mineralized rocks. The present norm calculation method permits the pre-selection of desired minerals of analyzed composition and treats those as known parameters. The chemical mass balance between a rock sample and its constituent minerals can be expressed as: Yt=2αん 、 ， ，??，•. ? 、、where Yi is the molecular proportion of an element in the rock sample， aj is the molecular proportion of a mineral， Xij is the moles of an element per mole of a mineral formula， m is the number of mineral phases present in the rock sample. If we select the same nurnber of elemental components as mineral phases， Eq. (1) can be expressed by using square matrix (Nishimura & Yanagi 2000; Pruseth 2009) as: XI Xlj X1m i{ al YI Xil X 日 Xim 1  aj Yj (2) Xm1 X 町/ Xmm 1¥a削 Ym Although conventional models have generally adopted oxides as whole rock component， there is no universally accepted method for the adjustment of iron-oxidation ratio as noted above. The present method use elements as whole rock componentかよinsteadof oxides. Equation (2) can also be expressed as: 。1 X1 X1j X1m i {YI 。j XiI x 日 Xim I I Yj (3) am} ¥ Xml Xmj Xmm J ¥Ym 66 Koshi NISHIMURA et al. The inverse matrix in the right-hand side can easily be calculated by using intrinsic functions of spreadsheet software such as MS-Excel. Once the inverse matrix is calculated， the molecular proportions of the al minerals can be determined simultaneously. The estimation error of the mineral proportion depends on the uncertainty in whole-rock and mineral analyses (Eq. 3). If the given whole-rock and mineral compositions are accurate， the estimation of the mineral proportion also becomes accurate. Once the mineral proportions are determined， whole-rock Fe-oxidation ratio can be calculated企omthe mineral compositions. When two or more minerals differ only in oxidation state (e.g.， magnetite (FeO・FeZ03)and hematite (FeZ03) in Fig. la)， we use the mean composition of those minerals. If the relative volume ratio between those minerals is obtained by using optical microscope or electron microprobe， the mean composition can be easily calculated as follows. For example， when the magnetite to hematite volume ratio is estimated as r， the following mass balance equation is formed: α= (rDM裕MwHem)/(DHem MwMag) (4) where αis the molecular ratio ofmagnetite divided by the molecular ratio ofhematite， DMag is the specific gravity of magnetite， DHem is the speci白cgravity of hematite， MwHem is the molecular weight of hematite and MwMag is the molecular weight of magnetite. By using Eq. (4)， the mean formula of magnetite and hematite can be defined asα(FeO・FeZ03)-Fez03・Adoptingthe values of these parameters in Table 1， Eq. (4) can be rewritten as α = 0.68r (5) Molecular ratio ofα(FeO・FeZ03)-Fez03can be calculated by using square matrix in the same way as described above. Weight percent of hematite (or magnetite) is obtained by multiplying the molecular weight of FeZ03 (orα(FeO・FeZ03))by the molecular ratio ofα(FeO・FeZ03)' F eZ03・Table 2 Comparison between normative and modal abundance ofminerals in fiuorite-hematite ore (sample P02) Mineral Hematit巴Fluorite Norm (wt %) Norm* (vol %) Mode (vol %) 64.75:t 1.26 56.65 :t 1.10 55.37:t 1.97 30.01 :t 0.58 43.35 :t 0.84 44.63:t 1.97 * Volume p巴rcentagesare normalized to total 100 % for comparison with modal analysis Norm Calculation Using Square Matrix 67 3. Application to natural rock The effectiveness ofthe present norm calculation is verified by comparison between the normative and modal abundances of natural rocks. Here we show three examples ofthe application ofthe present model to mineralized rocks. The whole-rock chemical compositions were determined by a lithium metaborate/tetraborate fusion ICP-MS package at Activation Laboratory， Ancaster， Ontario in Canada. The mineral formulas were estimated based on EPMA (JEOL JXA・8900R)analyses at the Geological Survey of Japan. The modal tthalyses were conducted following the procedure reported by Neilson and Brockman (1977). Figure 1 Photomicrographs of thin sections of ore samples under reflected (a) and transmited (b-d) light (a) Holocrystaline apatite ore (sample P700D) in the Mushgia Khudag hydrothernal deposits， Mongolia. The magneti旬 ispartly oxidized to hematite. (b) Fluorite-Hematite ore (sampJe P02) related to the activity of the Bushveld granite， South Africa. (c) Holocrystaline texture of the apatite ore， sampJe P700D. (d) Porphyritic apatite ore (sample P698A) in the Mushgia Khudag hydrothernal deposits， Mongolia. The matrix is composed of microscale crystals of quartz， goethite， phlogopite， apatite， monazite and celestine. We first apply the model to the rock composed of only two minerals as the simplest example. Figure 1 b shows the microphotograph of a fluorite-hematite ore (sample P02) formed through hydrothermal activity related to the Bushveld granite， South A企ica.A mass balance relationship between mineral and whole-rock composition (see general form in Eq. (2) can be expressed as: 68 Koshi NISHIMURA etal (~ ~)(す) = (~:) (6) where each element symbol shows molecular proportion of the element. The compositional matrix in the left-hand side is based on the chemical formula listed in Table 1. By substituting whole-rock element compositions and by premultiplying the inverse of compositional matrix， Eq. (6) can be rewritten as: (す)=(Yi)(::;) (7) The results of the norm calculation with the propagation of chemical analysis e町orare shown in Table 2 (The molecular ratios are converted into wt% and vol % using parameters listed in Table 1). The results of the modal analysis by point-counting are also shown for comparison. The point-count data are collected as 30 sets of N = 36 and the associated e町oris estimated in teαns of the sample standard deviation following the scheme of Neilson and Brockman (1977). It should be noted that the chemical analysis error depends on many factors and vary from laboratory to laboratory， whereas there is no interlaboratory difference in the modal analysis using same stochastic model. The calculated volume percent of normative minerals (hematite and fiuorite) agree with the modal analysis data within the margin of the modal analysis eror. A second example is the application of the calculation to a rock containing both magnetite and hematite: a holocrystalline apatite ore (sample P700D) formed through hydrothermal activity related to the intrusion of syenite， Mushgia Khudag， Mongolia (Fig. 1a and c). We used the r value of 1土0.1and took into account its e汀oras well as the chemical analysis e町or.The mass balance relationship between mineral and whole-rock composition can be expressed as: Ap Ca o 4.04 0 6 Mag+Hem :~ I = (8) o 0 6 0 Phl Mg o 0 2 2 Ann K The composition of the phlogopite crystal in Fig. 1c can be described by mixing of two end-members， phlogopite and annite (Table 1). By substituting whole-rock element compositions and by premultiplying the inverse of compositional matrix， Eq. (8) can be rewritten as: Norm Calculation Using Square Matrix 69 Ap ¥ (0.2 0 0 0 ¥(0.637 Mag + Hem I I 0 0.248 0.248 -0.74311 0.210 Phl I I 0 0 0.167 0 1 0.011 (9) Ann ) l 0 0 -0.167 0.5 H 0.007 The results of the calculation are shown in Table 3. All mineral volumes calculated by the present norm calculation agree with those determined by the point-counting method within the analytical errors. The present norm calculation can be applied to the rocks whose constituent minerals are too small to conduct point-counting analysis. Figure ld shows the microphotograph of a porphyritic apatite ore (sample P698A) formed through hydrothermal activity related to the activity of syenite， Mushgia Khudag， Mongolia. The rock is composed of large crystals of fluorapatite， phlogopite and goethite， and microscale crystals of quartz， goethite， phlogopite，自uorapatite，monazite and celestine. The point-counting method is not available for the microscale matrix. The mass balance relationship between mineral and whole-rock composition can be expressed as: o 6 6 000 Qtz Si 。o 6 0 0 0 Goe Fe o 0 6 0 000 Phl Mg 002 2 000 Ann = (10) 000 0 5 0 0 Ap Ca o 0 0 0 3 。Mnz P o 0 0 0 0 0 Cel Sr By substituting whole-rock element compositions and by premultiplying the inverse of compositional ma仕ix，Eq. (10) can be rewritten as: Qtz 。。-3 。。。0.308 Goe 。 -3 。。。0.031 Phl 。。0.167 。。。。0.018 Ann = o 0 -0.167 0.5 。。。 (11) Ap 。。。。0.2 。。0.669 Mnz 。 。。-0.6 。0.402 Cel 。。。。。o 1 0.015 70 Koshi NISHlMURA et al. The results of the calculation are shown in Table 4. The normative volume abundance for each crystal includes the crystal volume in the matrix. The relatively lower volume abundances of the point-counted crystals seem to be caused by the loss of counts in the microscale matrix. Prospects of ore generally contain the micro-scale minerals that are enriched in efficient elements. This norm calculation method would become a powerful tool to estimate proportion of ore minerals whose size is too small for point-counting analysis. Table 3 Comparison between normative and modal abundance of minerals in bolocrystaline apatite ore (sample P700D) Mineral Norm(wt %) Norm* (vol %) Mode (vol %) Fluorapatite 64.24 :t 0.50 82.88 :t 0.65 80.74土1.76Magnetite + Hematite 15.65 :t 0.27 12.48 :t0.22 13.89土1.49(Magnetite) 7.77 :t 0.26 6.23土0.21 7.04土0.80(Hematite) 7.88:t 0.27 6.26:t 0.22 6.85:t 0.69 Phlogopite + Annite 3.34:t 0.33 4.63:t 0.46 5.37:t 1.06 (Phlogopite) 1.59:t 0.05 2.20:t 0.07 (Annite) 1.75:t 0.28 2.43:t 0.39 * Volume percentages are normalized to total 100 % for comparison with modal analysis Table 4 Comparison between normative and modal abundance of minerals in porphyritic apatite ore (sample P698A) お1ineralQuartz Goethite Phlogopite + Annite (Phlogopite) (Annit怠)Fluorapatite Monazite劫(Ce)Norm (wt %) 17.1O:t 0.30 2.30:t 0.24 3.34:t 0.54 2.48:t 0.13 0.86:t 0.41 67.48 :t0.12 0.23:t 0.33 Norm* (vol %) Mode (vol %) 21.71 :t0.39 matnx 1.80:t 0.19 1.67 :t 0.45 3.76:t 0.61 3.33土0.452.79:t 0.15 0.97:t 0.46 70.27 :t 0.13 66.02 :t 2.26 0.15:t 0.22 matnx celestine 2.73 :t0.00 2.32:t 0.00 matrix * Volume percentages a陀 normalizedto total 100 % for ∞mparison with modal analysis Acknowledgements K.N.， Y.w. and T.T. thank to E. L. Tongu， S.Foya， N. S. Mayekiso， R. Yoshikawa and S. Kodama for the collaborative sampling of the fluorite-hematite ore in South Africa. Discussions with T. Ikeda contributed substantially to the content of this paper. This work was supported by a Grant-in-Aid for Young Scientists (B) (21740377)企omthe Japanese Ministry of Education， Culture， Sports， Science and Technology (MEXT KAKENHI). Norm Calculation Using Square Matrix 71 References Cox， K. G.， Bell， J.D. and Pankhurst， R. J. (1979) The Interpretation of Igneous Rocks. Allan & Unwin， London， 450p. Cross， C. W.， Iddings， J.P.， Pirsson， L.丸T.and Washington， H. S.(1902) A quantitative chemico・mineralogicalclassification and nomenclature of igneous rocks. J. Geol.， 10，555-690. Deer， w.A.， Howie， R.A. and Zussman， J.(1992) An Introduction to the Rock Forming Minerals (2nd edよLongmanScientific and Technical Publications， London， 696p. Johannsen， A. (1931) A Descriptive Pe仕ographyofthe Igneous Rocks. vol. 1， Chicago University Press， 267p. Kelsey， C. H. (1965) Calculation ofthe C. 1. P. W. norm. Mineral. Mag.， 34， 276-282. Le Maitre， R. W. (1976) Some problems of the pr句ectionof chemical data into mineralogical classifications. Contrib. Mineral. Petrol.， 56， 181-189. Middlemost， E. A. K. (1989) Iron oxidation ratios， norms and the classification of volcanic rocks. Chem. Geol.， 77， 19-26. Neilson， M. J.and Brockman， G. F.(1977) The error associated with point-counting. Am. Mineral.， 62， 1238-1244. Nishimura， K. and Yanagi， T.(2000) In situ crystallization observed in the Osumi granodiorite batholith， Earth Planet. Sci. Lett.， 180， 185-199. Pruseth， K. L. (2009) MATNORM: Calculating NORM using composition matrices. Comput. Geosci.， 35， 1785-1788. Ragland， P.C. (1989) Basic Analytical Petrology. Oxford University Press， Oxford， 369p. Verma， S.P.， Torres-Alvarado， 1.S. and Sotelo・Rodriguez，Z. T.(2002) SINCLAS: standard igneous norm and volcanic rock classification system. Comput. Geosci.， 28，711-715. Washington， H. S. (1917) Chemical analyses of igneous rocks. US Geological Survey Professional Paper， 99. 要旨正方行列を用いた鉱物ノルム計算法:微小鉱石鉱物の鉱量評価にむけて西村光史，渡辺寧，実松健造，星野美保子，高木哲一正方行列を用いて岩石試料の化学組成(全岩化学組成)から構成鉱物の体積比を見積も72 Koshi NISHIl¥征JRAetal. る手法(ノルム計算法)を開発した鉱物の種類と化学組成は既知のパラメータとして扱う.鉱物の体積比は全岩のFeO/FeZ03比が分かっていない場合でも計算可能である.本モデルをいくつかの完品質の岩石に適用し，鉱物の体積比を計算したところ，ポイントカウントデータと極めてよく一致することが確認された本モデルは原理的にすべての岩石種に適用することが可能である.鉱床中の鉱石鉱物には， しばしばポイントカウント分析ができないほど微小で、少量のものが存在する.そのような場合，本研究のノルム計算法は鉱物の体積比を見積もる有力なツールとなりうる.

A Norm Calculation Method Using Square Matrix: Implications for the Total Volume Estimation of Microscale Ore Minerals

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A Norm Calculation Method Using Square Matrix: Implications for the Total Volume Estimation of Microscale Ore Minerals

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