Numerical modeling of crystal shapes in thin sections: Estimation of crystal habit and true size

AmericanMineralogist,Volume79, pages 113-119,1994 Numerical modeling of crystal shapesin thin sections:Estimation of crystal habit and true size Mrcn...
Author: Charles Kelly
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AmericanMineralogist,Volume79, pages 113-119,1994

Numerical modeling of crystal shapesin thin sections:Estimation of crystal habit and true size Mrcnanr, D. HrccrNs Sciencesde la Terre, Universit6 du Qu6beci Chicoutimi, Chicoutimi, Qu6bec G7H 2BI, Canada

Ansrru.cr Although the size and shapeof crystalsin thin sectionshave been measuredin a number of studies, it has not been possible so far to calculate from these data the true, threedimensional shapeand size of the crystals.A numerical model, basedon orthogonal solids, has been developed to attack this problem. This model shows that crystal habit can be calculated from width to length ratio distributions for most crystals in massive rocks and for all crystals in laminated or lineated rocks. Variations in the habit of minerals can reveal aspectsof the physicochemicalconditions of crystallization. The same model has also been used to develop the equations necessaryto transform two-dimensional crystal size distributions into true crystal size distributions. Corrections for the cut effect and the intersection probability effect both require a knowledge of the crystal habit. INrnonucrroN Although most quantitative studies of igneous rocks concentrate on whole rock and mineral chemistry, recently there has beena growinginterestin quantification of the texturesof igneousrocks.Most work hasbeenconflned to the study of crystal size distributions (seereview by Cashman, 1990),with a little work on crystal habit and orientation (Higgins, l99l) and none on the spatial arrangementof crystals.All these studies have been limited by the lack of knowledgeof stereologicaleffects,measurementsare generallymade in two dimensionson thin sections,but crystals,and textures,are threedimensional. An exact solution to this problem is only possiblefor sphericalobjects,and empirical methods must be used for other shapes.Earlier stereologicalstudies were confined to spheresand equidimensionalobjects(seereviews in Cashmanand Marsh, 1988; Cashman, 1990; Royet, l99l), but the numerical model presentedhere can be applied to any orthogonal solid. The particular stereological problems addressedhere are the determination of crystal habit and the extraction of three-dimensional crystalsizedistributionsfrom two-dimensionalmeasurementsof crystalsin thin seclions. Crystal habit The study of crystal habit has a long history. In 1669 NicolausSteno(1968),wellknownforhislawconcerning the constancyofinterfacial angles,proposed that the external form of a crystal dependson the growth rates of the different faces.Since that time the factors controlling growth rate anisotropy, suchas temperatureand chemical potential gradient, have becomebetter known (seerecent reviews in Sunagawa,1987a).These studiesimply that observations of the actual habits of crystals can reveal 0003-004x/94 /0r024r l 3$02.00

something about their chemical and physical envrronment of formation. Many studies of the habits of silicate minerals have been concerned with the forms developed during rapid cooling, such as those seen in some experimental products and volcanic rocks (seefor exampleLofgen, 1980). Relatively little work has been done on the habits of the generally larger crystals in plutonic rocks or the more euhedral forms in some volcanic rocks (seethe review in Sunagawa,1987b).Crystal habit is easily measuredif a rock can be disaggregated,but that is not usually practicable. Generally, only two-dimensional slices through a rock are available,in the form ofthin sections. If a sufficient number of crystals are observed in thin section,then those with specialorientations, for example, those intersected parallel to their crystallographic axes, can be distinguishedand their dimensionsusedto establish the habit and true sizeof the crystals.However, this technique has several limitations: crystals with suitable orientationsmust be available,and they must be known to be representativeofthe whole population. A different approach is presentedin this paper. It will be shown that statisticalanalysisof crystalshapesand sizesin thin sections can be used to establishthe habits of most crystals and their true sizes. Crystal size distributions Recently, there has been a resurgenceof interest in crystal size distributions in igneous rocks (seereview by Cashman,1990).Suchstudiescanprovideimportantinformation on the kinetics of crystallization, such as nucleation and growth rates. In this way they can complement information from chemical and isotopic studies on theenvironmentofcrystallizationofthecomponentminerals in a rock.

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HIGGINS: NUMERICAL MODELING OF CRYSTAL SHAPES 1: 1 : 1

1:5:10

1:1:10

1 : 1 0 :01

Massivematerials

1i2i2t'i";;r"l;t 1:r:17t112 -Irr-v--

r:2:10 r:10:r0 1:1:10 15:10

for differentaspect Fig.2. Distributionofintersectionshapes ratiosof orthogonalsolids.

and Z coordinatesbetween - I and * l, until one point fell inside a sphereof radius I centeredon the origin. The point was projected from the origin onto the surface of the sphere,and the resulting X, Y, and Z coordinateswere Fig. l. Intersection shapes of I 00 randomlyorientedandpo- usedas the direction cosinesofthe equationofthe plane. sitionedplaneswith four orthogonalsolidsof differentaspect The addition of a randomly distributed number between ratios (shortest:intermediate:longest).In eachcasethe L pa- I and * I determinedthe position of the plane. Such (1 rameterof the solid or l0) is approximately equalto thespacplanes produceduntil one cut the solid, when the were ing betweenthe figures. number of sides and the area, length, and width of the intersection were calculated. Typical intersection shapes For want of suitable stereological studies, it has fre- are shownin Figure I for four shapesoforthogonal solid. quently been assumedthat the modal crystal length ob- Trials for 50000 planes were conducted for S : l, for I served in thin section approachesthe true length of the and L : 1,2,5, and 10,and for unconstrained and concrystal(e.9.,Cashmanand Marsh, 1988).Cashman(1990) strainedorientationsofthe planes.The program is writsuggestedthat the crystal width may be a better measure ten in Pascal,and a copy is availablefrom the author. of crystal size. Others have suggestedthat crystal area The intersectionof a plane with an orthogonal solid may be a better measurethan length. Clearly it is imporcan producethree-,four-, five-, and six-sidedfigures(Fig. tant to be able to relate all thesemeasurementsto a single 2). The most common figurehas four sides,and the least crystal-sizedistribution, where size is the true length of common six. Three-sidedfiguresare generallysmaller,in the crystals.Other parameters,suchas areaand volume, contrast to five- and six-sided figures, which are among can be calculated from the crystal habit and true length. the largest.The proportions of different shapesare most strongly dictated by the length of the longest dimension NuNrsnrcA,r, MoDELTNG (L) of the aspectratio. Isotropic materials Distributions of intersection lengthsare complex, with A numerical model has been constructed to simulate two or three peaks(Fig. 3A): the mode (highestpeak) is the variations in apparenttwo-dimensionalgrain shape at a lengthequal to the intermediatedimension,I, of the and size in isotropic materials for various different grain crystal, and there are subsidiarypeaksat L and Vtt + P. shapesof uniform size.This model is a developmentof Distributions of intersection widths are slightly simpler that describedby Naslund et al. (1986). In an isotropic than thoseof lengths(Fig. 3B): the mode is at S, and the material any plane intersects crystals with every orien- peak is strongly asymmetric, with few nanower intersectation. From the frame of referenceof the crystals, the tions and a long tail to wider intersections.There is a intersection can be considered to be that of randomly small subsidiarypeak at I. Distributions of intersection orientedand positionedplanes.This is the frame of ref- areas show two to four peaks (Fig. 3C). All habits show erenceusedin the model. The simplified crystalused in a narrow peak close to zero. This is not an artifact but these calculations has the form of an orthogonal solid, reflectsa real concentrationofintersectionsofsmall area, with dimensionsS : shortestdimension,I : intermedi- at the corners of the crystal. The most important peak is ate dimension,and L : longestdimension. at S.I, with subsidiarypeaksat S.L and, for high values A randomly oriented plane was produced as follows: of I, at I/2. Width to length ratio distributions have one randomly distributed points were generated with X, Y, or two peaks (Fig. 3D). The main peak for prisms (in

ll5

HIGGINS: NUMERICAL MODELING OF CRYSTAL SHAPES

IE]IGTH

WIDTH

WIIIIH / LEIIGTH

AREA

Fig. 3. Frequencyvs. (A) intersectionlengths,(B) intersectionwidths, (C) intersectionareas,and (D) intersectionwidth to length ratios for orthogonal solids with different aspectratios. The distribution ofthe intersection planesis isotroptc.

which I : L) is usually broad and in the region 0.5-1.0. For other aspect ratios the main peak is at S/I and is sharp. There is a subsidiary peak at S/L, but it is generally small. The results of this model can also be revealed in a qualitative way by an examination of the intersection shapes(Fig. l). The cube (l:1:l) has few squareintersections, but most intersectionsare equidimensional.However, most of the intersectionswith the prisms (1:l:10) are almost square,and few are elongated.Most intersections with the tablets(l:5:10 and l:10:10)are elongated, and few show the face of the tablet. Therefore, almost squareoutlinesin a thin sectionindicatesthe presenceof prismatic crystals, whereas elongated outlines indicate tabular crystals.

straints on the orientations of the intersection planes in sectionsnormal to the fabric are different from those in sectionsparallel to the fabric. It is assumedthat all crystals are perfectly aligned, and hence the fabric is developed to the maximum extent. All intersections have four sides. Crystals in rocks with a laminar fabric are aligned with their short axespointing in the samedirection. Sections normal to the lamination have modal lengths,areas,width to length ratios, and overall data shapessimilar to those of isotropic materials with crystalsof the samehabit (Table l). In contrast,sectionsparallelto the lamination are values TABLE 1. Equations relatingmodalvaluesandinvariant to the crystaldimensions of differentparameters

Linear and larninar fabrics The alignment of nonequant crystals gives linear or laminar fabrics or both. The numerical model for these materialsis similar to that for the isotropic materialsdescribed above, except that variation in the orientation of the planes of intersection is constrained. In contrast to isotropic materials, the orientation of the section with respect to the fabric is important. Two orientations are consideredhere:paralleland normal to the lamination or the lineation. For both linear and planar fabrics, the con-

Length Width lsotrooic Laminar(normal) Laminar(parallel) Linear(normal) Linear(parallel) Laminatedand linear (normal) Laminatedand linear (parallel)

I

;. I' L t. L-

S S t. SI S' t'

Note: S : shortest, | : intermediate,L : longest ' Denotesinvariantvalues.

Area SI SI tL' SI' SL SI' tL-

width/ length

YS

r/s L/rr/s. US r/s. ul.

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HIGGINS: NUMERICAL MODELING OF CRYSTAL SHAPES

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