Tectonophysics, 40 (1977) 257-285 0 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

257

THREE-DIMENSIONAL TEXTURE ANALYSIS OF THREE QUARTZITES (TRIGONAL CRYSTAL AND TRICLINIC SPECIMEN SYMMETRY)

H.J. BUNGE and H.R. WENK

Institut fiir Metallkunde und Metallphysik, Technische Universitb’t Clausthal, Clausthal-Zellerfeld (Germany) Department of Geology and Geophysics, University of California, Berkeley, Calif. (U.S.A.)

(Submitted March 5, 1976, revised version accepted October 12, 1976)

ABSTRACT

Bunge, H.J. and Wenk, H.R., 1977. Th ree-dimensional texture analysis of three quartzites (trigonal crystal and triclinic speciment symmetry). Tectonophysics, 40: 257-285.

The three-dimensional orientation distribution (ODF), a complete representation of texture (preferred orientation), is derived for three quartzite specimens. The first part describes the spherical harmonic analysis to calculate the ODF for 2 2/m crystal symmetry and 1 specimen symmetry from pole-figures measured with an X-ray pole-figure gonio- meter. The second part of the paper presents the results of the analysis for three quartz- aggregates deformed on thrust planes of Alpine nappes at different metamorphic grades. The three samples display moderate to strong preferred orientation with [ 00011 in the plane of foliation (normal to the lineation) at high temperature and [OOOl] normal to the foliation at low grade. In addition, X-axes [lOiO] are aligned and positive and negative rhombohedra have a different orientation, probably as a result of Dauphin6 twinning. Emphasis is on explaining how to read and interpret the three-dimensional ODF directly without having to rely on the more familiar pole-figure (fabric diagram). In the last sec- tion we discuss errors and reliability of the data. One source of error is the incompatibil- ity of pole-figures due to experimental difficulties, which is minimized by using a least- squares fit, a second source is the mathematical representation with spherical harmonics of limited order. In practice the order of harmonics to be used has to be decided in each individual case. Error analysis shows that all these textures show a significant deviation from hexagonal symmetry.

INTRODUCTION

Preferred orientation of crystals in rocks contains important information on the deformation history. On the one hand, it permits us to derive defor- mation mechanisms which were active in the crystal and thus on the tem- perature-pressurestrain rate conditions which governed during the defor- mation process. On the other hand, preferred orientation can be used to

obtain data on the stress-strain field in which a crystal aggregate was deformed. Therefore, an accurate and complete description of the relation between crystal and specimen orientation is the essential basis of p&rof&ri(: analysis. In the geological literature of fifty years, preferred orientation has usually been described in terms of pole-figures (fabric diagrams), i.e. repre- senting one crystallo~a~hjc direction relative to specimen coordinates. A more complete *and straightforward way to specify the relative orientation of two coordinate systems is by means of Eulerian angles. In metallurgy this method of representation, termed “orientation distribution function (ODF)“. im become a standard procedure (see Bunge, 1969, for further references).

Orientation distribution diagrams can either be constructed directly from orientation data of individual crystals measured on the universal stage (such an analysis has been clone for calcite marbles by Wenk and Wilde, 1972) or they can be calculated from X-ray pole-figure data using spherical harmonic analysis (Bunge, 1969). Geological applications of the latter technique have been scarce and this is mainly due to mathematical complications arising from low crystal and specimen symmetries, compared to metal systems. Baker and Wenk (1972) published an ODF of a quartz mylonite for 3 Z/m crystal and 2/m specimen symmetry. In this paper we present ODF’s for three quartzite specimens with i specimen symmetry, which appears to be most common in natural fabrics. Emphasis is on the presentation of the mathematical fundamentals used to prepare a computer program, a discus- sion of resolution and errors and an interpretation of the results. With this presentation of ODF data of quartzite specimens deformed on thrust planes of Alpine nappes under different conditions of metamorphic grade, we hope to make earth scientists more familiar with this method of representing preferred orientation. The computer programs are at present being rewritten into a standard format so that they can be used by every structural geologist-

MATHEMATICAL FUNDAMENTALS (for further details see Bunge, 1969)

Description of the crystal orientation

In order to describe the orientation of a crystallite within a polycrystalline specimen one may, for example, specify the orientation of a crystal coordi- nate system XYZ with respect to a specimen coordinate system ABC. Both coordinate systems are assumed to be Cartesian systems. Usually XYZ are chosen to be some distin~ished crystal axes or low index crystal directions and ABC are specimen directions such as rolling RD, transverse TD, and normal direction ND in metal sheets or mesoscopic fabric coordinates such as lineation b, normal to the lineation in the foliation a, normal to the folia- tion c, in naturally deformed rocks. In order to describe the orientation of the crystal coordinate system XYZ with respect to the specimen system ABC, we start with a position in which XYZ are parallel to ABC, respec- tively. Then the crystal coordinate system XYZ is rotated successively about

259

A Fig. 1. Definition of the Eulerian angles \P~#c&. XYZ are crystal coordinates, ABC speci- men coordinates.

the Z-axis through the angle (pl, about the X-axis through 4, and once more about the Z-axis in its new position through cpz. The resulting position of the crystal coordinate system XYZ is then uniquely described by the three Eulerian angles (pi&a (Fig. 1).

It should be mentioned here that there is another possible definition of Eulerian angles $@# where the second rotation 0 is taken about the Y-axis instead of the X-axis. The two sets of angles grtiz and $S$ are closely related to one another (cf. Baker, 19’71). It is:

O=g, (11

Qt = (P2+7rP

This definition of Eulerian angles has been used for example by Baker and Wenk (1972) in order to describe crystallite orientations in polycrystalline quartzite specimens.

There are many other possibilities to describe crystal orientations and some of them have already been applied to preferred o~en~tion ~texture) studies. For example, one can specify at first the orientation of a specific crystal direction by two angular parameters a/3 and then a rotation about this direction by a third parameter y. Another possibility is to give the direction cosines of the crystal XYZ-axes with respect to the specimen axes ABC in the form of an orientation matrix. Because of the mathematical treatment of the orientation distribution functions in terms of generalized spherical harmonics it is most convenient, however, to use the Eulerian angles as orientation coordinates.

“60

7’hc three-dimensional orientation distribution function

An orientation distribution function f‘(ipr&Q is defined by the volume part dV(p,@q,) of the specimen having an orientation between (p1@q2 and

PI + dp,, 0 -t dQ, PF~ + dp,:

The weighting function sin o/8rr2 has been chosen in order for the function f(pl$qz) to be expressed in multiples of the random density. This function can be developed into a series of generalized spherical harmonics:

f(q19pc2) = C C C C;““P;““($)exp(imq2)exp(inp1) (3) 1=0 m= 1 ti= I

An alternative representation is obtained if one terms by angular functions and combines terms and negative values of m and n. Thus one obtains:

f(q10q2) = 5 k i: N(m)N(n) X 1-o m=O n=O

X{cosmq, cosnp, [~;““(Q)A;“” - yj”“(~)R;‘“]

+ sinmq2sinnqI [p;n”(~)B;“‘L - y;1”(@)A~“]

+ i~osm~,sinn~~Ip;R”(~)D;l” --q;‘“(fp)E;““]

+ isinmys ~osn~~]~;11’~(o)E;1” -- q~“(o)D;““]j

where:

I = 1 for m # 0 and N(m) = f for m = 0

is a normalization factor:

p;““(S) =$[Pr”“(qq + P;~‘““(@)l

q;1”(9) = +[PYW -p; ‘““(@)I

and:

&n” = +ql” + ,,-n-11 + c; r?I” + ,r,-,,

B1 mn=__ - + +

Dl mn=+ - +

E, mn = + - - +

expresses the exponential of corresponding positive

(4)

(5)

(6)

(7)

(8)

261

The two representations eqs. 3 and 4 are equivalent and they both are gen- eral, that is to say they are apt to represent any distribution function by an approp~ate choice of coefficients A to E. The functions pi”“(@) and hence also py”(@) and qTn($) are real or imaginary according to whether m + n is even or odd. Hence for the function f(pi&.J to be real it is necessary that

Al mn, Bf”” be real and Or”, ET” imaginary or vice versa according to whether m + rz is even or odd. According to eq. 3 this requires:

(c;m-n)* = (-,),,+“cy””

where * means the conjugate complex quantity.

(9)

The axis distribution function

If we choose a certain direction By in the crystal coordinate system (Fig. 9) and another direction BP in the specimen coordinate system (Fig. Z), then the general axis distributions function is defined by the integral over the three- dimensional distribution function along a path in the orientation space, that is, over all those orientations, for which the crystal direction Oy is parallel to the specimen direction ~$3:

hII@ (10) Thus we define:

A(@?, ~0) = ~~f~~~~~~)d~ (111

where ti is the angle of rotation about the common direction By, c$. If in eq. 11 the distribution function f is expressed by the series eq. 3 then we obtain (c.f. Bunge, 1969, p. 27):

where~(~) are the normalized associated Legendre functions. Equation 12 may be transformed in a similar way as that in which eq. 3 has been trans- formed into eq. 4. We thus obtain the general axis distribution function in terms of angular functions:

X {Gyncosnzy sin@ + Gimnsinmy sin no +

ST” eosmy cos 92@ + Simn sin my co.5 no>

where Gr”, G;““, Sf”” and Simn are defined according to eq. (21). (13)

262

Pole-figures

If we assume the crystal direction By to be constant and the specimen direction c$I to be variable, then the axis distribution function describes the distribution of a specific crystal direction of all the crystallites with respect to the different specimen directions c$. This is the poie-figure corresponding to the crysal direction 0,~ which is usually (but not necessarily) assumed to be a low index direction 1 hhll OK [hkil 1 *. Hence the (hhl)-pole-figure is defined by

PhJ</(CYii:) = A(0r, @ii J (14)

where 8 and y are the pole distance and azimuth describing the orientation of the crystal direction [hhl] with respect to the reference axes XYZ of the crystal coordinate system. We introduce the normalized surface harmonics by:

with :

1 for n = 0

en Tz J2 for n f 0

Then we obtain from eq. 13 for the pole-figure:

Ph,zl(@p) = 6 iJ [Fyh; (e/J) + F;“h;“(cYp)] I=0 n=O

with the coefficients:

47r & !Qv@) [syk?r”(ey) + $““fp(&y)] F; =21+1,=o E,E,

where :

(15)

(16)

(17)

(18)

(19)

(20)

* The index I in this case must not be mixed up with the order 1 of the spherical har-

monics.

263

Sl ??I”= Sl ‘mn = G;““= G ‘rnrl = form n

+ AT”

-DT””

-i Ey”

-iB;“”

+ iD;t”

- iA;“”

--By*

--ET”

even

even

even

odd

-E;“” + iAT” + iBy” -DT”” odd even

--BY” -iDy”” + iEf”” +Ar”” odd odd (21)

Inverse pole-figures

If, on the other hand, the specimen direction C$ is assumed to be constant and the crystal direction Oy is variable then the axis distribution function represents the inverse pole-figure corresponding to the specimen direction (rp:

K&W = A(ey, 06) (22)

The inverse pole-figure may be expressed according to eq. 13 with the defini- tions eq. 15 to 17 in the form:

L 1

KY&Y) = c c [Hf” hr”(~r) + fG” vY~~)l (23) I=0 m=O

with the coefficients:

-!% 6 N(m)N(g [si”“/$(@) + Gr”“h;“(@)] V=21+1.=o E,E, (24)

(25)

Inverse pole-figures generally are calculated for the three main specimen directions (rolling RD, tranverse TD, and normal direction ND in sheets; lineation b, normal to the lineation in the foliation a, normal to the foliation c in naturally deformed rocks, for example). According to eqs. 15 and 16 for these directions the functions kr and hi” take on values given in Table I (see p. 267). Because of the systematic vanishing of some of these values the inverse pole-figures of the three main directions do not contain the full infor- mation about the texture. Generally they exhibit a higher symmetry than the texture as a whole. This additional symmetrification does not occur in the inverse pole-figure for an arbitrary specimen direction. But generally inverse pole-figures of other directions than the three main axes are not used.

264

Representation of the generalized spherical harmonics

If the functions ppn($) and q;ln(@) eqs. 6 and 7 are to be expressed by a Fourier development, then a distinction between m + n even or odd is neces- sary (c.f. Bunge, 1969, p. 223):

m + n even m + n odd

41n”($) = I6 aimns cow@; = lb a;m”Ssinsf$ (27) s= l(2) s= l(2)

The coefficients aimns in turn may be expressed by a lower number of coef- ficients Q;” by the relations:

a;mnO = Q;“” Qf”

for m + n even a;“‘ns = 2Qy”Q;fi

aimns = 2 iQf”” Q;” for m + n odd

The coefficients Q;” have been tabulated (Bunge, 1974).

(28)

Symmetry considerations

The representation of the three distribution functions, the threedimen- sional one, the pole-figures and the inverse pole-figures by sine- and cosine- terms leads to much more complicated expressions than the representation by the exponential function. There are two reasons why we chose the trigo- nometric representation. Firstly, in ALGOL programming language complex calculus is not directly available and one has to decompose the exponential functions. Secondly, the real representation of the spherical harmonics eqs. 15 and 16 expresses more directly the symmetry in as far as the k? exhibit the mirror symmetry and the hi” corresponds to the antimirror part of the dis- tribution function. Hence in this representation it is easy to consider explic- itly the symmetry elements of the crystal symmetry (inverse pole-figures) as well as those of the specimen symmetry (normal pole-figures).

We consider now the special case of the trigonal crystal symmetry 3 2/m having a threefold axis in the direction 0 = 0 and a mirror plane passing through this direction and the direction y = 0 (Fig. 11) as well as a center of symmetry. According to the first symmetry element the “selection rule”:

m = 3m’ (29)

applies. Ml coefficients with m-values which do not fulfill this rule must

265

vanish. Because of the mirror plane, the anti-mirror part of the inverse pole- figure eq. 23, that is the coefficients N;“, must vanish. Hence also the eoef-

ficients Si”” and G’,,’ must vanish (eq. 25):

H;” = ,;mn zz G;““’ = 0 (30)

and eqs. 19 and 20 are reduced to:

(31)

(32)

and the inverse pole-figure eq. 23 is expressed by the equation:

where the Hy are given in eq. 24. From eqs. 30 and 21 it follows:

mn - E, -B;““=O for m even

(34) Al -Q

mn - mn = 0 for m odd

According to eq. 8 this corresponds to the relation:

c;- mn = (-1)” Cy”” (35)

The orientation distribution function eq. 4 is reduced to:

f(~~@#s)= 6 k 6 acts) X r=o. m=O n=O

m even m odd

X {cosmpzcosnrpl [+pjnn(rb)Af”“] [--41n”(@J) VT

+ sinmqzsinncpl [-qjn”($)ATn] [+dYrPfm”l

+ icosmqasinnrpi [+~~~(~)~~~I [~q~~t~)~~nl

+ isinm~,cosn~pl [+@“(f$)Dy”]} [+p;““(@) JV-7 (36)

where the terms in parentheses are to be replaced by the ones in the second column for m odd. In the trigonal system the lattice planes (hlza’l) and (hhil), using Bravais index- ing, are not symmetrically equivalent although they have the same lattice spacing and are therefore overlapped in an X-ray powder pattern. Pole-

266

figures of (hhil) and (/Ail) cannot be measured separately. With the pole figure goniometer we record a superposition of both of them:

P(aB) = ahkilPhkiL(aP 1 + a/zh,/ Ykhil(QP) (37

where chhil is a known weighting factor which is proportional to the square of the corresponding structure factor. If the composite pole-figure eq. 37 is developed into a series according to eq. 18 then the coefficients F;’ and F;” corresponding to those of eqs. 31 and 32 are given by:

(39)

where h;“(hhil) means h;l(eh,zilrhkil ). There are some special pole-figures which are superposed for example those for (h h 2h I), (hlziO), or (0001). For these pole-figures eqs. 31 and 32 are to be used. Equations 38 and 39 are systems of linear equations for the unknowns S;l” and G?” with known coef- ficients corresponding to eqs. 31 and 32 for the non-superposed pole-figures. Both these systems can be combined to one system from which the unknowns S;l” and Gy” are to be determined. When the quantities S?” and Gf”” are known, it is possible to calculate the pole-figures (hkil) and (hhil) separately according to eqs. 18, 31 and 32.

The (OOOl)-pole-figures and the C-axis inverse pole-figure

In the section ‘the axis distribution function’ the orientation distribution function was integrated over all those orientations for which a certain crystal direction falls into a certain specimen direction. The result may be either the pole-figure of that specific crystal direction or the inverse pole-figure of the chosen specimen direction. We did, however, not specify the path in the pi&&-space along which the integration over d$ had to be carried out. This path depends on O-&3 in a rather complicated way. There are however two special choices of 0rolp for which the relationship becomes very simple.

The crystal direction Z = [OOOl] corresponds to 0 = 0 (y is not defined in this case). The distribution of [ OOOl] -axes over the specimen directions (~0 is the (OOOl)-pole-figure P c0001~(ar/3). If one recollects the definition of the Eulerian angles given in the first section then it is obvious that the position of the crystal Z-axis [OOOl] with respect to the specimen axes is described by 4 and cpr and the relationship of these angles with the pole-figure angles (Y and 0 in this pole-figure is given by:

a=Q

P = Pi-n/2 (40)

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268

Hence from eqs. 11 and 14 it follows immediately:

That means that the (0001)~pole-figure is the qZ-projection of the t,hrer- dimensional orient,ation distribution function.

Similarly the specimen C-axis has the coordinates a = 0 (/3 not defined). The inverse pole-figure of this direction describes the distribution of C-axes over the crystal directions Or- From the definition of the Eulerian angles it follows in the same way:

0 z (,J>

7 = -‘rr.J + 7i/2 (42)

where liy are the coordinates in the C-inverse pole-figure. From eqs. 11 and

22 it follows:

The inverse pole-figure of the specimen C-direction is the q,-projection of the three-dimensional distribution function.

MATERIAL AND DATA COLLlSCTIOK

The analysis described above was applied to three quartzite samples

deformed on thrust-planes of nappes during the Alpine orogeny. Oriented specimens have been collected in the Bergell Alps (East Central Alps) in a

region characterized by strong metamorphic gradients (Wenk, 1974; Wenk et

al., 1974), thus permitting us to evaluate the influence of metamorphic grade

(particularly temperature) on preferred orientation. The same specimens for which we are presenting texture data in this paper have been investigated with transmission electron microscopy techniques and the dislocation micro-

structure provides further information on deformation mechanisms. The quartzites (for localities see Table II) were deposited as sedimentary

beds in Triassic times (Sci. 638, 690) or formed as veins during metamorphic

recrystallization (Sci. 293). They were then deformed on thrust-planes of the Pennine nappes in early Tertiary during folding of the Alpine belt. Meta- morphic facies varied between high amphibolite (Sci. 293) and low green- schist (Sci. 690). Deformation and subsequent annealing in a series of rocks with a metamorphic gradient produced a variation in microstructure (Liddell et al., 1976) and texture.

Sci. 293 is a very fine-grained quartzite mylonite with flattened grains,

ribbon texture and very high preferred orientation (25-30 mrd.) and ]OOOl] in the plane of schistosity and normal to the lineation (c.f. Fig. 8) from the high-metamorphic Gruf complex. The microstructure with smooth

269

grain-boundaries, often meeting in 120” triple junctions and a low disloca- tion density indicate that these very highly deformed rocks have recrystal- lized, Well-developed hexagonal networks of dislocations in the recrystal- lized grains show that deformation and recovery continued during grain growth. Sci. 293 can be described as hot-worked.

Sci. 638 is a Triassic muscovite quartzite with equidimensional grains and polygonal grain boundaries in thin sections. It is associated with chloritoid schists and thus of medium to high greenschist metamorphic facies. Preferred orientation is much lower than in Sci. 293 although the rocks have been strongly deformed on a thrust-plane between nappes. [OOOl] is oriented normal to the foliation as is typical for greenschist quartzites (e.g. Sander, 1950). The microstructure of this, on microscopic evidence, moderately deformed specimen shows high dislocation densities (>101’ cme2) and is heterogeneous as is typical of cold-worked material. Slight recovery is indi- cated by dislocation interaction, loops and networks. In addition in heavily deformed regions there are signs of recrystallization with small crystallites (O-25-2 pm) which have [OOOl] preferentially oriented in the plane of foliation.

Sci. 690 is from the same geological sequence as Sci. 638 but further away from the region of high-grade metamorphism. Metamorphic grade of this rock never exceeded low-greenschist facies. Grains are flattened and grain boundaries are interfingering. Occasionally deformation lamellae are visible. Preferred orientation is similar as in Sci. 638 but weaker. Electron micro- graphs exhibit dense tangles of dislocations forming tightly packed cell walls. Their distribution is heterogeneous and the density is high (>lOg cmY2). First signs of recovery are present but the sample has the general features typical of cold-worked material. No recrystallization has been observed in Sci. 690.

All three specimens show well-developed mesoscopic fabric coordinates which are clearly displayed in hand-specimens mainly by the orientation and arrangement of mica. Lineation 1 and foliation = schistosity s are constant over large areas in the field and of Alpine age (Wenk, 1974). It was first assumed that they would represent symmetry axes for the pattern of pre- ferred orientation and slabs on which the pole-figures were measured have been cut parallel to these mesoscopic axes. Later it was found that the over- all symmetry of the pole-figure did not match with these axes and a new coordinate system was introduced for the ODF representation.

X-ray pole-figure data were collected on a commercially available Philips pole-figure goniometer with modified motor drives and collimator system in a continuous spiral scan in reflexion and transmission mode. Ni-filtered Cu-K, radiation was used. Advance speeds on the spiral scan were 18” per minute for the azimuth and p per minute for the pole-distance; counts were integrated over 8 sec. During the spiral motion the specimen is rotated in its own plane and the normal to the specimen plane is tilted away from the plane of the incident and diffracted beam. This tilt causes an intensity decrease due to defocusing of the beam. The full pole-figure in Sci. 293 was

210

constructed from a reflection (o-70”) and a transmission (60-90’) scan on the same thin section. In Sci. 638 and 690 reflection scans (out to 60-70” ) taken on three perpendicular cut slabs were combined. In order to obtain better grain statistics, particularly in the medium to coarse-grained samples Sci. 638 and 690, slabs were rapidly translated, thereby achieving an average over an area of about 2 cm X 2 cm.

Following the procedure of Baker et al. (1969) and using the FORTRAN computer program 0VLAP72 to process raw data, measured counts were first corrected for the background and specimen tilt. The correction factor for the latter was obtained from scans with undeformed flint which has random preferred orientation as an external standard. The pole-figure is represented numerically by a rectangular array of 1461 elements (49 col- umns and 41 rows) placed over an equal-area projection of the hemisphere. Corrected intensity data are entered into this array but fill it only partially. Values for the remaining elements are obtained by a weighted averaging and smoothing of the surrounding cells. The resulting continuous pole-density distribution which extends over the whole hemisphere is then normalized in order to express densities in multiples of a random distribution (mrd-units).

Eight pole-figures were measured in each of the three samples and crystal- lographic data are given in Table III. A first inspection showed that meso- scopic fabric coordinates did not conform with the approximate symmetry axes of the texture. In interpreting textural properties it is expedient to use distinguished coordinates of the pole-figure. In the case of metal sheets these coincide with the rolling, transverse and normal direction. Jn our case a new

TABLE III

Crystallographic data of the pole-figures of quartz ___~ .___~ -__ __- ____._.

Diffraction Indices Crystallographic Relative Difference

peak /l/ii/ angles intensity U/l I? il akhil

20(CuK,) -- ahkil

0 Y _-- _---

26.7 lOi 51.8 0.0 0.70 0.40 26.1 Olil 51.8 60.0 0.30 36.6’ 1120 90.0 30.0 1 42.5 2020 90.0 0.0 1 45.8 2021 68.5 0.0 0.30 45.8” 0221 68.5 60.0 0.70

i 0.40

50.2 1122 47.7 30.0 1 50.7 0003 0.0 0.0 1 60.0 2181 73.4 19.1 0.54 60.0 1231 73.4 40.9 0.46

i 0.08

64.1 1123 36.2 30.0 1 73.5” 1014 17.7 0.0 0.22

0.56 73.5” 0114 17.7 60.0 0.78

j . ..-.-__-

obs

obs

Fig. 2. Experimentai and recalculated pole-figures (fabric diagrams) of the specimen Sci. 293. Stereographic projection. Contours are in multiples of random detlsity (mrd-units).

coordinate system was chosen and we found it convenient to place it so that C coincides with the single maximum of (OOOl)-poles. All pole-figures have been rotated into this new orientation. They are shown in Figs. 2-4. In the (llZO)-pole-figures mesoscopic fabric coordinates I and s are indicated. It should be noted that in the high-grade specimen Sci. 293 [OOOl]-axes are oriented roughly in the plane of foliation (s) and in the low-grade specimens Sci. 638 and 690 they are normal to it. This indicates that different deforma- tion mechanisms have been active.

The pole-figures shown in Figs. 2-4 served as data in the spherical har- monic analysis to calculate the orientation-distribution function. Orienta- tion-density values on concentric rings with 5” intervals in azimuth and pole-

ohs

Fig. 3. Experimental and recalculated pole-figures of the specimen Sci. 638

distance served as input for the ALGOL computer programs. 14th order har- monics were used in all calculations. They expressed the topography ade- quately and did not cause any significant series termination errors (ref. next section). The orientation~~stribution function in 3 2/m crystal and 1 speci- men symmetry requires 467 coefficients to represent it in 14th order. With eight pole-figures the system is sufficiently overdetermined to permit a least,- squares approximation.

In the first program ANALYS coefficients Fi’ and ,;I’ of the pole-figures are calculated. In the second program COEFF the system of linear equations 31, 32, 38 and 39 is solved for the coefficients Sf”” and G;“” of the orienta- tion distribution function using the coefficients F;’ and F;” of the single and

273

obs

obs

talc

Fig. 4. Experimental and recalculated pole-figures of the specimen Sci. 690

superposed pole-figures as input. From the coefficients Sy” and Gp” the pole-figure coefficients Fr and F;” and the pole-figures themselves are recal-

culated and the inverse pole-figures and their coefficients c are obtained. This program also determines error quantities (Bunge, 1969). In a third program SYNTH the three-dimensional orientation distribution is calculated according to eq. 36. A fourth program was used to draw the three-dimen- sional distribution function, the pole-figures and the inverse pole-figures with a computer-operated plotter.

Data reduction with OVLAP was done on a CDC 6400 computer at Berkeley, all other calculations were done on a NE 503 computer at Dres- den.

271

RESULTS AND INTERPRETATION OF THE TEXTURE:

In Figs. 2-4 measured pole-figures (in stereographic projection) are com-

pared with pole-figures recalculated from the orientation distribution func-

tions. The good qualitative agreement indicates that the measured pole- figures are consistent with each other (see also the following section).

Especially for the specimen Sci. 293 they show a pseudohexagonal sym- metry, but with some significant deviations from it.

The three-dimensional ODF (Figs. 5-~-Y) is a complete representation of preferred orientation. It gives the explicit and complete answer to the ques- tion which volume fraction of the crystallites have any given orientation

(prO& of the crystal axes XYZ with respect to the chosen specimen axes -ABC. Admittedly this function is more complicated to read than, for

example, the classical fabric diagrams. But these latter diagrams do not allow to distinguish between crystal orientations having the same orientation of the reference crystal direction, for example the [OOOl]-axis in the (OOOlj-

fabric diagrams. Hence these diagrams, although they are easier to read generally are more difficult to intcrpwt in terms of the complete crystal

orientations. The interpretation usually is accomplished by comparing differ-

ent fabric diagrams corresponding to different crystal axes. The result, a

qualitative description of the preferred orientation in terms of all three orientation parameters, is, however, usually subject to serious uncertainties

which are avoided by mathematically calculating the ODF. The main diffi-

culty in visualizing the orientation-distribution function. is its three-dimen-

sional character. This difficulty cannot be avoided because crystal orienta-

tion is a quantity depending on three parameters.

The ~airz feature of the three textures, according to the chosen settings of axes ABC, is a high concentration close to the cP1pz-plane in the vicinity of

Q = 0. According to the definition of the Eulerian angles, 0 = 0 means that the crystal %-axis ] OOOl] is parellel to the specimens C-axis which is the

result of the specific: choice of coordinate systems. The structural geologist can obtain the more familiar (OOOl)-fabric dia-

gram by projecting the ODF along the p2-axis according to eq. 41. ipr is the

azimuth and $ the pole distance of the pole-figure. Diagrams which display in all three specimens a single maximum of [OOOl] parallel to C (at 4 = 0) are shown in Fig. 8 in stereographic projection. These (OOOl)-pole-figures, which are most widely used in the petrofabric analysis of quartz, cannot be measured by X-rays, since the very weak (0003)-reflection is close to and

partially overlapped with the strong (1122)-peak. Similarly a projection along the cp,-axis eq. 43 (with ~2 as the azimuth and

cp as pole distance) gives an inverse pole-figure of the specimen C-axis with respect to crystal coordinates XYZ. As can be seen in Fig. 9 the inverse pole- figures also display a single maximum of C-axes in the [OOOl] direction which is in agreement with the (OOOl)-pole-figure and the ODF. Specimen C-

axes are roughly parallel to [ 00011.

275

Fig. 5. Orientation distribution function (ODF) of the specimen Sci. 293. Orientation coordinates are the Eulerian angles (PI&~. The density is in multiples of the random density (mrd-units). Traces o, b, c, corresponding to Fig. 10 are indicated.

Fig. 6. Orientation distribution function of the specimen Sci. 638.

A second feature of the textures can also easily be read from the ODF, Figs. 5-7. The orientation density is not constant along the line p1 = 0 in the $ = 0 plane. The distribution on this line taken from the ODF (trace a in Figs. 5-7) is shown in Fig. 10. It depends on the angle (pz that is the angle between the crystal X-axis [lOiO] and the specimen A-axis. *

* According to the definition of the Eulerian angles, the crystal orientation for # = 0 depends only on the sum of the angles cp1 + ~2, because the first Euler rotation through cp1 and the third one through ~2 are carried out about the same axis of rotation which is in this case the crystal Z-axis and at the same time also the specimen C-axis. Hence points in the section Q, = 0 having the same value of ~71 + 92 represent the same orientation. Accordingly, the density distribution along the line $I = 0, 91 = 0 is repeated also along the lines @ = 0”) ip1 = lo”, pl = 20” ,_ . , only shifted by lo”, 20”, . . , and so on, degrees.

Fig. 7. Orientation distribution function of the specimen Sci. 690.

/

/’ :5

211 0 \ ,I’

J b ‘\, / ”

‘j* j;

,‘2 >, I,

‘* 1 o 3

z t5 4 A \ ,” ‘1 : \ ;

!, ‘\ \ / \_

‘\\ _/,’ ‘, _\ ’ -.a__/ , 1,’

so16$3

‘, .____ --’ SCI 293 SCI 690

Fig. 8. (OOOl)-pole-figures (fabric diagrams) calculated from the orientation-distribution functions. Stereographic projections, mrd-units.

Traces at Q-90”

638

690

1 2.5 @b 1

2.5 .5

1

Ed?? I5

1.5

+ 2.8 E

c 20 x

c” 1.2

g 04 _ g 0” 20” 40” 60” ad _

g 1.6

0.8

0

I%+! A

’

0” 20” 40” 6)” Iloiol- ~++[oliol

Fig. 9. Inverse pole-figures for the speciment directions A, B, C, and traces at 6 = 90 for A

and B.

The density distribution along this line (or these lines) is bimodal, particu- larly clearly in Sci. 293 Fig. 10, indicating two preferred orientations of the crystallites with respect to a rotation about the [ OOOl]-axis. The correspond- ing orientation of the trigonal prism is shown on the right side of Fig. 10. The two orientations I and II occur with comparable but not equal fre- quency in an angular distance of about 60”. This gives rise to an approximate two-fold axis parallel to the specimen C-axis and since the crystal Z-axis [OOOl] is nearly parallel to C also to an approximate two-fold axis parallel to [OOOl]. The result appears as two trigonal quartz crystals (3 2/m), one in a positive and one in a negative orientation, which are twinned according to the Dauphin4 twin law and combine to a pseudo-hexagonal crystal (6/mmm). Notice that, due to the Friedel’s law X-ray pole-figures do not resolve the true 32 symmetry of quartz. They possess a center of symmetry. Furthermore positive and negative reflections (hhil) and (hhil), although not symmetri- cally equivalent, have the same Bragg-angle. Consequently the corresponding pole-figures cannot be measured separately. The (hkil)- and (/Ail)-planes of the same crystal are separated from one another by a 60” rotation about the crystal Z-axis [OOOl]. Hence the two crystal orientations in the Dauphine twin position cannot be distinguished immediately in the pole-figures because the (hkil)-plane of one of them always coincides with the (hhil)-

I

II

690 ~ 2 level of uncertmly

___-~-----_-_“--~-.

/

0' 20' 10" 60" 80" 100" 120"

.b

~

-_

--.- q+ -_--+ _-@-__r

Angle between ilOiO1 and s~ecmw" A-ax's Angle between 10~XJ11 and specmen c-0x1s

Trace 0 rrom bi

A

Fig. 10. Traces a, b, c of the orientation distribution functions (compare Figs. 5-7) and the two orientations I and II of the trigonal crystal.

plane of the other one. The somewhat complicated symmetry situation is explained in Fig. 11.

The two orientations can, however, be distin~ished in the calculations leading to the ODF. They give rise to the two peaks of different height in the curves of Fig. 10. This is based on the different structure factors of the reflections (hhil) and (h&l) corresponding to Table III, indicated as big and small symbols in Fig. 11, which enter the formulae eqs. 38 and 39. From the ODF it is then possible to calculate the (hkil)- and (h/Cl)-pole-figures sepa- rately as is shown for the (1031) and (Olll)-pole-fi~re in Fig. 12. In these pole-figures the pseudo-hexagonal symmetry found in the superposition

279

True Symmetry

Frledel’s

Law

hkll + khil

Orientation I

32 a3

@

a

\ l

0 l a2

Y

\

0. OliOl

01 [1o1ol [iiZol X

Orientation I 0 0.

Upper hemisphere l . m 1

hkil •O~CI

Orientations I + II

Orientation II n 0 n

Lower hemisphere 0 0

khil l I Fig. 11. Symmetries and pseudo-symmetries of quartzite pole-figures due to the super-

position of (hhiZ)- and (khi[)-reflections and the two orientations I and II.

(1011) + (Olil) is separated into two approximately trigonal symmetries, 60” apart from one another (because the separation relies on the differences in the structure factors of (hhil) and (khil) respectively and on the differ- ences of the intensities in the superposed pole-figures at points 60” apart from one another the accuracy of the separation is much lower than the accuracy of the recalculated superposed pole-figures, see also the following section).

280

(lOi 1 4 a i

! _./ /

j .

(Olil)

Fig. 12. (7 Oi 1). and (Olil)-polr-figures calculated separately from the orientation distrj-

bution function. Stereographic projection, mrd-units.

Finally also the deviations from the ideal maximum density orientations can easily be read from the ODF chart. The spread of crystal orientations with respect to a rotation abotit the [OOOl]-axis may be estimated from the half-maximum width of the curves in the left of Fig. 10. In Sci. 293 it is about 17” for both orientations I and II. The deviation of the [OOOl]-axis from the specimen C-axis may be estimated from the traces q1 = const., q2 = const. which are drawn as traces b and c respectively in the ODF charts Figs. 5-7. For cpl = 0 these traces in the @-direction correspond to a rotation of the crystal about the specimen A-axis that is they indicate the deviation of the [OOOl]-axes from the C-axis in the direction towards the specimen axis B. The half-maximum width of the spread in this direction is about 17”. For ppl = 90, traces h’ and c’ in the ODF correspond to a rotation of the crystals about the specimen B-axis, hence they allow the spread of [OOOl]-axes in the A-direction to be estimated. This spread is smaller, the half-maximum width is about 8”. The same anisotropy of the spread about the specimen C-axis is obvious in the (OOOl)-pole-figures, Fig. 8.

Thus we have seen that many features of the texture can easily be read from ODF charts. Some of these features - the concentration of [OOOl]- axes in the C-direction and the anisotropic spread of [OOOl]axes about C - can also be seen in the (OOOl)-pole-figure which can be measured with the universal stage-microscope but not with the X-ray pole-figure goniometer. Other features such as the two orientations in Dauphin6 twin relation could

also be seen from separated (lOrl)-pole-figures but these can only be deduced from the threedimension~ ODF. We tried to demonstrate that the ODF is advantageous not only in being a more complete representation but it is also not more difficult to read than a fabric diagram if we become familiar with it.

ESTIMATION OF ERRORS

If reliable conclusions are to be drawn from the results of measurement and calculation it is necessary to estimate the possible errors. A first check of the reliability of the measurements is obtained by comparing the experi- mental pole-figures with the corresponding ones which were recalculated from the ODF. Since we used more pole-figures than are mathematically necessary to determine the ODF, the ODF is not exactly compatible with each pole-figure, it is rather an optimum fit to all the pole-figures. The same holds also for the recalculated pole-figures which are not identical to the measured ones but in contrast to those they are mathematically exactly com- patible among themselves. Hence comparing the experimental pole-figures with the recalculated one means to check the degree of compatibility of the experimental measurements. As can be seen in Figs. 2-4 the agreement is good as far as the general features are concerned, but noticeable deviations do exist in details. There are two possible causes for these deviations. First there are unavoidable experimental errors in data collection. It was the aim to use as many pole-figures as possible to get a good least-squares fit and thus to reduce the effect of these errors in the result of the calculations, the ODF.

The second reason is a mathematical one. It can be seen that the calcu- lated pole-figures generally show less topography than the experimental ones. This is due to series truncational effect. In the present calculations all series were extended to L = 14 which is obviously not sufficient to represent the true shape of rather sharp peaks in the distribution functions. The peaks are flattened out to a degree which may be estimated by comparing the experimental and recalculated pole-figures. An artificial flattening of the same order of magnitude must be assumed also in the ODF so that the true peak-height in this function, as well as in all the other calculated distribution functions, must be assumed about 20% higher.

The experimental errors combined with a series truncation effect give rise to negative values in the calculated distribution functions. Since, per defini- tion, all the distribution functions must be positive or zero the magnitude of the errors can be estimated to be at least as high as the highest negative value. The so determined level of uncertainty of the three ODF’s is shown in Fig. 30.

A quantitative measure of the errors may be obtained when solving the overdetermined system of linear equations 31, 32, 38, 39. Along with the coefficients Sr” and Grn the probable errors AS;In and AGrn of these quantities can be calculated. The mean absolute values of AS?” and AG;ln

282

-- taken over m and n, that is AC, (cf. Hunge, 1969) are compared with the same mean values G of the coefficients S;l” and G;“” (Fig. 13). Because oi the orthogonality of the spherical harmonics and the generalized spherical harmonics with respect to the index 1 the coefficients Sy” and CT” as well as their respective errors A&“” and AC;,” are independent of the degree I, of the series development. They do not change their values if further terms arch -. ._. added to the series. Hen,:e the dependence of the c and AC, curve on 1 permits one to assess the ~onvergeIl~e of the series devel~~pment. For speci- men Sci. 690 and Sci. 638 the coefficients have already decreased to the level of the errors at 1, = 14 whereas for Sci. 293 they seem to be slightly above this level at L = 14. Hence for the latter specimen an extension of the series beyond I, = 14 could have somewhat improved the results. For the other two specimens the series truncational effect is of the same magnitude as the expe~mental errors so that a higher degree L woufd not have added any improvement.

293

536

590

OR 1

IJ? c m

” !r.

Pegwe I 7oq1 F?

Fig. 13. The error quantities AC, for all the coefficients and for the even and odd values of m and n separately.

283

As was mentioned above the separation of (h/d)- and (Ml)-pole-figures depends on the magnitude of the difference ahkil - ekhil compared with the value 1 for the non-superposed pole-figures. The mean value of this quantity for the four superposed pole-figures was 0.36. Hence the error of the separa- tion should be about three times larger compared with the overall error. If we can separate these pole-figures it means that we can determine the exis- tence or non-existence of a two-fold axis of symmetry in the crystal Z-direc- tion [OOOl] and, because of the preferred orientation of [OOOl]-axes in the specimen C-direction, also of a two-fold axis in C-direction. A two-fold axis in the crystal Z-direction gives rise to the selection rule:

m = 2m' (44)

and in the specimen C-direction to:

n = 212’ (45)

Hence the existence of such an axis is expressed in the coefficients Sy”” and GT” with odd values of m or n. The two-fold axis is absent if these coeffi- cients are larger than their respective errors. In Fig. 13 the same error quanti- ties were therefore calculated for even and odd values of m and n separately. It is seen that the margin between the coefficients and their errors is con- siderably smaller for the odd values than for the even ones. But in each case at least part of them exceed their corresponding errors sufficiently so that their deviation from zero must be assumed to be statistically significant. Hence, the deviation of the coefficients from the selection rules eqs. 44 and 45 is statistically significant and this, in turn, means the absence of the two- fold axis. The deviations of the distribution function from this two-fold symmetry thus have proved larger than the experimental errors. The differ- ence of the orientation densities in the two twin-related positions must be assumed to be statistically significant.

CONCLUSION

The three quartzite textures can thus be “decomposed” into separate parts. Of major importance is the orientation of [OOOl] normal to the plane of schistosity in Ski. 638 and 690, possibly through some mechanism of basal glide. In the high-temperature mylonite the orientation of [OOOl] in the plane of foliation may have been achieved by some mechanism of pris- matic glide or recrystallization in an anisotropic stress field. Whichever mechanisms acted they produced in addition to strong preferred orientation of [OOOl] = 2 a secondary orientation of X-axes. It is conceivable that at a later stage Dauphin6 twinning, which has recently been established as a practically strainfree deformation mechanism in quartz (Tullis, 1970) has been active and caused positive and negative forms to be unequally repre-

sented. There is an overall similarity of the three textures, if they arc’

regarded with respect to the specimen coordinate system ARC. However these individually chosen specimen coordinates ABC have a different relation to the mesoscopic fabric coordinates a, b = 1, c = ! s in Sci. 293 and in the

two specimens Sci. 638 and Sci. 690 and we attribute this to a different gee- logical history, particularly different temperature conditions during deforma- tion. Only more comparative studies of ODF’s of quartz combined with

applications of theories which predict preferred orientations (e.g. Taylor, 1938 and application to quartz by Lister, 1974) and an independent deter- mination of acting deformation mechanism (e.g. electron microscopy) will allow to interpret the texture quantitatively in terms of t,he deformation his-

tory. A puzzling feature is still the small but significant difference in orienta- tion between mesoscopic fabric coordinates aird the approximate sym-

metry axes of the pole-figures. One possible explanation might be that a

shear component was active during deformation which is expected to give

rise to a rotation about an axis in the plane of shear perpendicular to the

shear direction. We have discussed some special features of the quartz fabrics. An impor-

tant parameter is naturally the degree of preferred orientation which can be

expressed with the texture index, the integral over the square of the texture

function (Sturken, 1960; Bunge, 1969, p. 59). Values in Table IV describe

quantitatively the big differences. We have presented for the first time a spherical harmonic analysis which

permits us to represent triclinic specimen symmetry and we could demon- strate that deviations from monoclinic symmetry are real. The triclinic

analysis seems to apply to most naturally deformed rocks even though many rock-textures show at first glance monoclinic or orthorhombic symmetry. Thus the procedure described in this paper and explained with three quart- zite samples should be a useful addition to make the three-dimensional orien- tat,ion distribution representation more easily adaptable to geological prob- lems. particularly the analysis of the important trigonaf rock-forming

minerals quartz, calcite, dolomite and hematite.

TABLE IV

The texture index

Specimen Sci 293 Sci 638 Sci 690 ._____-. ___c____-_ -.. ____.___-..-

J 11.0 2.4 1.6 -. .___

285

ACKNOWLEDGEMENT

We are indebted to DR. C. Esling, Metz, for valuable discussions which led to the cancellation of an error in the formulae.

HRW acknowledges support through NSF grant DES 75-14213.

REFERENCES

Baker, D.W., 1971. Symmetry of the orientation distribution in crystal aggregates and mechanical twinning. In: J. Karp et al. (editors), Quantitative Analysis of Textures. Proc. Int. Seminar Cracow.

Baker, D.W. and Wenk, H.R., 1972. Preferred orientation in a low-symmetry quartz mylonite. Geology, 80: 81-105.

Baker, D.W., Wenk, H.R. and Christie, J.M., 1969. X-ray analysis of preferred orientation in fine-grained quartz aggregates. Geology, 77 : 144-172.

Bunge, H.J., 1969. Mathematische Methoden der Texturanalyse. Akademie-Verlag, Berlin, 330 p.

Bunge, H.J., 1974. Calculations of the Fourier-coefficients of the generalized spherical functions. Kristall Technik, 9 : 939-963.

Liddell, N.A., Phakey, P.P. and Wenk, H.R., 1976. The microstructure of some naturally deformed quartzites. In: H.R. Wenk (editor), Electron Microscopy in Mineralogy. Springer, Berlin, pp. 419-427.

Lister, G.S., 1974. The Theory of Deformed Fabrics. PhD thesis, Australian National University, Canberra.

Sander, B., 1950. Einfiihrung in die Gefiigekunde der geologischen Kiirper, 2. Springer, Wien, 409 p.; translated in 1970, Introduction to the Study of Fabrics of Geological Bodies. Pergamon Press, Oxford, 641 p.

Sturken, E.F., 1960. In: US AEC Rep. NLCO 804 (1960) 9. Taylor, G.I., 1938. Plastic strain in metals, J. Inst. Metals, 62: 307-324. Tullis, J., 1970. Quartz: preferred orientation in rocks produced by Dauphine twinning.

Science, 168: 1342-1344. Wenk, H.R., 1974. The structure of the Bergell Alps. Eclogae Geol. Helv., 66: 255-291. Wcnk, H.R. and Wilde, W.R., 1972. Orientation distribution diagrams for three Yule

marble fabrics. Geophys. Monogr., 16 : 83-94. Wenk, H.R., Wenk, E. and Wallace, J.H., 1974. Metamorphic mineral assemblages in

pelitic rocks of the Bergell Alps. Schweiz Mineral. Petrogr. Mitt., 54: 508-554.

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