American Mineralogist, Volume 63, pages 98i,-990, 1978 Triclinic feldspars:angular relations and the representationof feldsparseries Jnuns B. THorvrpsoN,Jn. Department of Geological Sciences,Hqruurd Uniuersity Cambridge, M assachus et t s 02 I 38 nNo Guy L. Hovls Department of Geology, Lafayette College Easton, Pennsyluania I 8042 Abstract The rhombicsection angles and obliquitiesof triclinicfeldspars are simpletrigonometric functionsof a* and 'y. Alkali-exchangeseries and short-termheating experiments show wide variationin a* but littlevariation in 7. Differingdegrees of Al-Si orderatconstant composi- tion and temperaturecause wide variations in 7 and relativelyminor onesin a*. Graphical methodsare developed by whicha*, ,y,rhombic section angle, and obliquitycan be readeasily from a singleplot. The sameplot alsodifferentiates the varioustypes of feldsparseries at least as well as any other in currentuse. The obliquity,@, is a directmeasure of the departureof a triclinicfeldspar from the dimensionalrequirements for monoclinicsymmetry. The quantity (l - cos@)must vanish for monoclinicfeldspars and is a nearlylinear function of temperaturefor isochemicalfeldspar seriesthat approacha monoclinicstate on heating.A newconvention is proposedfor defining therhombic section of potassicfeldspars in orderto maketheir values consistent with thoseof contiguousK-Na feldspars. Feldsparsof a givencomposition that are in full internalequilibrium should lie along uniquepaths in termsof a* and 7. Sufficientinformation is now at handto showthe general form of suchpaths lor K-Na-Ca feldspars,and to providepreliminary formulations of these pathsfor the end-memberfeldspars. Introduction suffice,although it may seemaesthetically unappeal- The idealizedstructure of feldsparis topologically ing,at first,to mix directand reciprocal quantities. If consistentwith monoclinicsymmetry, and certain po. a* and 'y or otand 7* arechosen, mathematical and tassiumand bariumfeldspars are indeed monoclinic. trigonometric advantagesare gained (seelater dis- The mostabundant feldspars in rocks,however, are cussion),since angles are determinedby axial ele- triclinic,but even in thesethe departuresof lattice mentsthat lie in two mutually orthogonalplanes. parametersfrom the requirementsfor monoclinic This is not generallytrue if both anglesselected are symmetryare small. Many such feldsparscan be director both reciprocal. transformedcontinuously into monoclinicforms, ei- We shallshow that a* [whichalso may be regarded ther by heattreatment or by alkali exchange. as the dihedralangle between the (001) and (010) Departuresof a triclinic crystal from the dimen- cleavagesland 7 [whichis a measureof the obliquity sionalrequirements of monoclinicsymmetry may be of the (001) face of the unit celll provide unique describedcompletely in terms of two independent advantagesin the treatmentof triclinic feldspars. angularmeasurements. With theb-axis chosen (in the Other anglesof specialinterest are relatedto them by tricliniccrystals) to correspondtopologically to the simple trigonomeftic equations,much more simple D-axisof a possiblemonoclinic variant, a sufficient ones,in fact, than the onesobtaining if wholly direct or pair would be eitherthe a and 7 anglesof the direct wholly reciprocal cell-angles are used. Changes cell, or the a* and 7x anglesof the reciprocalcell. brought about by short-termheating, or by short- Pairssuch as a* and1, or otand 7*, however,will also term alkali exchange,cause wide variationsin a* 0003-004x/ 78 /09 I 0-098l $02.00 981 982 THOMPSON AND HOVIS: TRICLINIC FELDSPARS with but slight variationsin 7 (nearly negligible,in ing Don whichthe traces of the forms(l l0) and(l l0) fact, in many instances).The effectsof long-term form a rhombus.Inasmuch as the diagonalsof a heating, in feldsparsas subsequentlyexamined at rhombusintersect at right angles,an alternativeand room temperature,cause wide variations in 7 and more practical definition is to define the rhombic relativelyminor variationsin a*. Although we are section as the plane definedby b and the line r in concernedprimarily with triclinic feldspars,many of (010)that is perpendicularto b. As a sectionof the the methodsdeveloped are applicableto any mineral unit cell the rhombic section is thus a rectangular group containingtriclinic crystalsthat showcontinu- section,which might indeedbe a bettername for it. If ous variationsabout a monocliniclimit. 0 + 0 thereis a uniquerhombic or rectangularsec- tion, but if : 6, all sectionscontaining D arerhom- Obliquity 6 bic sectionsand the rhombic section,as a unique In a monocliniccrystal, conventionally oriented, it plane,is indeterminate.Even in this last instance, is necessary(but not sufficient)that the b-axisand the however,we may haveinterest in a limiting rhombic pole, b*, to (010) coincide,and that b : l/b* : sectionwhen dealingwith a phasetransformation in d(010).In a tricliniccrystal, on the otherhand, b and which @ - 0 continuously.The orientationof the b* do not, in general,coincide. We shall therefore rhombic section may be specifiedin two alternate follow Donnay (1940) and designatethe angle,e, ways: either as the dihedralangle, which we shall betweenthem as the obliquity.[t follows that designateas d*, betweenthe rhombic section and (001); d(010):r/b*:6cosd (l) or by the angleo, betweenr and a, which is thusan anglein (010).Feldspars have low obliquities Inasmuchas the cell volume,V., of a monoclinic (lessthan 5o), hencethe distinctionbetween o ando* crystalis abc sinB,that for a triclinic crystalis given is of little practicalimportance and doesnot exceeda by few hundredthsof a degreeat most. We V" : abc sinp cos@ (2) shallfind it usefulto designatethe line normal to r in (010)as s, and the poleto therhombic section Similarly for the reciprocalcell volume Z| ass*; r isthus normalto J,J*, b andb*.In additionb is V\ : q*['x.* sinp* cos@ (3) normal to J*, c* or a*, and bx to s, a or c, hence Although it is necessaryfor monoclinicsymmetry sAs*:b6tb*=6 (4) : : that4 0 or cos@ l, it is possible(though we know Also of no examples)that a tricliniccrystal may alsohave zero obliquity, despitethe origin of the word "tri- r lta= o: sA[Iain (010)] (5) clinic". The vanishingof the obliquity thus insures s* A c* = d* : r ltfLb in (001)l (6) only that thegeometric form of the unit cell meetsthe requirementsfor monoclinicsymmetry, and doesnot as shownin Figurel.If A # 0 in a triclinicfeldspar prove that the crystalis indeedmonoclinic. Such a and 7 is 90o,the rhornbic section must coincide with crystal,for example,might have an Al-Si distribution (001);and if a*, whichis alsothe anglebetween the that is inconsistentwith monoclinicsymmetry. (010)and (001) cleavages, is 90o,the rhombic section The obliquityas used here, in thesense of Donnay, must be normal to (001).In eachof thesespecial shouldnot be confusedwith the quantity[d(l3l) - caseso and o* are identical. d(13l)l which has beencalled the "triclinicity" by Goldsmith and Laves (1954) and "obliquity" by Orientation of triclinic feldspars McConnelland McKie (1960;see also Martin, 1969, The crystallographicreference axes for triclinic 1970).Although the quantity td(l3l) - d(l3l)lmust feldsparsare chosenso as to correspondtopologi- vanishwhen @ vanishes, the reverseis not true.The callyto thoseof monoclinicfeldspafs. In centrosym- vanishingof @is thusthe morestringent requirement metric monoclinicfeldspars, because the axesare for monoclinic symmetry,though not in itself suf- non-polar,there are two entirelyequivalent ways of ficient. selectingthe positiveends of theseaxes so asto form a right-handedarray. In centrosymmetrictriclinic Rhombicsection feldspars,though the axesare still non-polar,the The rhombicsection (Rath, 1876)of a triclinic feld- correspondingtwo waysare non-equivalent. An early spar is defined,traditionally, as the sectioncontain- conventionwas to choosethe orientationthat made THOMPSON AND HOVIS: TRICLINIC FELDSPARS 983 [ro in(Olo)] =32.57o (o) (b) Fig. l. Angularrelationships in Ameliaalbite (Fl0l of Grundyand Brown, 1969). (a) Axialelements normal to b asviewed in positive directionalong b. (b) Axial elementsnormal to D* as viewedin positivedirection along b*. (c) Axial elementsnormal to r asviewed in positivedirection along r. The crystalis in conventionalorientation (a obtuse).Arrows showpositive ends of axes. d* acute,but Laves(195 l) demonstratedthat this The angularrelations aI 25"C for a typical low orientationplaced microclines out of propercontext albite(Fl0l of Grundy and Brown, 1969)are shown relativeto other closely-relatedtriclinic feldsparsand in Figure I for the crystal in conventionalorienta- proposeda new conventionsuch that a for all tri- tion.The arrowsindicate what we shalltake as posi- clinic feldsparsbe taken as obtuse.We shall refer tive directionsfor r, J, and s*. In a K-Na exchange hereinto the latter orientationas conuentional, andto seriesfrom low albiteto microcline,6 ando* increase the orientationof feldsparsso that a is acuteas anti- continuouslyto betweenl00o and l05o as measured conuentionaLThe principaiambiguities that ariseare at 250oC, and a similar effect may be observed with K-rich triclinic feldsparsof very low obliquity throughshort-term, reversible heating of low albite. suchthat the departureof a from 90o lies within the Long-term heating of albite, however,reduces the experimentalerror. If this occurs,the orientationis valuesof o ando* continuouslyto a limitingvalue of consistentwith thosefor relatedfeldspars if y is taken about -3o if measuredat