Closure in crystal size distributions (CSD), verification of CSD calculations, and the significance of CSD fans

Transcription

1 Amercan Mneralogst, Volume 87, pages , 2002 LETTERS Closure n crystal sze dstrbutons (CSD), verfcaton of CSD calculatons, and the sgnfcance of CSD fans MICHAEL D. HIGGINS 1, * 1 Scences de la Terre, Unversté du Québec à Chcoutm, Chcoutm, G7H 2B1, Canada ABSTRACT Crystal sze dstrbuton (CSD) measurements are susceptble to the closure problem, just lke chemcal compostons. In ts smplest form ths means that the total crystal content of a rock cannot exceed 100%. Where chemcal or thermal effects lmt the total quantty of a sngle phase, closure can occur at lower volumetrc phase proportons. Ths means that parts of the CSD dagram [ln(populaton densty) vs. sze] are not accessble. If the volumetrc phase proporton s constant, then straght CSDs wll appear to rotate around a pont at small szes gvng a fan of CSDs. These fans are sgnfcant and do show changes n crystal szes that can be nterpreted n terms of magmatc processes. However, the correlaton between the slopes (or characterstc lengths) and ntercepts of ndvdual CSDs n a famly s not sgnfcant, but just a consequence of the constant phase proporton effect. Many other graphs, such as characterstc length vs. volumetrc phase proporton, can gve more nformaton on magmatc processes. INTRODUCTION Crystal sze dstrbuton (CSD) studes of gneous and metamorphc rocks have become more popular recently partly because t has been realzed that they are a quanttatve method of lookng at gneous processes that s complementary to geochemcal studes (e.g., Cashman 1990; Hggns 1999; Marsh 1998). Another reason may be the recent avalablty of smple methods of calculatng CSDs from measurements n two dmensons, such as thn sectons (e.g., Hggns 2000; Peterson 1996). The crystal content of a rock cannot exceed 100%, hence, as wth chemcal analyzes, we must always be aware of the closure problem. Closure for an ndvdual phase can also occur at less than 100% crystals. For nstance, f a rock s made of 50% plagoclase and 50% olvne then closed-system processes, such as smple textural coarsenng, cannot change the phase proportons they are each fxed at a maxmum of 50%. Closure has not, so far, been dscussed n publshed CSD studes. However, I wll show that t must be consdered n all CSD studes, both of volcanc and plutonc rocks. The closure effects descrbed n ths paper are unrelated to the Inherted correlaton effect of Pan (2001). That study was flawed by napproprate use of CSD equatons [see comment by Marsh et al. (2002)]. * E-mal: mhggns@uqac.ca CLOSURE AND CONSTANT PHASE PROPORTION IN CSD MEASUREMENTS We are concerned here wth the effects of constant phase proporton on CSDs, however, I wll show later that even qute large varatons n volumetrc phase proportons can show the same effects. Clearly, closure s just a specal case of ths more general problem, where the volumetrc phase proporton s equal to one. The volumetrc proporton of phase, V, s calculated by ntegraton of the volume of all the crystals: V = σ n( L) LdL 0 where σ = shape factor of phase, equal to the rato of the crystal volume to that of a cube of sde L (see below), and n (L) = populaton densty of crystals of phase for sze L. There are many dfferent defntons of crystal sze, but n ths paper I wll follow Hggns (2000) and other authors, and defne sze as the length of the longest axs of the smallest rectangular parallelepped that encloses the crystal. The shape factor, σ, s expanded nto a more applcable form: σ = [1 Ω (1 π/6)]is/l 2 (2) where Ω s the roundness factor, whch vares from 0 for rectangular paralleleppeds to 1 for a traxal ellpsod. Ths defnton accords wth the roundness factor mentoned n Hggns (2000). S, I, and L are the short, ntermedate, and long dmen- (1) X/02/ $

2 172 HIGGINS: CLOSURE IN CRYSTAL SIZE DISTRIBUTIONS sons (or axes) of the parallelepped or ellpsod. Obvously, most crystals have a more complex shape, but we are concerned here just wth a statstcal measure. However, Equaton 2 should not be appled to crystals wth concave surfaces or holes, as they do not necessarly have volumes that le between that of an ellpsod and an equvalent parallelepped. Equaton 1 apples to all CSDs. Marsh (1988) showed that CSDs n many smple magmatc systems are descrbed by the equaton: N (L) = n 0 e L/C () where n 0 = fnal nucleaton densty and C = a constant called the characterstc length. It s equal to the mean length of all the crystals n a straght CSD that extends from zero to nfnte sze. Ths dstrbuton s lnear on a graph of ln(populaton densty) {ln[n (L)]} vs. sze (L; Fg. 1a). The ntercept s ln(n 0 ) and the slope s 1/C. Combnng Equatons 1 and gves L/ C V = n e LdL (4) σ 0 0 whch can be ntegrated to gve (Marsh 1988) V = 6σn 0 C 4 (5) It should be emphaszed that ths equaton s for straght CSDs and that volumetrc phase proportons calculated n ths way are senstve to errors n the shape factor. A smple straght lne CSD s shown n Fgure 1a. At low volumetrc phase proporton, for example 1%, the slope of the CSD can change ndependently of the ntercept. For example, f all crystals grow at the same rate and nucleaton ncreases exponentally, then the CSD wll move up wthout changng slope. Once the crystal content reaches 100%, then closure s reached and the CSD s locked. It can only move f the rato between crystals of dfferent szes s changed. That s, some crystals must become smaller f others are to enlarge. A famly of straght CSDs for 100% crystallzed materal defne a fan (Fg. 1b ). Portons of ths fan appear to descrbe rotaton of the CSD around a pont commonly close to the left of the dagram. The CSDs together outlne a concave-up envelope. Straght CSDs cannot exst n the area above ths lne. Fgure 1c shows another vew of the same effect n terms of characterstc length vs. ntercept. All straght CSDs can be defned by ther ntercept and characterstc length. Closure lmts descrbe a curve for each crystal shape. Straght CSDs can only exst below ths lne. Closure for a sngle phase can occur at less that 100% volumetrc crystal content, f there are chemcal or physcal constrants on further crystallzaton. For nstance, f a rock contans 50% plagoclase and 50% olvne, then closure for each phase s at 50%. Thermal condtons may also lmt crystal content to a maxmum value. If the crystals of a phase n a sample are far from ther closure lmt, then the ntercept and slope can change ndependently gvng two degrees of freedom. However, as the crystal content approaches the closure value, then any process that changes the slope of the CSD must also change the ntercept n ths stuaton, there s essentally only one degree of freedom. Therefore, crystals must dssolve (melt) or dvde nto sub-grans to en- FIGURE 1. (a) The closure problem n CSDs, llustrated for cubc crystals. A straght CSD wth a slope of 0.57 (characterstc length 1.7 mm) and an ntercept of 8.6 has a volumetrc phase proporton of 1%. If the ntercept s ncreased to 4.0, for example by crystal growth and exponental ncrease n nucleaton rate, then the crystal content wll reach 100% and textural changes wll stop. At 100% crystals, the CSD s locked and cannot move nto the grey area of the dagram. (b) A fan of straght-lne CSDs wth 100% crystals s tangental to a concave-up curve. Straght CSDs cannot exst n the grey part of the dagram. (c) Straght CSDs can be represented by a pont on a graph of characterstc length ( 1/slope) vs. ntercept. Possble CSDs for 100% crystals proscrbe a curve. CSDs can exst below, but not above ths lne. The poston of the lne dffers for dfferent shapes, here ndcated by the aspect ratos of rectangular paralleleppeds and spheres. It s assumed here that the crystals completely fll the volume.

3 HIGGINS: CLOSURE IN CRYSTAL SIZE DISTRIBUTIONS 17 able changes n the slope and ntercept of the CSD. Natural examples of fannng CSDs may not exactly resemble Fgure 1b, because the volumetrc phase proporton may vary (Cashman and Marsh 1988; Hammer et al. 1999). However, as wll be seen later n the example, even n ths stuaton the correlaton between characterstc length and ntercept s stll evdent and s lttle perturbed even by qute large varatons n volumetrc phase proportons. VERIFICATION OF CSD DATA Equatons 1 and 5 are a useful way to verfy that CSDs have been correctly converted from two-dmensonal data, by comparng the measured value of the phase proporton wth that defned by the CSD. However, cauton must be exercsed as the errors n phase proporton estmated from CSDs can be sgnfcant, especally for CSDs n whch much of the phase volume s contrbuted by the largest crystals. Such crystals are not numerous and hence ther populaton densty s not well known. Equaton 5 can be used drectly for straght CSDs. For complex, curved CSDs the correspondng volumetrc proporton of the phase can be calculated from the shape factor, the wdth and md-ponts of the sze ntervals, and the populaton densty of those ntervals. For these calculatons, a dscrete form of Equaton 1 s as follows: V = σ n( Lj) LjW (6) j where j s the number of nterval, W j s the wdth of the nterval, and L j s the mean sze of the nterval j. It should be remembered that ln[n I (L j)] s plotted n a CSD dagram. Ths calculaton has been ncorporated nto verson 1.2 of the program CSDCorrectons (Hggns 2000; ~mhggns/csdcorrectons.html). The volumetrc proporton of the phase n a rock s equal to the area proporton of the phase n any secton (Delesse 1847). The fabrc of the rock and the orentaton of the secton are not mportant any plane wll gve the same area. Crystal ntersecton areas are commonly measured at the same tme as the ntersecton length and wdth of the crystals. They can also be determned by pont countng or by automatc mage analyss. The total crystal area s ndependent of the length or wdth measurements used to construct CSDs. Volumetrc phase proportons can also be calculated n some cases by mass balance of chemcal compostons or even densty (for two-phase rocks). EXAMPLE LAVAS FROM MOUNT TARANAKI A seres of 14 samples of andeste lava and tuff from Mt. Taranak (Egmont volcano), New Zealand, provde a good test of the methods developed above (Hggns 1996). Although ndvdual CSDs are slghtly curved, they are straght enough that both Equatons 5 and 6 can be used to calculate the volumetrc phase proportons, and the results of the two methods can be compared wth the actual volumetrc phase proporton. The orgnal plagoclase ntersecton data were recalculated usng the methods of Hggns (2000; Fg. 2a). The crystal shape was determned followng Hggns (1994): the crystals are euhedral hence the roundness equals zero (paralleleppeds). The mode of the ntersecton wdth/length ratos s 0. hence the rato I/S =.0. The skewness of the ntersecton wdth/length ratos suggested that the rato L/I s close to one, gvng an overall aspect rato of 1::. The actual volumetrc phase proportons were determned from the total area of the plagoclase crystals n thn secton. There s a good correlaton between the actual volumetrc phase proporton and that determned from the CSD usng Equaton 6 and the CSD calculated wth the program CSDCorrectons (Fg. 2b). Most of the CSDs are almost straght, hence the volumetrc phase proporton also can be calculated wth Equaton 5, usng the slope and characterstc length from the regresson of the actual CSD. Agan there s a good correlaton between the actual volumetrc phase proportons and the calculated values, except for two slghtly more curved CSDs (Fg. 2c). The overall agreement between the two methods of calculatng the volumetrc phase proportons from the CSD and the values determned ndependently from the crystal areas, ndcates that the CSDs have been correctly measured and can now be nterpreted. It s dffcult to compare a seres of samples on a classc CSD dagram such as Fgure 2a as they as they all tend to overlap. Another way of lookng at the varaton between straght or almost straght CSDs s to regress the CSD and plot the ntercept aganst the characterstc length (Fg. 2d) these parameters were used to calculate the volumetrc phase per cent used n Fgure 2c. It s clear that despte the relatvely low volumetrc phase proporton of plagoclase, and ts varablty (10 0%), the correlaton between ntercept and characterstc length s the most mportant component of the varaton n ths dagram. The varaton normal to the closure lmt s much less evdent and s controlled by the volumetrc phase proportons. There are several better ways of lookng at these data. One such graph plots characterstc length aganst volumetrc phase proporton (Fg. 2e). Ths dagram has the advantage that t s completely accessble. That s, there are no forbdden zones, as s the case for the slope vs. characterstc length dagram (e.g., Fgs. 1b and 2d). Ths accessblty s because the two parameters are completely ndependent. The Taranak CSDs fall nto three groups. Fanthams Peak lavas have the smallest characterstc lengths and volumetrc phase proportons. Ths area s the youngest part of the volcano, hence t would be expected that the lavas have lower resdence tmes and are crystal poor. The Burrell ash (sample 12) also falls n ths group suggestng that the current magma chamber may be smlar or dentcal to that whch produced other lavas of Fanthams Peak. The Starcase lavas have the greatest characterstc lengths. Ths result ndcates that the magma chamber was evolvng slowly at ths tme, ether because t was large or because the erupton rate was slow. The Summt and Castle lavas have ntermedate characterstc lengths, but the greatest range n volumetrc phase abundance. The most crystal-rch samples are from the base of thck flows, suggestng ether that there was crystal settlng n the lava flows or that the frst part of the magma chamber to be sampled was rcher n plagoclase.

4 174 HIGGINS: CLOSURE IN CRYSTAL SIZE DISTRIBUTIONS FIGURE 2. CSD data from 14 samples of andeste lava, Egmont volcano, Mt. Taranak, New Zealand (Hggns 1996). Orgnal 2D data have been converted to D values usng the program CSDCorrectons 1.2 (Hggns 2000). (a) Crystal sze dstrbuton dagrams. Most ndvdual CSDs are almost straght. However, the envelope of CSDs descrbes a curve. Ths s the effect descrbed as CSD fannng. (b) Comparson of volumetrc phase proporton n percent determned from the total area of crystals n thn secton (measured) vs. that calculated from the CSD and crystal shape usng Equaton 6. (c) Comparson of measured volumetrc phase proporton vs. that calculated for a straght CSD wth ntercept and slope derved from a regresson of the actual CSD usng Equaton 5. The two samples that le above the correlaton lne are more strongly curved that the other samples. (d) Characterstc length vs. ntercept of the regressed CSDs. Samples cannot le above the closure lmt. Although the Taranak samples are far from the closure lmt, there s a strong correlaton between characterstc length and ntercept. The prefx NZ has been omtted from the labels. (e) Volumetrc phase proporton vs. characterstc length. The dfferent groups of lavas are clearly separated n ths dagram. CONCLUSIONS AND RECOMMENDATIONS (1) Closure lmts form an easly calculated, convenent, and useful reference for CSD studes. They ndcate how much lberty s possble for CSD movement durng petrologcally mportant processes. They can also show f correlated characterstc lengths (or slopes) and ntercepts of straght CSDs are sgnfcant or not. Closure lmts should be ndcated on publshed CSDs where possble. (2) Calculaton of volumetrc phase proportons from CSDs can be used to verfy that CSDs have been correctly determned from two-dmensonal measurements. Common errors, such as n the determnaton of crystal shapes or by the use of napproprate equatons, can be easly recognzed and corrected. () It s commonly observed that there s a strong correlaton between the characterstc length and ntercept of CSDs. Restrcted varatons n volumetrc phase proportons can account for ths effect. A more nformatve dagram may be that of characterstc length vs. volumetrc phase proporton.

5 HIGGINS: CLOSURE IN CRYSTAL SIZE DISTRIBUTIONS 175 ACKNOWLEDGMENTS I thank other practtoners of quanttatve textural analyss for ther deas and presentatons, both professonals and students. Comments by Sarah Barnes, Jula Hammer, and Alan Boudreau mproved ths paper. These deas were ntated durng a sabbatcal at Unversté Blase-Pascal, Clermont-Ferrand, France. Ths research was funded by the Natural Scence and Engneerng Research Councl of Canada. REFERENCES CITED Cashman, K.V. (1990) Textural constrants on the knetcs of crystallzaton of gneous rocks. In J. Ncholls, and J.K. Russell, Eds., Modern methods of gneous petrology: understandng magmatc processes, p Revews n Mneralogy, Mneralogcal Socety of Amerca, Washngton, D.C. Cashman, K.V. and Marsh, B.D. (1988) Crystal sze dstrbuton (CSD) n rocks and the knetcs and dynamcs of crystallsaton II. Makaopuh lava lake. Contrbutons to Mneralogy and Petrology, 99, Delesse, M.A. (1847) Procedé méchanque pour détermner la composton des roches. Comptes Rendus de l academe des scences (Pars), 25, Hammer, J.E., Cashman, K.V., Hobltt, R.P., and Newman, S. (1999) Degassng and mcrolte crystallzaton durng pre-clmactc events of the 1991 erupton of Mt. Pnatubo, Phlppnes. Bulletn of Volcanology, 60, Hggns, M.D. (1994) Determnaton of crystal morphology and sze from bulk measurements on thn sectons: numercal modellng. Amercan Mneralogst, 79, (1996) Crystal sze dstrbutons and other quanttatve textural measurements n lavas and tuff from Mt Taranak (Egmont volcano), New Zealand. Bulletn of Volcanology, 58, (1999) Orgn of megacrysts n grantods by textural coarsenng: A Crystal Sze Dstrbuton (CSD) Study of Mcroclne n the Cathedral Peak Granodorte, Serra Nevada, Calforna. In C. Fernandez, and A. Castro, Eds., Understandng Grantes: Integratng Modern and Classcal Technques. Specal Publcaton 158, p Geologcal Socety of London, London. (2000) Measurement of Crystal Sze Dstrbutons. Amercan Mneralogst, 85, Marsh, B. (1988) Crystal sze dstrbuton (CSD) n rocks and the knetcs and dynamcs of crystallzaton I. Theory. Contrbutons to Mneralogy and Petrology, 99, (1998) On the nterpretaton of Crystal Sze Dstrbutons n magmatc systems. Journal of Petrology, 9, Marsh, B.D., Hggns, M.D., and Pan, Y. (2002) Inherted correlaton n crystal sze dstrbuton. Geology, n press. Pan, Y. (2001) Inherted correlaton n crystal sze dstrbuton. Geology, 29, Peterson, T.D. (1996) A refned technque for measurng crystal sze dstrbutons n thn secton. Contrbutons to Mneralogy and Petrology, 124, MANUSCRIPT RECEIVED JUNE 1, 2001 MANUSCRIPT ACCEPTED SEPTEMBER 1, 2001 MANUSCRIPT HANDLED BY ROBERT F. DYMEK