Fe-tremolite. Fet. CazFesSi8Ozz(OH) 2. 163.5. 6.759. Pargasite. Par. NaCazMg~A13Si6Ozz(OH)2. 160.0. 6.537 a ~o in cal/mol-deg b ~ in cal/bar. #N2o: see text ...
Contributions to Mineralogy and Petrology
Contrib Mineral Petrol (1983) 83:348-357
9 Springer-Verlag 1983
Quantitative P-T Paths from Zoned Minerals: Theory and Tectonic Applications Frank S. Spear and Jane Selverstone Department of Earth and Planetary Science, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Abstract. An analytical approach to the analysis of zoning
profiles in minerals is presented that simultaneously accounts for all of the possible continuous reactions that may be operative in a given assemblage. The method involves deriving a system of simultaneous linear differential equations consisting of a Gibbs-Duhem equation for each phase, a set of linearly independent stoichiometric relations among the chemical potentials of phase components in the assemblage, and a set of equations describing the total differential of the slope of the tangent plane to the Gibbs free energy surface of solid solution phases. The variables are the differentials of T, P, chemical potentials of all phase components, and independent compositional terms of solid solution phases. The required input data are entropies, volumes, the compositions of coexisting phases at a reference P and T, and an expression for the curvature of the Gibbs functions for solid solution phases. Results derived are slopes of isopleths (dP/dT, dJ2/dT or dX/dP) which can be used to contour P - T diagrams with mineral composition. To interpret mineral zoning, T and P can be expressed as functions of n independent composition parameters, where n is the variance of the mineral assemblage. The total differentials of P and T are differential equations that can be solved by finite difference techniques using the derivatives obtained from the analytical formulation of phase equilibria. Results calculated from Zone I and Zone IV garnets of Tracy et al. (1976) indicate that Zone I garnets grew while T increased (A T ~ + 72 ~ C) and P decreased sharply (AP,~--3 kb). Zone IV garnets zoned in response to decreasing T ( A T ~ - - 1 7 ~ C) and P ( A P ~ - I kb). A P - T path calculated for a zoned garnet from the Greinerschiefer series, western Tauern Window, Austria, also indicates growth during decompression ( A P ~ - - 3 kb) and heating (AT~ + 15 ~ C). A P - T path calculated for the Wissahickon schist (Crawford and Mark 1982) indicates growth during cooling and compression ( A T ~ - 2 5 ~ C, A P ~ + 2.2 kb). The calculated P - T paths differ according to structural environment and can be used to relate mineral growth to tectonic processes.
Introduction
One of the principal goals of metamorphic petrology is deciphering the P - T evolution of metamorphic rocks. Geothermometry and geobarometry have contributed
Offprint requests. F.S. Spear
greatly to our understanding of peak metamorphic conditions; however, it is clear that the "snapshot" view of metamorphism derived from geothermometric/barometric studies is oversimplified in that metamorphism is not an instantaneous event, but an evolutionary process. The array of P - T conditions deduced from a sequence of metamorphic rocks cropping out in the field (the "metamorphic geotherm", using the terminology of England and Richardson 1977) may, in fact, be quite different from the P - T paths followed by individual samples. For example, in the thermal model of England and Richardson (1977) for overthrust terranes, the P - T path of individual samples is a clockwise loop, which intersects the "metamorphic geotherm" at a relatively high angle. In contrast, P - T paths of samples from terranes dominated by large scale igneous intrusives may be characterized by counterclockwise loops. Clearly, knowledge of P - T loops for individual samples collected across an orogenic belt would provide considerable insight into the tectonic processes responsible for orogenesis, and the relationship between metamorphism and heat flow. There are many ways that P - T loops can be deduced for metamorphic rocks. Geothermometry/barometry and phase equilibria considerations can provide information on peak metamorphic conditions. Analysis of retrograde textures and late-stage fluid inclusions can help delineate the P - T conditions of the uplift path. Studies of mineral zoning and mineral inclusions often provide information on portions of the pre-preak crystallization history. This report focuses on the last of these methods: deducing P - T paths of individual samples from mineral zoning studies. Mineral zoning is generally thought to result from the interaction of two processes: (1) porphyroblast growth or exchange via continuous reactions (e.g. Loomis 1975; Tracy et al. 1976; Thompson et al. 1977; Tracy 1982); and (2) diffusional processes (e.g. Anderson and Olimpio 1977; Lasaga et al. 1977; Tracy and Dietsch 1982). This paper primarily addresses the first process, mineral zoning via a reaction mechanism, with specific reference to garnet zoning. The effects of diffusional processes will be considered later. Modeling of garnet zoning by a reaction mechanism has been discussed by a number of authors (see Tracy 1982, for a review). Qualitative and semiquantitative models have been presented by Thompson (1976), Tracy et al. (1976), Trzcienski (1977), and Thompson et al. (1977) (the Rayleigh fractionation model of Hollister, 1966, is also a type of reaction model). Quantitative treatments have been given by Loomis (1975; 1982) and Loomis and Nimick (1982).
349 The model presented in this report differs from those mentioned above in that it is quantitatively rigorous and it simultaneously accounts for all of the possible continuous reactions that may be operative in a mineral assemblage. It is also easily expandable to account for systems containing any number of components and with any variance.
d(,u2- ~1) = F(T, P, X2). These equations take the form (for a binary solution): 0 = -- (d/22 - d# 1) -- (82 - - S1) dT+ (~'2 - ~'1)dP
(4)
Theory
The model involves analytically formulating the phase equilibria of a given mineral assemblage in such a way that changes in the composition of coexisting minerals in the assemblage can be monitored as functions of changing T and P. The procedure has been discussed in detail by Rumble (1974, 1976) and Spear et al. (1982) and will be only briefly reviewed. Consider a mineral assemblage containing P phases in a rock that can be described by NC system components. For each phase there is a Gibbs-Duhem equation of the form 0 = S d T - ~dP + Y, X~dl2~
(1)
(where i refers to the phase components in the phase) that provides a constraint on the system. There are also conditions of heterogeneous equilibrium that must be satisfied. These equations take the form 0=X
!2it i
(2)
where vi is the stoichiometric coefficient of component i in a balanced reaction relationship among the phase components and/~f is the chemical potential of component i. These equations can be differentiated to yield 0=2
vld#i.
(3)
There may be a large number of possible equilibrium constraints such as (3), but not all will, in general, be linearly independent. To formulate analytically the phase equilibria of a mineral assemblage, only a linearly independent set of equilibrium conditions is needed because all other equilibrium conditions (and possible continuous reactions) can be derived from this linearly independent set. The number of linearly independent equilibrium constraints (NR) is given as: N R = N P - NC Where NC is the number of system components and NP is the number of phase components (Thompson 1982) (the number of phase components describes the permissible chemical variability of a phase; for example, NP = 4 in a quaternary garnet). Derivation of a linearly independent set of equilibrium conditions [Eq. (3)] can be done using the algorithm of Thompson (1982), which utilizes a GaussJordan reduction to determine the row-nullity and columnnullity of the system of mass balance equations that define the phase components in terms of the system components. A third set Of equations introduces the derivatives of measurable compositional variables, dX r These are not necessary to describe the phase equilibria of the rock, but are required to monitor compositional changes 'as a function of P and T. The equations are derived by considering the total derivative of the slope to the tangent plane to the Gibbs free energy surface of the phase of interest. The slope of the tangent is a function of P, T and a linearly independent set of compositional terms. That is, for a binary solid solution
Where S~ and V~ are partial molar entropies and volumes and (~2G/~X2)v ' T is the second derivative of the Gibbs function of the phase with respect to composition. For ternary and higher order solid solutions, additional terms involving cross curvatures of the Gibbs function are also required (see Spear et al. 1982). Collectively, Eq. (1), (3) and (4) comprise a system of simultaneous linear differential equations, the variables of which are the differentials of T, P, chemical potential and composition. Solution of these equations can yield the slopes of isopleths of mineral composition, (~P/~T) x, or the change in mineral composition with temperature at constant P, (~X/~T)p, or with pressure at constant T, (~X/~P)r. The type of result that can be derived depends on the variance of the mineral assemblage because the variance of the system of equations is the same as the variance of the mineral assemblage. If the assemblage is invariant, the only solution is the null solution, dT= dP= dJ21=dX,.= O. If the assemblage is univariant, a dP/dT slope can be derived, as can a dX/dT or dX/dP slope representing the change in mineral composition along the univariant curve. In divariant regions it is necessary to hold one parameter constant to obtain a solution, i.e., the method solves for partial differentials such as (OP/OT)x or (~X/OT) e. In assemblages of higher variance it is necessary to hold additional parameters constant to obtain solutions. As an example, consider the assemblage garnet ( G a r ) + biotite (Bio) + kyanite (Kya) + muscovite (Mus) + quartz (Qtz)+fluid (H20) in the system S i O 2 - A 1 2 0 3 - F e O MgO - K 2 0 - H 2 0 (KFMASH). Garnet and biotite are assumed to be binary F e - M g solid solutions and all other phases are pure, fixed compositions; solid solution in muscovite could also be taken into account, but has been ignored for the sake of simplicity here (see Table 2 for a list of mineral compositions used in this paper). There are a total of 6 system components and 8 phase components so the number of independent equilibrium conditions is 2. The system of equations for this assemblage is shown in Table 1. The first 6 equations are Gibbs-Duhem equations, one for each phase. The next two equations are the conditions of heterogeneous equilibrium and the last two equations are the total differentials of the tangent line to the G-surfaces for garnet and biotite. Because there are 10 equations and 12 unknowns, the variance of the system of equations is 2, the same as the variance of the mineral assemblage. Solutions can be obtained for the slopes of isopleths of constant garnet composition, (~P/~r)Zl~Alm, or biotite composition, (~P/~T)XAnn, and for the spacing between these isopleths, (OXAlm/OT) e o r (~XMm/OP)T. The necessary input parameters to solve the system of equations are 1) the compositions of coexisting phases at a reference P and T; 2) the molar entropies and volumes of these phases at the reference P, T and composition; 3) the partial molar entropies and volumes of the components of the solid solution phases; and 4) an expression for the curvature of the Gibbs free energy surface
350 Table 1. System of homogeneous equations for the assemblage garnet (Gar) + biotite (Bio) + kyanite (Kya) + quartz (Qtz) + mus-
covite ( M u s ) + H 2 0 ) in the system K F M A S H -~Qt~
-
V Qtz
1
0
0
0
0
0
0
0
0
0
dT
0
~us
_~u~
0
1
0
0
0
0
0
0
0
0
dP
0
S-KYa
-- [~Kya
0
0
1
0
0
0
0
0
0
0
d#Q tz
0
~n2o
-- 17H2~
0
0
0
1
0
0
0
0
0
0
d# M.s
0
V Gar
0
0
0
0
0
0
0
d # Kya
0
0
0
0
0
yGar ~ Aim 0
0
-- ~7Bi~
xGar Pyr 0
yBio ~Phi
yBio 0 zx Ann
0
d#n2 ~
0
1
0
0
0
0
0
1
0
0
d HGar P~Pyr d Gar #Aim
~Gar ~Bio
- -
0
0
2
-2
4
0
0
0
2
-2
4
0
-2 0
0 -2
0
{~2 ~Gar~
o-
2o.)
o
o
o
o
o
o
o
o
-1
1
0
0
0
-1
0
1
T
0
0
Bio
0
I
Bio
d#Ann
0
d X Ga' Aln _
d X Bi~ Am
Table 2. Thermodynamic data
Phase components
Abbreviation
Formula
go,
Quartz Muscovite K-feldspar Water Kyanite Sillimanite Andalusite Anorthite Albite Pyrope Almandine Spessartine Grossular Phlogopite Annite Mn-biotite 14A-Chlinochlore 14A-Daphnite Mg-staurolite Fe-staurolite Mn-staurolite Tremolite Fe-tremolite Pargasite
Qtz Mus Ksp HzO Kya Sil And Ano Alb Pyr Alm Sps Gro Phi Ann Mnb Chl Dap Mgs Fes Mns Tre Fet Par
SiO2 KA13Si301 o(OH)2 KA1Si308 H20 A12SiO5 A12SiOs AlzSiO 5 CaA12SizO 8 NaA1Si30 s Mg3AlzSi3Oa2 Fe3AlzSi3012 Mn3AlzSi3012 Ca3A12Si3012 KzMg6AlzSi6Ozo(OH)4 KzFe6AlzSi602o(OH)4 K2Mn6A12Si6Ozo(OH)4 MgloA14Si602o(OH) l 6 F%oA14Si6Ozo(OH) 16 Mg2A19Si3.75022(OH) 2 FezA19Si3.75022(OH)z Mn2A19Si3.75022(OH)2 CazMg 5Sis O 22(OH)2 CazFesSi8Ozz(OH) 2 NaCazMg~A13Si6Ozz(OH)2
9.88 68.80 51.13 45.10 20.00 23.13 22.20 49.10 49.51 61.162 75.60 74.50 60.87 152.22 190.20 201.00 222.4 285.0 96.2 109.0 112.60 131.19 163.5 160.0
/7ob 0.542 3.363 2.602 ~ 1.054 1.193 1.232 2.4091 2.3961 2.7036 2.7552 2.8180 2.9950 7.1538 7.3764 7.6104 9.901 10.202 5.294 5.363 5.44 6.523 6.759 6.537
a ~o in cal/mol-deg
b ~ in cal/bar #N2o: see text for discussion for solid solution phases. Derivation of t h e r m o d y n a m i c quantities used in calculations are presented in the next section. It should be pointed out that this m e t h o d o l o g y differs from m o r e conventional t h e r m o d y n a m i c treatments, such as those employing K D expressions, by using the differential forms o f the t h e r m o d y n a m i c equations rather than the integrated forms. Thus, this procedure determines changes in T and P as a function o f changes in mineral chemistry, rather than absolute values o f T and P as a function o f composition. It is also for this reason that enthalpy d a t a are not required, which is a considerable advantage as these
are usually the most p o o r l y constrained t h e r m o d y n a m i c data. If, however, a reference set o f mineral compositions at a reference P a n d T are known, they can serve, in a way, as the constant o f integration for the t h e r m o d y n a m i c equations. In this manner, absolute values of P and T can be derived from the analysis. T h e r m o d y n a m i c data
Phase compositions, m o l a r entropies and molar volumes at 1 b a r and 298 ~ K are listed in Table 2. All o f the d a t a were taken from the compilation o f Helgeson et al. (1978),
351 with the following exceptions. An entropy value for Fe staurolite was taken fore Pigage and Greenwood (1982; average of range for model D). Mg staurolite entropy was estimated assuming A S ( F e - M g ) = 6 . 4 eu/atom. Entropies and volumes for Mn biotite, Mn chlorite and Mn staurolite were estimated by assuming A g ( F e - M n ) = - 1.8 eu/atom and A IT"(Fe- Mn) = - 0.039 cal/bar-atom. These values represent the average for F e - M g and F e - M n exchange between several pairs of silicate minerals. The entropy of pyrope was calculated by subtracting SOlm, NOun, and S~ from the experimentally determined AS for the garnet-biotite exchange reaction of Ferry and Spear (1978). l?n~o at the initial P and T was taken from Burnham et al. (1969) for the construction of Figs. 1-3, and a modified RedlichKwong equation of state (Holloway 1977; Flowers 1979) was used to increment VH~o along the calculated P - T paths. All molar entropies and volumes used at T and P for the solid phases were assumed to be the standard state values; that is, ACp = e = f l = 0. Molar volumes were calculated as linear combinations of end member molar volumes,
Contouring P - T Space One application of the results obtained from the analytical formulation of phase equilibria is the construction of quantitative T - X (at constant P) and P - X (at constant 7) diagrams (e.g. Thompson 1976), or a projection of the two onto the P - T plane. P - T diagrams contoured with mineral composition can be quite useful for deducing P - T paths from mineral zoning profiles. The assemblage garnet + biotite + AlzSiO 5 + quartz + muscovite + H 2 0 _+plagioclase has been chosen for analysis because (1) it is a common assemblage of pelitic rocks observed over a wide P - T range and (2) continuous reactions in this system do not evolve H20 and it is therefore difficult to ascertain the behavior of this assemblage with respect to P and T (e.g. see Thompson 1976). This assemblage will be examined in the context of three chemical systems, K F M A S H , K N C F M A S H (i.e. + C a + N a ) and K N C F M M A S H (i.e. + Mn).
KFMASH System
or
P=x x,~, Molar entropies of solid solution phases were calculated assuming ideal mixing. S=Z XiS~
Z X~ lnX i
where n is the site multiplicity. Differences between the partial molar entropies between phase components were computed by recognizing that ( S c - S ) is the slope on an S - X diagram, which is (SS/~X)P, T, Xk~ j. This yields (S~ - S ) -- ~o _ So _ n R (ln X f - in Xj) The quantity (12i - l?) was computed as (I7~- 17~j). To calculate the curvature of the Gibbs free energy function, it was assumed that solid solution phases are ideal, so that
where Xj is the dependent compositional term (Xj= 1-Y~ Xi). Cross curvature terms, which are required in equations such as (4) for ternary and higher order solutions, were computed as
where X s is, again, the dependent compositional parameter. It should be pointed out that these simplifying assumptions do not seriously affect the results. The reason is that this method requires knowledge of the compositions of phases at a reference P and T (e.g. the T and P of last equilibration of the garnet rim) and then changes in composition away from this point are calculated. Because it is generally only necessary to integrate over several tens of degrees, 1-2 kbar and a few tenths of a mole fraction in composition, the final result is rather insensitive to errors in estimates of volumes, entropies and Gibbs function curvatures. Hence, another advantage of this methodology is that it provides quantitative results even with rather poorly constrained thermodynamic data and activity models.
In the K F M A S H system the assemblage G a r + B i o + AI2SiO 5 + Qtz + Mus + H20 is divariant and hence P - T space can be contoured for mineral composition directly. Results are presented in Fig. 1. The reference P and T was chosen to be 525 ~ C, 5 kb and was taken from Loomis and Nimick (1982) who show these to be the conditions of the discontinuous reaction garnet + chlorite = biotite + kyanite. The composition of garnet at this point was chosen to be XAlm=0.87, as given by Loomis and Nimick (1982, Fig. 1). Biotite composition was chosen as Xann= 0.5 to be consistent with a temperature of 525 ~ C at 5 kb from the garnetbiotite geothermometer of Ferry and Spear (1978). Chlorite composition at this point was taken as Xch1= 0.4 from Loomis and Nimick (1982, Fig. 1). Hence the lower limit of the assemblage Gar + Bio + AlzSiO 5 + Qtz + Mus + H 2 0 is taken at the discontinuous reaction G a r + C h l = B i o + AlzSiO 5. The upper limit is taken at the upper thermal stability of muscovite+quartz (Althaus etal. 1970). It should be noted that the assemblage Gar + Bio + A12SiO 5 + Q t z + M u s + H 2 0 is probably metastable with respect to other assemblages containing staurolite, chloritoid or cordierite over some of the contoured area of Fig. 1, but this does not affect the validity of the derived contours. The entire diagram of Fig. 1 A was constructed from only (1) a knowledge of the compositions of coexisting garnet, biotite and chlorite at the reaction Gar + Chl = Bio + Kya, (2) the T and P of this reaction, (3) the phase boundaries in the AlzSiO 5 phase diagram (taken from Holdaway 1971), (4) the position of the muscovite breakdown curve (taken from Althaus et al. 1970) and (5) the data in Table 2. First the slope of the univariant equilibrium Gar + Bio + Chl + Kya + Qtz + Mus + H 2 0 was computed and the curve contoured with XAI m and Xgnn. Slopes derived at the reference P and T are shown in Table 3. The univariant curve has a steep positive slope with XA~m and JfA,,n decreasing with increasing P and T. Using the slope (dX~/dP), the compositions of garnet, biotite and chlorite were calculated at the sillimanite-kyanite boundary; the slope of the univariant curve in the sillimanite field was calculated using sillimanite rather than kyanite as the AlzSiO 5 polymorph. As can be seen in Fig. 1 A, the slope is slightly less steep than in the kyanite field. This procedure was repeated on the
352
"%'
l
'
l
I j'
I
'
I
'
'
I
'
I
II
+oI 6
++
SiIP
/
/And ,' 5oo
5so
6oo
65o
7oo
1", ~
/
5001--'~ 1.0
I
0.8
Chr
T~
Ij,
0.6
XFe
I
0.4
Gaar +C~ i
I
0.2
i
0
Oil 1.0
,
rl Ksp 0.8
l
0.6 X Fe
~
l 0.4
diagram for the assemblage Gar+ Bio +A12SiO 5 + Q t z + M u s + H 2 0 in the system KFMASH contoured for XA~min garnet (solid lines) and JfAn, in biotite (dashed lines). Heavy solid lines are discontinuous reactions. A12SiO5 phase diagram from Holdaway (1971) and muscovite breakdown curve from Althaus et al. (1970). The low T discontinuous reaction is Gar+Chl=Bio + A12SiO5. Note that contour spacing for Bio is AXAn, = 0.02. For Gar, contours of AXA1~ are 0.01 for the assemblage Gar + Bio + AI2SiO 5. In the chlorite field A X A I m = 0.05 and above muscovite breakdown AXA~m = 0.01. B T - - X diagram at P = 5 kb constructed from Fig. 1 A. C P - X diagram at 600~ C constructed from Fig. t A Fig. l. A P - T
Table 3. Calculates slopes at the reference T and P (525 ~ C, 5 kb) Assemblage:
Gar + Bio + Chl + Kya + Qtz + Mus + H20 a (dP/dT) = 369 bar/deg (dP/dXAI~) = -76103 bar/mole fraction (dP/dXAnn) = -33816 bar/mole fraction
Assemblage:
Gar + Bio + Kya + Qtz + Mus + H20 a (OP/OT)XAI m = - 27.8 bar/deg (~P/~T)XAn n = 7.0 bar/deg (~P/~XAIm) T = -- 81839 bar/mole fraction (~P/~XA..)T = - 33171 bar/mole fraction
a XAlm = 0.8 7
XAnn = 0.5 0
XChl = 0 . 4 0
V~ o = 0.5045 cal/bar andalusite-sillimanite boundary and slopes were found to be even less steep in the andalusite field. Slopes for the divariant assemblage Gar + Bio + Kya + Q t z + M u s + H 2 0 were then computed (Table 3) and the kyanite field was contoured for JfA]m (solid lines) and XA. . (dotted lines). Isopleths for )[Aim show negative slopes whereas those for XAnn show positive slopes with both phases decreasing in Fe content with increasing P. This produces the unusual configuration shown on the T - X section (Fig. 1 B) where garnet becomes more Mg-rich and biotite becomes more Fe-rich with increasing T at constant P in the kyanite field the two phases approach each other in Fe/Mg. This result is consistent with the calculated T - - X section of Loomis and Nimick (1982, Fig. 1), but is contrary to that predicted by T h o m p s o n (1976), who concluded that both garnet and biotite should both become more Fe-rich in this paragenesis. Using the slopes shown in Table 3, the compositions of coexisting garnet and biotite on the kyanite-sillimanite boundary were computed and used to contour the sillimanite field. Slopes of both XAIm and XA.n isopleths in the sillimanite field are positive with Xve decreasing with increasing P, consistent with the prediction of T h o m p s o n (1976). This procedure was repeated at the andalusite-silli-
manite boundary and it was found that slopes of Zl/Alm isopleths in the andalusite field are practically flat with ( ~ P / ~ ~f) Z~(Al m = - - 0.44 bar/deg. Contours are also drawn for XA~m and XA. n for the assemblages Gar + Bio + Chl + Qtz + Mus + H 2 0 and Gar + B i o + A 1 2 S i O s + Q t z + K s p + H 2 0 . Isopleths are steep for these assemblages because both involve dehydration, with XVe decreasing with increasing T, as predicted by T h o m p s o n (1976). It should be pointed out that garnet isopleths for the chlorite-bearing assemblage are spaced approximately 5 times more closely than are those in the Ksp-bearing assemblage. This arises because for any given increment of T, the chlorite assemblage dehydrates to a greater extent that the Ksp-bearing assemblage. Figure 1 A clearly has considerable application in geothermometry/barometry as well as in deciphering garnet zoning profiles. The contours for XAI,, and XAn~ intersect at specific P - - T points, indicating that, for the system K F M A S H , the Fe/Mg of coexisting garnet and biotite in the assemblage Gar + Bio + A12SiO 5 + Q t z + Mus + H 2 0 uniquely define both the P and T o f crystallization. This is particularly true in the kyanite field where contours intersect at high angles. An analytical error of +_0.005 in XA]m and XA. n would convert to an estimated error in P and T of _+15~ 300bars in the kyanite field, _ I 0 ~ 200 bars in the sillimanite field and _+_15 ~ C, 150 bars in the andalusite field. Application of Fig. 1 to geothermometry/barometry is, however, predicated on the validity of the assumptions, discussed above, required to construct the diagram. Potentially serious problems arise because (1) the compositions of coexisting garnet, biotite and chlorite at the reference P and T may not be correctly specified and (2) the presence of other components in the system will shift the absolute positions of the isopleths. These problems, although seriously limiting the applicability of Fig. 1 A to geothermometry/barometry, do not affect its applicability in the interpretation of garnet zoning profiles. The reason is that, although the absolute positions
353 of the contours are subject to considerable error in assumptions made, the spacings between the isopleths are independent of these assumptions and are subject only to the validity of the activity models used and the assumption that ACp=O. Another diagram, identical to Fig. 1 but contoured with AXAIm and AXAnn could be constructed that would be completely valid even in the presence of additional components. Thus, if a garnet is zoned from rim to core in J~Alm (where XA~m= Fe/Fe + Mg) such that AXA~m = 0.02, then the P - T conditions of the rim and core must be separated by two contours such that AXA1m between the two contours is equal to 0.02. If the P and T of last equilibration of the rim is known, and if the garnet contains an inclusion of biotite in the core for which XA, . can be measured, then the unique temperature and pressure conditions of crystallization of the core can also be calculated.
ix \ \ \ ' \ \ \ , \
b / \ / \ / \ k \& - ~
500
System K N C F M A S H M a n y pelitic schists contain CaO and N a 2 0 as extra components and plagioclase as an additional phase. In these rocks, garnet contains the additional phase component, grossular. Modeling of the assemblage Gar + Bio + A12SiO 5 + Plag + Qtz + Mus + HzO has been chosen to determine the behavior of XGro in garnet and XA, o in plagioclase in such rocks. As in the first example, the reference P and T were taken to be 525 ~ C, 5.0 Kb. The biotite and chlorite compositions were assumed to be the same at XA.~=0.5 and Xch~= 0.4. The plagioclase composition was arbitrarily set at Xa,o=0.30 and XG~o was computed to be consistent with a pressure of 5.0 Kb at 525 ~ C from the calibration of the equilibrium a n o r t h i t e + 2 k y a n i t e + q u a r t z = 3 grossular by Ghent etal. (1979) (X~ro=0.08). The Fe/Mg in the garnet was then calculated to be consistent with a temperature of 525 ~ C from the calibration of Ferry and Spear (1978) ( X a l m = 0 . 8 0 , Xey r = 0.12). The resulting diagrams are shown in Figs. 2 and 3. Contours are shown for X'AIm (solid lines) versus X~o (dashed lines, Fig. 2) and XA, o (dashed lines, Fig. 3). Because assemblage Gar + Bio + A12SiO s + Plag + Qtz + Mus + H 2 0 is trivariant in the system K N C F M A S H , it is necessary to hold one variable constant in order to construct a contour diagram. In Fig. 2, XA,o was held constant and the diagram can be thought of as a slice through P - T - - X A , o space. In Fig. 3, XGro was held constant. Complete characterization of the assemblage would require a 3-dimensional representation. As can be seen in Fig. 2, isopleths of XAj m have positive slopes in all three aluminosilicate fields with X-AIm decreasing with increasing T. The difference in slope for these isopleths between Figs. 2 and 1 is directly a consequence of the change in 2"~ro in the garnet. However, even though contours for XA~~ in Fig. 2 show different slopes from those in Fig. 1, contours of Fe/(Fe + Mg) have slopes identical to those of Fig. 1. This can be seen in Fig. 3, which is a portion of the p2_ T diagram contoured by holding Xa~ o constant. I n this diagram, isopleths of Xa~m are identical to isopleths of F e / ( F e + Mg) and it can be seen that they have the same slope and spacing in Fig. 1, although the absolute positions of the isopleths are displaced because of the dilution by Xaro. Isopleths for Xc~ o in Fig. 2 have gentle positive slopes with Xr increasing with increasing P. The slopes of isopleths of XA, o in Fig. 3 are similar to those of Xc~o in Fig. 2 but with XA,o decreasing with increasing P.
~.,--"-
~ } , - -
7 -
~-'5----~
550
600
~
. ~
650
Ill
700
T, ~
Fig. 2. P - T diagram for the assemblage Gar + Bio + AI2SiO5 + Plag + Qtz + Mus + H20 in the system KNCFMASH. Contours are shown for XAXm (solid lines) and Xar o (dashed lines). Diagram is drawn at constant plagioclase composition (AXA.o= 0)
61 / ~ 1 " ~ 5
/
l
//
9
/
500
550
600
T, ~
Fig. 3. P - - T diagram for the assemblage Gar+Bio+AlzSiO 5 + Plag + Qtz + Mus + H 2 0 in the system KNCFMASH constructed at constant grossular composition (AXGro=0). The kyanite field only is shown. Contours are ~Alm in garnet (solid lines) and XAno in plagioclase (dashed lines) One of the assumptions made in constructing Fig. 1 is that the slopes of isopleths are constant over the P - T area being contoured. This is not a bad assumption where the P - T area being contoured is small and the mole fractions of components being contoured are large (i.e., no components are minor or trace components). In Fig. 2, it was found that this assumption was invalid for the grossular component in garnet and it was necessary to integrate numerically over the P - T diagram, reevaluating the slopes for each increment of P and T. The result is that the contours for Jk'~Grobecome more widely spaced as J(O,o becomes small (e.g. less than 0.05) in the garnet at low pressures. Figs. 2 and 3 can be used for interpreting garnet zoning profiles in the relevant assemblages. In general, decreasing XGro from core to rim signifies a decrease in P depending on how Xal m and XA. o vary. A mathematically rigorous procedure for handling assemblages with variance of 3 or more will be described in a later section. System K N C F M M A S H
Most pelitic garnets contain Mn as spessartine component and any application to real systems must include this vari-
354 able. The assemblage Gar + Bio + A12SiO 5 + Plag + Qtz + Mus + H 2 0 in the system K N C F M M A S H is quadravariant and a P - T contour diagram can only be constructed by holding two compositional variables constant. The possible choices are XAlm, XGro, XSps, XAno, XAnn and XM,BI SO there are 15 possible diagrams that can be drawn. A diagram drawn at constant X ~ o and Xsp, would look like Figs. 1 and 3; a diagram at constant Xsp~ and XAno would look like Fig. 2. Because of the high variance of this assemblage, contour diagrams are of limited use in interpreting garnet zoning profiles; the mathematical approach described in the next section is more generally applicable. Interpretation of Garnet Zoning: An Algebraic Approach
The discussion in the previous section was directed at deriving P - T diagrams contoured for mineral composition, which can be used in the interpretation of garnet zoning profiles. However, for assemblages with variance of 3 or more the graphical approach is inadequate to account simultaneously for all of the possible degrees of freedom of the assemblage. An algebraic approach is required. Consider an assemblage of variance n. There are n independent composition parameters that need to be specified to uniquely determine T and P. For example, the assemblage Gar + Bio + K y + Q t z + M u s + HzO in the system K F M A S H shown in Fig. I has a variance of 2 and it is necessary to specify two composition variables, e.g., XA~m and XAnn, to determine T and P. Therefore, T and P can be specified as functions of n composition parameters, or T = f (X~, X 2, X3 ... Xn), P = f ( X l , X2, X 3 . . . X,).
The total differential of T and P can be expressed as
and
This is the mathematical formalism necessary to calculate A T and A P for assemblages of any variance. The slopes, (OT/OX3x . and ( ~ P / S X i ) x . . , can be computed from the algebraic i~ormulatlon of ~}~ase equlhbrm discussed earlier. If these slopes are constant (i.e. not a function of P, T, and X ) then A T and A P can be calculated .j~
.
j
t
.
.
,
as
and
If the slopes are not constant, then A T and A P must be computed by a finite difference technique. The validity of the assumpition of constancy of slopes depends on the magnitude of the zoning (i.e. the magnitude of AXI) and the absolute value of X~ (i.e. the smaller the values of Xi, the more non-linear are the slopes; see, for example, contours of Xaro in Fig. 2). A computer program has been written to perform the finite difference approximation. For the ex-
amples that have been studied, convergence to within _+5~ C and _+50 bars occurred within 20 iterations. Results
Quantitative evaluation of garnet zoning profiles must be done by consideration of the specific assemblage and associated continuous reactions which proceeded in that assemblage. If a discontinuous reaction is encountered during garnet growth, this must be taken into account in the model. In addition, the final P and T of equilibration of the garnet rim must be known and a sufficient number of composition parameters (AXI) must be specified, commensurate with the variance of the assemblage. There are several examples in the literature of garnet zoning studies where the zoning profile and coexisting assemblage have been well documented although sufficient composition data are not always reported. Two examples from Tracy et al. (1976), one from Crawford and Mark (1982) and one from Selverstone et al. (in press), will be discussed to illustrate the potential of this technique and the types of data required for its application. Zone I garnets of Tracy et al. (1976) are believed to have formed from continuous reactions in the assemblage Gar+Bio+Sta+Plag+Qtz+Mus+H20 in the system K N C F M M A S H . Garnet, from core to rim, shows enrichment of ~Alm (AX'Alm= 0.09) and depletion of XCro(AXG,:o = --0.05) and Xsps(AXsps= -0.08) (Tracy et al. 1976), Fig. 2 and Table 3). Staurolite composition was assumed to be XFe=0.87, XMn = 0.01; the results are relatively insensitive to the actual staurolite composition chosen. The reference T and P for the garnet rim was specified at 600 ~ C, 6.0 Kb. Zone IV garnets of Tracy et al. (1976) are believed to display retrograde rims formed by continuous reaction in the assemblage Gar + Bio + Sill + Plag + Qtz + Ksp + H 2 0 during cooling. Garnet, from core to rim, shows enrichment in XAlm(AXAlm = 0.04) and Xsps(AXsp s = 0.01) and minor depletion in XC,r (AXcl. = --0.01). Because both of these assemblages are quadravariant, an additional composition parameter is required to monitor A T and AP. Because no additional composition data were reported, tests were run assuming AXA~ o in plagioclase was 0.0, + 0.05 and -0.05. The finite difference approximation was employed in each case.
The results given in Table 4 and Fig. 4 show that the calculated A T is rather insensitive to assumptions about AXAn o. Calculated values of AP, however, are more sensitive to assumptions about plagioclase zoning because of the strong correlation between XC~ro, XAno and P. The P - T path for Zone I garnets indicates growth with increasing T ( A T m 7 2 ~ C), consistent with the conclusions of Tracy et al. (1976). Growth during marked decompression (AP = --2.8 to --3.8kb) is also indicated. These calculated changes in P and T are consistent with the growth of Zone I garnets during the doming stage of the Bronson Hill anticlinorium as suggested by P. Robinson (personal communication). Zonation of Zone IV garnets occurred during cooling ( A T ~ - 1 7 ~ and decompression ( A P = - I . 8 to - 0.5 kbar), which is consistent with the interpretation that these are retrograde rims produced after the metamorphic peak. Crawford and Mark (1982) report evidence from the Wissahickon schist in southeastern Pennsylvania for two superimposed metamorphic events, an early high-tempera-
355 Table 4. A T and AP calculated from two garnet zoning profiles
of Tracy et al. (1976). AT and AP are core to rim values
!
Crawford
AXA, o= 0.05 AXA. o= 0.0 AXA, o = -- 0.05
AP (bars)
67 72 78
--2830 -3289 -- 3871
Zone IV Garnet: Gar + Bio + Sill + Plag + Qtz + Ksp + HzO A T (~
0.05
z~XAn o =
0,0
--16 - - 17 --18
dXa,o = --0.05 i
,
i
-
-
i/
/
1I
4
2
P
i
I
500 600 TEMPERATURE
I
700
Fig. 5. P-- T diagram showing the path followed during the growth
of garnet in sample 5a from Crawford and Mark (1982). P - T conditions for the early low-pressure (I) and later high-pressure (II) metamorphic events are also indicated 12
10
Selverstone
8
Zone I
et al.
10
~o. 6 42
/
8
AP (bars)
--1778 1203 -- 523
t
& Mark
400 AXA,o=
i
[19821
Zone I Garnet: Gar + Bio + Stau + Plag + Qtz + Mus + HzO d T (~
t
10
R
CORE
IM 8
~
Z
o
n
e
IV
~8 Tracy i
et al. 119761
400
i
I
500 600 TEMPERATURE
i
700
Fig. 4. P - T diagram showing P - T paths following during garnet growth for Tracy et al. (1976) Zone I and Zone IV garnets. Solid line is for the assumption of AXAn o = 0.0, dashed line for AXa, o +0.05 and dotted line for AXA,o= --0.05 ture ( T > 550 ~ C) and low-pressure ( P < 5 kbar) event followed by a later high-temperature (550-650 ~ C), mediumpressure (6.5-7 kbar) event. Sample 5a was used in the model calculation, which contains the assemblage garnet + biotite + sillimanite + orthoclase + quartz + plagioclase. The T and P estimated from the garnet rim composition are 580 ~ C, 5.5 kbar; the garnet is zoned with decreasing Xpyr and increasing X ' A l r n , X G r o and XspS from core to rim. The calculated P - T path, together with the estimates by Crawford and Mark (1982) for the P - T conditions for the first (I) and second (II) metamorphic events, are shown in Fig. 5. The path shown is for AXAno = 0. Calculations made assuming AXAn ~ = + 0.05 and - 0.05 change A T by ~ 1~ C and change A P by +_0.5 kb, respectively. Sample 5 a is from a low-pressure locality and is reported to have been little affected by later recrystallization. The calculated P - T path indicates garnet growth during minor cooling (A T = -- 25 ~ C) accompanied by increasing pressure ( A P = 1.7-2.6 kb). This P - T path is entirely consistent with the conclusions of Crawford and Mark (1972) and indicates that garnet growth occurred in response to the proposed overthrusting event. The fourth example is from a garbenschiefer sample from the Greinerschiefer series, western Tauern Window, Austria (Selverstone et al., in press). The assemblage is gar-
400
500 600 TEMPERATURE
Fig. 6. P - T diagram showing the path calculated from the zoning profile of a garnet from a hornblende-kyanite-staurolite garbenschiefer from the western Tauern Window, Austria (Selverstone et at., in press) Boxes indicate P - T conditions calculated from geothermometry/barometry for the garnet core and rim net + hornblende + staurolite + kyanite + chlorite + biotite + plagioclase + quartz. The garnet is zoned with Xar o and Jfsp~ decreasing and Xal m and Xpyr increasing from core to rim. Garnet-biotite geothermometry (Ferry and Spear 1978) and plagioclase(An34)-garnet-kyanite-quartz geobarometry (Ghent et al. 1979; Hodges and Spear 1982) indicate that the garnet rim equilibrated at 550 +_20 ~ C, 7 _+ 1 kbar. Inclusions o f more sodic plagioclase (Anls) within the garnet also permit calculation of the pressure of crystallization of the garnet core (P=9.5_+1 kbar at an assumed T of 550 ~ C). The calculated P - T path from garnet zoning is shown in Fig. 6. Because the assemblage is divariant, no assumptions about plagioclase zoning were necessary as input. As can be seen, the calculated path corresponds very well with the P - T conditions estimated for the garnet rim and core from geothermometry and geobarometry; the kink reflects a change in the curvature of the zoning profiles midway across the garnet. In addition, the model permits calculation of the expected plagioclase composition that should be in equilibrium with the core of the garnet. The calculated value
356 is Anl6 , in excellent agreement with the observed plagioclase composition in the core of the garnet of An18. Discussion
Although the method presented above is completely general, P - T paths calculated from this technique involve simplifications and assumptions that may affect the results. Specifically, these include the effects of (1) the development of zoning profiles in response to resorption versus growth processes; (2) failure to maintain equilibrium between phases during mineral growth; (3) diffusional reequilibration; (4) minor components not taken into account; and (5) the use of simplified solution models. Growth versus resorption. The primary assumption of the technique is that zoning profiles preserve a record of continuous reactions that proceeded during mineral growth under conditions of changing P and T. Garnet growth, as opposed to resorption, is therefore assumed. Although this is probably reasonable for the Zone I garnet of Tracy et al. (1976) and the examples from Crawford and Mark (1982) and the Tauern Window, it is clear that not all zoning profiles are the result of porphyroblast growth. In particular, Zone IV garnets of Tracy et al. (1976) were probably completely homogenized by internal diffusion processes at peak metamorphic temperatures. The observed zoning profiles must therefore be the result of diffusional exchange and/or resorption during cooling and do not reflect garnet growth. As long as the garnet was completely unzoned at peak P - T conditions, however, the technique described in this paper accurately predicts the total changes in P and T over which the observed zoning profile developed, regardless of whether this profile resulted from growth, diffusion, or resorption. In the latter two cases, however, the curvature of the profile will not reflect the true P - T path followed by the rock; only the core and rim endpoints will provide accurate P - T information. Where there is evidence of garnet resorption, therefore, only the total values of A T and AP between core and rim should be determined, rather than a path based on the entire zoning profile. Disequilibrium. Another requirement is that local equilibrium was maintained between the edges of all phases at all times. This assumption is difficult to evaluate and impossible to prove. One expected result of disequilibrium, however, would be the formation of depletion haloes in the matrix around garnet; such haloes are not observed, to our knowledge, in any of the examples discussed here. Diffusion. In general, partial homogenization of garnet by volume diffusion at high temperatures will simply result in shortening of the calculated P - T path by reducing the absolute values of the A X i parameters. If all species within the garnet diffuse at the same rate, the orientation of the P - T vector will remain unchanged. If, on the other hand, species diffuse at vastly different rates or if cross coefficient terms are important, the direction of the calculated P - T path will deviate from the true path followed by the rock. Such deviation will also result if garnet is partially homogenized but another monitor phase, such as plagioclase, retains its complete zoning profile. Diffusional reequilibration is probably not a significant problem for samples from the amphibolite facies, but becomes increasingly important at
high temperatures. A study of the magnitude of the effects of diffusion on calculated paths is in progress. Minor components. In theory, minor components such as Ti in mica or Mn in ilmenite are easily accounted for by the technique presented here. In practice, however, this proves to be more difficult owing to lack of thermodynamic data on the relevant endmember components and the absence of adequate multicomponent solution models. The effects of such components on calculated P - T paths will, in any case, be minor as long as the host phases show no significant zonation in these components. For example, calculated paths where MnO was included in staurolite and biotite showed agreement to within a few degrees and a few tens of bars of paths calculated when this component was ignored. Solution models. Because this technique utilizes the differential rather than the integrated forms of the thermodynamic equations, the effects of using simple versus complex solution models for the phases are minimized. For the trials we have run, use of an ideal mixing model versus a Margules formulation for garnet has had no significant effect upon the derived P - T paths. While more complex and accurate models are desirable, their effects upon calculated P - T paths and the resultant tectonic interpretations will be relatively minor. A rigorous error analysis incorporating all of these effects is beyond the scope of this paper. However, an empirical estimate of the magnitude of the errors resulting from the above assumptions can be obtained by comparison of the calculated P - T paths with independent estimates based on geothermometry/barometry. For the examples where such independent data exist (Crawford and Mack 1982; Selverstone et al., in press; see Figs. 5 and 6), the good agreement between the two techniques suggests that the cumulative effect of these uncertainties is small. Conclusions
Results from the four examples discussed above indicate that quantitative P - T paths can be calculated from garnet zoning profiles from the approach outlined in this report. Moreover, for the examples from Crawford and Mark (1982) and Selverstone et al. (in press), the results are entirely consistent with results obtained by independent means, lending considerable credence to the viability of the technique. Perhaps the most important result of this type of modeling is that it enables distinction among different tectonic processes by examination of the P - T path followed during mineral growth. The marked decompression path followed by the Zone I garnets of Tracy et al. (1976) suggests considerable unloading during the doming phase. Similar unloading is implied by the path followed by the Greinerschiefer series in the western Tauern Window. Moreover, both paths indicate heating during decompression. These P - T paths support the model of England and Richardson (1977) whereby P - T paths for metamorphic rocks are controlled by uplift and erosion. The P - T path followed by the Wissahickon schist (Crawford and Mark 1982) reveals cooling in conjunction with loading and is consistent with a tectonic environment characterized by emplacement of a cool thrust sheet over hot rocks.
357 It is clear that the m e t h o d o l o g y discussed in this report is not restricted to analysis o f garnet zoning from m e t a m o r phic environments, but m a y have useful application in other areas. F o r example, studies of zoned amphiboles, garnets and pyroxenes from blueschist/eclogite terranes might reveal P - T paths during suduction a n d / o r subsequent emplacement in the crust. Changes in other intensive variables such as fH2o or fo2, in addition to P and T, can also be monitored, as has been done in studies by R u m b l e (1978), F e r r y (1979), G r a m b l i n g (1981) and Spear (1977). The same technique can be applied to the interpretation of zoned minerals from mantle xenoliths and the resulting P - T paths might shed light on mantle tectonics. Applications to the interpretation o f phenocryst zoning in igneous rocks are also possible. In conclusion, we wish to reemphasize the importance of determining P-Tpaths along which rocks have recrystallized and reequilibrated during metamorphism. Only by detailing complete P - T loops for rocks across an orogenic belt can the dynamics o f tectonics, heat flow and orogenesis be fully understood.
Acknowledgements. The careful reviews and comments of J. Ferry, B.W. Evans, J. Cheney, J. Rice, D. Hickmott and L.S. Hollister are gratefully acknowledged. Financial support was provided by a grant from the National Science Foundation (81-08617-EAR), a Joseph H. DeFrees grant of the Research Corporation, and an NSF Graduate Fellowship. References Althaus E, Karotke E, Nitsch KH, Winkler H G F (1970) An experimental reexamination of the upper stability limit of muscovite plus quartz. Neues Jahrb Mineral Monatsh: 325-336 Anderson DE, Olimpio JC (1977) Progressive homogenization of metamorphic garnets, South Morar, Scotland: evidence for volume diffusion. Can Mineral 15:205-216 Burnham CW, Holloway JR, Davis NF (1969) Thermodynamic properties of water to 1000 ~ C and 10,000 bars. Geol Soc Amer spec pap 132, pp 96 Crawford ML, Mark LE (1982) Evidence from metamorphic rocks for overthrusting, Pennsylvania Piedmont, USA. Can Mineral 20:330-347 Dietvorst EJL (1982) Retrograde garnet zoning at low Water pressure in metapelitic rocks from Kemio, SW Finland. Contrib Mineral Petrol 79:37-45 England PC, Richardson SW (1977) The influence of erosion upon the mineral facies of rocks from different metamorphic environments. J Geol Soc Lond 134:201-213 Ferry JM (1979) A map of chemical potential differences within an outcrop. Am Mineral 64:966-985 Ferry JM, Spear FS (1978) Experimental calibration of the partitioning of Fe and Mg between biotite and garnet. Contrib Mineral Petrol 66:113-117 Flowers GC (1979) Correction of Holloway's (1977) adaptation of the modified Redlich-Kwong equation of state for calculation of the fugacities of molecular species in supercritical fluids of geologic interest. Contrib Mineral Petrol 69, 315-318 Ghent ED, Robbins DB, Stout MZ (1979) Geothermometry, geobarometry, and fluid compositions of metamorphosed calc-silicates and pelites, Mica Creek, British Columbia. Am Mineral 64:874-885 Grambling J (1981) Kyanite, andalusite, sillimanite, and related mineral assemblages in the Truchas Peaks region, New Mexico. Am Mineral 66 : 702-722 Helgeson HC, Delany JM, Nesbitt HW, Bird DK (1978) Summary and critique of the thermodynamic properties of rock-forming minerals. Am J Sci 278-A: 1229
Hodges KV, Spear FS (1982) Geothermometry, geobarometry and the AlzSiO s triple point at Mt. Moosilauke, New Hampshire. Am Mineral 67 : 1118-1134 Holdaway MJ (1971) Stability of andalusite and the aluminum silicate phase diagram. Am J Sci 271:97-131 Holloway JR (1977) Fugacity and activity of molecular species in supercritical fluids. In: DG Fraser, ed., Thermodynamics in Geology. D Reidel, Holland 161-181 Hollister LS (1966) Garnet zoning: an interpretation based on the Rayleigh fractionation model. Science 154:1647-1651 Lasaga AC, Richardson SM, Holland HD (1977) The mathematics of cation diffusion and exchange between silicate minerals during retrograde metamorphism. In: SK Saxena and S Bhattacharji, eds, Energetics of Geological Processes, 354-387, Springer-Verlag, New York Loomis TP (1975) Reaction zoning of garnet. Contrib Mineral Petrol 52:285-305 Loomis TP (1982) Numerical simulation of the disequilibrium growth of garnet in chlorite-bearing aluminous pelitic rocks. Can Mineral 20:411423 Loomis TP, Nimick FB (1982) Equilibrium in M n - - F e - M g aluruinous pelitic compositions and the equilibrium growth of garnet. Can Mineral 20:393-410 Pigage LC, Greenwood HJ (1982) Internally consistent estimates of pressure and temperature: the staurolite problem. Am J Sci 282: 943-969 Rumble D, III (1974) Gibbs phase rule and its application in geochemistry. J Wash Acad Sci 64:199-208 Rumble D, III (1976) The use of mineral solid solutions to measure chemical potential gradients in rocks. Am Mineral 61:1167-1174 Rumble D, III (1978) Mineralogy, petrology, and oxygen isotopic geochemistry of the Clough Fol:mation, Black Mountain, western New Hampshire, USA J Petrol 19:317-340 Selverstone J, Spear FS, Franz G, Morteani G (in press) Highpressure metamorphism in the SW Tauern Window, Austria: P - T paths from hornblende-kyanite-staurolite schists. J Petrol. Spear FS (1977) Phase equilibrium of amphibolites from the Post Pond Volcanics, Vermont. Carnegie Institute of Washington Yearbook 76: 613-619 Spear FS, Ferry JM, Rumble D, III (1982) Analytical formulation of phase equilibria: the Gibbs Method. In: JM Ferry, ed, Characterization of Metamorphism through Mineral Equilibria. Min Soc Am Rev in Mineral 10:105-152 Thompson AB (1976) Mineral reactions in pelitic rocks: I. Prediction of P-T-X(Fe-Mg) phase reactions. Am J Sci 276:401424 Thompson AB, Tracy RJ, Lyttle P, Thompson JB, Jr (1977) Prograde reaction histories deduced from compositional zonation and mineral inclusions in garnet from the Gassetts schist, Vermont. Am J Sci 277:1152-1167 Thompson JB (1982) Composition space: an algebraic and geometric approach. In: JM Ferry, ed, Characterization of Metamorphism through Mineral Equilibria. Mineral Soc Am Rev in Mineral 10:1 31 Tracy RJ (1982) Compositional zoning and inclusions in metamorphic minerals. In: JM Ferry, ed, Characterization of metamorphism through Mineral Equilibria. Mineral Soc Am Rev in Mineral 10:355-397 Tracy RJ, Dietsch CW (1982) High temperature retrograde reactions in pelitic gneiss, central Massachusetts. Can Mineral 20:425437 Tracy RJ, P Robinson, Thompson AB (1976) Garnet composition and zoning in the determination of temperature and pressure of metamorphism, central Massachusetts. Am Mineral 61 : 762-775 Trzcienski WE, Jr (1977) Garnet zoning - product of a continuous reaction. Can Mineral 15:250-256 Received March 10, 1983; Accepted May 23, 1983
