PHYSIC8 CHEMISTRY MINERAIS

Phys Chem Minerals (1985) 12:217-222

© Springer-Verlag 1985

A1, Si Ordering in Cordierite Using "Magic Angle Spinning" NMR. II: Models of AI, Si Order from NMR Data Andrew Putnis and Ross J. Angel Department of Earth Sciences, University of Cambridge, Cambridge, England, CB2 3EQ

Abstract. A method of determining the number of A1 - O - A 1 bonds per unit cell from 29Si nuclear magnetic resonance (NMR) data of synthetic cordierites with increasing Si, A1 order is described. The number of A 1 - O - A 1 bonds is found to vary linearly with the logarithm of the annealing time. This may be correlated with previously published heat of solution data on similar samples (Carpenter et al. 1983) to determine the enthalpy change A h, associated with a single AI~--Si interchange in cordierite. Ah is found to be 8.1 kcal/mole. The N M R data show that the short range A1, Si order cannot be described in terms of twin domains of ordered orthorhombic cordierite. An ordering model derived from group theoretical constraints on possible A1, Si distributions within the hexagonal symmetry of the cordierite is found to provide a better fit to the N M R data.

Introduction In the previous paper (Putnis et al. 1985 hereafter referred to as Paper I) silicon-29 nuclear magnetic resonance (NMR) data were published for a sequence of progressively ordered synthetic cordierites, and the changes in the local Si environment (in terms of the number of A1 nearest neighbours, Si(n A1)) determined as a function of annealing time. In this paper the data in Table 2 of paper I are considered in further detail, and it is shown how the linkages between T1 and Tz sites in the cordierite structure enable various models of A1, Si distributions to be directly compared with the observed N M R data. The data also define the number of A 1 - O - A 1 bounds per unit-cell, which may be correlated with heat of solution data from the same material (Carpenter et al. 1983) to estimate the enthalpy change associated with a single AI~--Si interchange in cordierite.

Tetrahedral Linkages in Cordierite Cordierite differs from many other aluminosilicate structures which have previously been investigated by solid state NMR, such as zeolites (Fyfe et al. 1983) in that the framework topology allows the direct calculation of the number of certain tetrahedral linkages. The only assumption made is that the stoichiometry is maintained and that the N M R data yield the space

plus time average of the populations of the five possible silicon nearest neighbor configurations, Si(nA1). The calculation is then based on a structural unit whose structure is that of this average "seen" by NMR. A structural unit of nine tetrahedra is employed; this has the tetrahedral content A14Si 5 distributed over the five types of sites of the orthorhombic structure (Table 1). The N M R spectra do not resolve all of these types of tetrahedral sites (Paper I), but instead show two sets of peaks which are allocated to T 1 and T2 sites, the two types of sites in the hexagonal structure of high (disordered) cordierite. The calculation below, however, is carried out using the orthorhombic sites and is shown to be independent of the true symmetry of the material. The various inter-tetrahedral linkages are listed in Table 1, and it is the fact that T1 sites are connected only to T2 sites that allows the calculation of the number of A 1 - O - A 1 inter-tetrahedral linkages within the structure from the N M R data. The method of calculation, which is described in detail in the Appendix, may be outlined as follows. The N M R data give the ratio of the total number of Si on T1 sites to that on T2 sites (Paper I). The total number of Si on T 1 sites and on T2 sites within each structural unit, denoted by Ns~1 and Ns~2, respectively, are determined directly from the N M R spectra and these are listed for each cordierite sample in Table 2. The N M R data also define the proportion of total Si in each type of tetrahedral site which is involved in specific groupings Si(nA1). From this the total number of Si T 1 - O - A I T2 linkages (i.e. linkages between Si on T1 sites and A1 on T2 sites) may be calculated, as well as the total number of Si T2-O-A1T~ plus Si T2- O-A1Ta linkages. These are listed in T a b l e 2 as 5Zn[Si(nA1)] T~ and

Table 1. Tetrahedral framework topology Site

Multiplicity in A14Si 5

Linkages per tetrahedron Orthorhombic

T 1 T1 T16 ['I"21 T2~T23 tT26

2 1 2 2 2

Hexagonal

2T21 +2T23 43"26 } 4T2 2T~1 +T23+T26] 2T~1+T21 +T~ 6 ~2T~+2T 2 2T26 +T21 +T23/

218 Table 2

Annealing time

NT1

NT2

5 Zn [Si(nA1)] T~

5Zn[Si(nA1)]T~

N(AI-A1)

2 min 6.5 rain 20 min 6h 23.5 h 48.5 h 96 h 408 h 2,000 h Fully ordered

0.605 0.595 0.665 0.780 0.810 0.751 0.840 0.865 0.925 1.000

4.395 4.410 4.335 4.220 4.190 4.249 4.160 4.135 4.075 4.000

1.642 1.770 1.992 2.526 2.742 2.748 2.964 3.090 3.426 4.000

11.33 11.25 11.34 11.99 11.78 12.13 12.09 12.40 12.21 12.00

1.51 1.49 1.33 0.74 0.74 0.56 0.47 0.26 0.18 0.00

5ZnESi(nA1)] T2, respectively. Finally the number of A1 - O - A 1 linkages in the structural unit, denoted N(A1 -A1), may be calculated for each cordierite sample. This is also shown in Table 2. Clearly, the total number of tetrahedral linkages of all types within the structural unit is 18.

I

1"8

I

No. A I - O - A I bonds per 9 tetrahedra

I-2

Results 0"9

Figure 1 shows a plot of the average number of A 1 - O - A 1 bonds per formula unit (i.e. N(A1-A1)) as a function of the logarithm of the annealing time for the nine experimental samples investigated by N M R . The plot is linear with a slope o f - 0.31 and a product moment correlation coefficient of 0.989 (a value of 1.0 indicating perfect linearity). The effectiveness of N(A1-A1) as an order parameter lies in the fact that it is independent of the degree of long-range A1, Si order. The existence of a short-range ordered domain structure in cordierite (Putnis 1980a, b) precludes the use of any order parameter based on average site occupancies. We may conclude from the above data that the A1, Si ordering process in cordierite is essentially one of eliminating A 1 - O - A 1 bonds. As can be seen from Table 2 the site occupancies of T 1 and T2 sites show no regular variation with annealing time, while the correlation of N(A1-A1) with annealing time is extremely good. Similar cordierite samples prepared from the same batch of glass have been used to determine the changes in AHso~n associated with A1, Si ordering (Carpenter et al. 1983). The plot of Agsoln against the logarithm of annealing time (Carpenter et al. 1983, Fig. 4) is linear with a slope of 2.51 and correlation coefficient of 0.998. Although the ordering process clearly involves changes in bonds other than those between tetrahedra, we may in the first instance make the assumption that the difference in the A Hsoln arises from the energy associated with the reaction (A1 - O - A1) + (Si - O - Si) --*2 (A1 - O - Si). This reaction involves a single AI~-Si interchange with which we may associate an enthalpy change A h. Following Carpenter et al. (1983) we may write OAH t

Ot

ON Ah Ot

where AH t is the enthalpy change due to ordering after time t, and N is the number of A1, Si pairs in "wrong" sites.

o ~

0.3 i

i

I

I

I

-I'0

0

I'0

2"0

3'0

log time (h)

Fig. 1. The number of A 1 - O - A I bonds per unit cell of cordierite as a function of the logarithm of the annealing time

OHt _ 0t

OAHsoln 0t

'

since a decrease in the enthalpy of cordierite due to ordering will result in an increase in A Hsoln.

OAHsoin=--.ON Ah. Ot Ot

i.e.

The experimental result that AHsoln varies linearly with log (time) may be written

OAHsoln 2.3034) 0t

t

where q5 is the slope of the AHso~n-logt plot. Thus 2.3034) ON . . . . Ah. t 0t But ON 0t

2.303~b t

where ~b is the slope of the N ( A 1 - A 1 ) - l o g t plot. Substituting q5= 2.51 (Carpenter et al. 1983) and ~ = - 0 . 3 1 yields a value of Ah= - 8 . 1 kcal tool -1. A less rigorous way of arriving at this same result is to equate the change in enthalpy (2.5 kcal tool -1 for

219 Table 3. Comparison of the proportions of Si(nA1) configurations obtained from NMR data with those calculated from various models

NMR data Disordered model McConnell model Fully ordered

T1 sites

T2 sites

Si(4A1) Si(3A1) Si(2A1) Si(1A1) Si(0A1)

Si(4A1) Si(3al)

0.0198 0.0025 0.0271 0.2

0.0747 0.0392 0.0889

0.0532 0.0417 0.0062 0.0198 0.0592 0 . 0 7 9 0.0395 0.0593 0.0593 0.0356 0.0148 . . . . .

every unit log annealing time) with the change in N(A1 -A1) (0.31/formula unit for every unit log annealing time). The quotient yields the enthalpy change associated with the elimination of each A 1 - O - A 1 bond, i.e. Ah. The linear plots of both AHsoln and N(A1-A1) with log annealing time result in a value of A h which is independent of the degree of order. This suggests the dominance of nearest neighbor interaction terms in the overall enthalpy due to ordering. Data with which to compare the above value of A h =8.1 kcal mo1-1 are sparse. De Jong and Brown (1980) have carried out molecular orbital calculations for small aluminosilicate clusters and conclude that for approximately optimum geometries the reaction ( A 1 - O -A1)+(Si-O-Si)~2(A1-O-Si) is exothermic by some 120 kcal mol 1. Navrotsky et al. (1982) suggest a value of 10-20 kcal tool -1 based on calorimetric data on aluminosilicate glasses, indicating that the magnitude of the stabilization energy obtained from small clusters cannot be applied to glasses. Enthalpy data exist for A1, Si ordering in a number of minerals (Carpenter 1985) and may be used to calculate Ah if the state of order is known. For example in albite, NaA1Si3Os, a value of AHora-~3.0 kcal tool 1 is widely quoted (Holm and Kleppa 1968; Newton et al. 1980; Carpenter et al. 1985). If we assume that this value refers to the enthalpy difference between totally disordered monalbite and fully ordered albite we may calculate A h as follows: In disordered albite the site occupancy for each tetrahedral site is AI: 0.25, Si: 0.75. The number of each type of bond per A1Si 3 structural unit (8 bonds) is therefore N(A1-A1)=0.5, N ( A 1 - S i ) = 3 . 0 , N ( S i - S i ) =4.5. In fully ordered albite N ( A 1 - A 1 ) = 0 , N ( A I - S i ) =4.0, N ( S i - S i ) = 4 . 0 per A1Si a structural unit. If we refer to the enthalpy of the T - O - T bond as E ( T - T ) , the enthalpy difference between ordered and disordered albite is 0.5 E(A1 - A 1 ) + 3E(A1 - Si) + 4.5 E(Si - S i ) - 4E(A1 - S i ) - 4E(Si - Si) = 0.5 E(A1 -A1) + 0.5 E(Si - Si) - E ( A 1 - S i ) = 3 kcal tool- 1. Thus the enthalpy change associated with a single A I ~ S i interchange, A h=6.0 kcal tool -1. Allowing for the effects of interaction between tetrahedral and nonframework cations and the assumption made about the structural state of the albite, this value of A h is in good agreement with that obtained here for cordierite. This approach may be extended to consider A1, Si ordering in anorthite for which the estimated A Hora

Si(2AI)

Si(1A1) Si(0A1)

0.408 0.347 0.0489 0.1976 0.3256 0.1976 0.039 0.3061 0 . 2 3 7 0 . 1 1 8 5 0.0494 0.8 -

=3.7_+0.6 kcal tool 1 (Carpenter et al. 1985). Using the same method as that above for albite, the enthalpy change associated with total disorder-~ total order may be equated with the enthalpy for the reaction 2 ( A 1 - O -A1)+2(Si-O-Si)~4(A1-O-Si) per mole. Using Ah=6.0 kcal tool -1 yields as AHor d for anorthite of 12 kcal mol-1. Allowing for the effect of C a substitution for Na, the large difference between measured and calculated AHora suggests, as expected, that the "disordered" anorthite had a considerable degree of short range order.

Models of Short Range Order As has been described in previous papers (Putnis 1980a, b; Putnis and Bish 1983; Carpenter et al. 1983; McMillan et al. 1984; Paper I) freshly crystallized cordierite is hexagonal and has no long range order, although the techniques described in the above mentioned papers suggest a considerable degree of short range order. An estimate of the degree of order may be made by comparing the maximum measured enthalpy change on ordering in cordierite (9.76 + 1.56 kcalmol 1, Carpenter et al. 1983) with the maximum possible value from the relation A Hora=T~ASord. Assuming a first-order transformation at 1,450°C (Putnis 1980a) and A Sor~ of 12.0 cal m o l - l K (Navrotsky and Kleppa 1973; Putnis 1980b) then AHord-~21 kcal tool -1, indicating that the first-formed cordierite has a configurational entropy of about half of that for total disorder. The N M R data are entirely consistent with the calorimetric data. The number of A 1 - O - A 1 bonds in the first-formed cordierite is 1.51 per formula Unit (Table 2). If we assumed a disordered model in which the partitioning of Si and A1 between T1 and T2 sites was the same as that in the ordered orthorhombic form (i.e. in which the occupancy of T 1 sites is ½Si, }AI and of T2 sites is 2Si, ½A1) the calculated number of A 1 - O - A 1 bonds is 3.3 per formula unit. Furthermore, the number of each of the local Si environments, Si(nA1) for both T1 and T2 sites can be calculated for this disordered model and compared to the N M R results (Paper I, Table 2). This has been tabulated in Table 3, from which it can be seen that the number of observed Si(4A1) and Si(3A1) configurations is considerably in excess of those expected from the disordered model. Clearly, increasing the number of S i - O - A 1 bonds reduces the total number of A 1 - O - A 1 linkages. An alternative model for the early cordierite is to assume a domain structure in which individual domains are ordered on the orthorhombic ordering

220 scheme, but the domains are twin-related. The disorder therefore exists on domain boundaries between essentially well-ordered domains. The initial attraction of this hypothesis is that a domain structure has been observed by transmission electron microscopy in the later stages of annealing (Putnis 1980a), that it appears to fulfil the requirement that a high degree of local order exists despite the absence of long range order, and that it reproduces the partitioning of Si, A1 between T 1 and T2 sites. However, we have considered the number of A 1 - O - A 1 bonds generated on the {110} and {310} twin planes between ordered orthorhombic cordierite (see Putnis 1980a) and in order to account for the observed number the size of the domains becomes unrealistically small ( ~ u n i t cell size). We conclude that a simple model based on orthorhombic ordered domains is not tenable. As we shall see below a modulated structure based on this ordering scheme is also not consistent with theory. The problem of describing this short range order is constrained by considering the possible modes of behavior within the hexagonal symmetry of the cordierite as it attempts to reduce its free energy by ordering. The initial disturbance associated with ordering must be compatible with this symmetry, i.e. must accord with the normal thermal fluctuation behavior of the system. The group theoretical approach, which describes the permissible distortions of the system, will not be described here, as a full treatment is given in a recent paper by McConnell (1985). Here we give merely a brief outline of the McConnell model. The loss of symmetry compatible with local ordering is associated with four two-dimensional representations of the hexagonal point group. One of these (Ezg) contains an even representation which corresponds to the orthorhombic subgroup of low cordierite. Associated with this representation is a second odd representation which is degenerate with the first. It is possible to show (McConnell 1985) that each of the two representations defines, in the limit of maximum order, the two A1, Si distributions illustrated in Fig. 2. The notation +½, --~, etc. refers to the fractional occupancy of Si in excess of the mean for that site, e.g. for T 1 2 1 sites + 51 means 5Si, 5A1, - 51 is A1, etc. For T2 sites + 51 is Si, - ½ is ½Si, }A1, etc. The even ordering scheme involves no disorder, i.e. zero configurational entropy and minimum enthalpy. The odd ordering scheme is degenerate under hexagonal symmetry but involves an increase in both entropy and enthalpy. This degeneracy implies that we cannot promote one ordering scheme without the other. In considering how the transformation will occur in practice, we note that any A1, Si ordering will involve lattice displacements which must be consistent with the fluctuation modes of the crystal, and further, that of the two ordering schemes above, it is only the odd function that will interact with transverse acoustical waves. The presence of transverse modulations in cordierite has been observed by transmission electron microscopy (Putnis and Bish 1983) and we may conclude that therefore it is only the odd ordering scheme which is compatible with such a modulation. We cannot promote a continuous change in the degree of order with the even function directly, even though this corresponds

Fig. 2a, b. The cordierite structure showing only the tetrahedra. The rings of Tz tetrahedra are linked to each other via T1 tetrahedra (see also Fig. 1 of Paper I). The average occupancy of each tetrahedron is indicated on the figures by the following notation: T2 sites (av. occupancy ½A1,2Si) 0: ½A1, 2Si +½: Si

T~ sites (av. occu'pancy ~A1, ~Si) 0: ~AI, ½Si +~: ½A1,2Si

- - i A1

+

Si

(a) The odd ordering scheme. (b) The even ordering scheme. The arrows in Fig. 2(a) indicate the net average movement of ½Si required to convert the odd structure to the even structure to the ordering scheme of ordered, orthorhombic cordierite. The transverse modulations which exist within the structure from the earliest stages of crystallization may be described in terms of the odd ordering scheme and its inverse in the modulation pattern (Fig. 3). The inverse merely involves changing the signs in Fig. 2(a). When the maximum permitted ordering on this scheme is reached (at which point the system is still hexagonal), the modulation provides a "template" which will define the way in which the fully ordered structure may ultimately be obtained. Examination of the odd structure shows that a relatively small number of atomic interchanges are required to produce the orthorhombic ordering scheme. The arrows in Fig. 2(a) show the net

221

ODD

-

and define an order parameter based directly on the number of A 1 - O - A 1 bonds. The rate of change of N(A1-A1) may be compared to the rate of change of d/-/soln of these samples to provide data on the energetics of Si~-A1 interchanges. The group theoretical approach imposes constraints on the possible ordering schemes within the initially hexagonal structure and provides a model which is a starting point for considering the nature of the short-range order.

Acknowledgements. A.P. acknowledges the financial support of Fig. 3. Spatial distribution of the odd structure and its inverse within the modulation average movement of 1Si required to convert the odd to the even structure. Alternatively, one third of those sites joined by arrows require an AI~--~-Si interchange. The resultant structure will be cross-hatched twinned from the _+ odd structure. At this point the original orthogonal displacements are no longer orthogonal and the phase is therefore no longer hexagonal. This description is in accord with the TEM observations (Putnis 1980a) but no experimental data have previously been available to compare with the theoretical prediction of the McConnell model. There are two aspects of the model which we will consider here: (i) that in the early stages of ordering the configurational entropy is defined by the maximum permitted ordering of the odd ordering scheme (Fig. 2a) and (ii) that the number of A 1 - O - A 1 linkages and the number of Si(nA1) groupings can be calculated for this ordering scheme and compared with the N M R data. (i) The calculated configurational entropy for the odd ordering scheme is 6.3 calmol -* K. As stated above, the heat of solution data suggest that the first formed cordierite has a configurational entropy of approximately half of that for total disorder (12 cal m o l - ~ K). The odd ordering scheme is thus consistent with these data. (ii) The calculated number of A 1 - O - A 1 linkages per formula unit of the odd ordering scheme is 2.2 per 9 tetrahedra, compared with the observed value of 1.5 for the shortest crystallization time. This may suggest that even in the early stages local A1, Si order may be somewhat better than that predicted by the model. In Table 3 the number of Si(nA1) configurations for the odd ordering scheme is compared to that for the disordered structure and that derived from the N M R data. The proposed model provides a considerably better fit to the data than the disordered structure, particularly in predicting much higher values of Si(4A1) and Si(3A1), in accord with the observations. It should be noted that in calculating the proportion of each type of Si(nA1) configuration for the odd ordering scheme, the total SSi(nA1) for the T, and T 2 sites has been normalized to 0.2 and 0.8, respectively, whereas the N M R data suggest that in the early stages the T 2 sites are richer in Si.

the Natural Environment Research Council of Great Britain (NERC), and RJA a NERC research studentship. We thank Dr. J.D.C. McConnell for stimulating discussions on symmetry constraints on ordering and Professor C.A. Fyfe for his continuing involvement in the cordierite project. This is E.S. Contribution No. 581.

Appendix

The total number of Si on T 1 sites and on T2 sites (denoted Ns~1 and Ns~2 respectively) may be obtained directly from the N M R data and are listed in Table 2. The individual site occupancies are constrained by 2Xs~~+

x ~ i 6 = N s T1

91 23 __ T2 2X~" i q - R X s i - } - 2 x 2 i 6 - - N s i

where Xsll1, etc. is the fraction of the T, 1 site etc. occupied by Si. Since all sites are totally occupied (i.e. XAl +Xsi for any given site is unity): 2 x ~ + x ~ 6 = 3 - N TI 21 23 26 1 /~/T2 XAI -}- XAI -}- XAI - - 3 - - 2 * ' S i "

(l)

The N M R data represent the proportion of total Si in the structure which is involved in specific groupings of Si(nA1), and in Table 2 of Paper I, these data are normalized to a total Si content of unity. The number of each type of Si(nA1) group per A14Si s unit is obtained by multiplying these proportions by five. Each Si(nAl) group contains n S i - O - A1 linkages, and thus the total number of such linkages in the structural unit may be determined by summing terms of the form 4

5 ~ n[Si(nA1)] for each of the T 1 and T2 sites. The T 1 n=l

sites comprise two crystallographically distinct sites T 11 and T16, thus: 4

5 ~ n[Si(nA1)] T1 n=l

= N(Sit 6 _A126) + N(Si 11 _ A I 21) + N(Si 11 _A123) where terms of the form N ( S i 16 -A126) are the number of linkages between Si on a T16 site and A1 on a T26 site etc. The terms on the r.h.s, may be grouped together as the total number of Si xl - O - A 1 x2 linkages:

Conclusions 4

We have shown that for cordierite, 29Si N M R data provide a method of characterizing the structural state

5 X n [Si(nA1)] T*= N(Si wl - AIT2). n-1

(2)

222 8 ( x ~ + x ~ + x~ 6) = 2N (A1T~ - A1T~)+ N (A1Tz - A1T~) + N(A1 T2 - siT9 + N(A1 f~ - siT1).

A similar summation over the ring sites gives 4

5 ~, n[Si(nA1)] T~

The 1.h.s. of the equation may be written as 2 4 - 4 N s T2 (from Eq. 1), N(A1T2--siT 9 is given by Eq. (4) and N(A1 r~ - S i rl) by Eq. (2). Rearranging gives

= N(Si 26 - AP 6) + N(Si 2, _ A I ' 1) + N(Si 23 _ A111) q- N (Si 21 _ A123) q_N (Si 21 _ A126) q_N (Si 23 _ A121) q- N(Si 23 - A 1 2 6 ) q- N(Si 26 -A121) + N(Si 26 - A123).

4

The first three terms of the r.h.s, may be grouped as N(siT~--A1Tg; the last six terms as N(SiT~-A1T2) to give 4-

5 ~ n[Si(nA1)]Ta=N(SiI~-A1T')+N(SiT2--A1TO.

(3)

n=l

These summations are presented for each experimental run in Table 2. They both contain terms counting the various types of S i - O - A 1 linkages within the structure, while here we are concerned with the number of A 1 - O - A 1 linkages. These may be related by considering the various linkages from each type of tetrahedron in the structure. Consider a tetrahedron T which has an aluminium occupancy x~l, and occurs N T times in the AI~Si 5 unit. The number of A 1 T - O - ( S i or A1) bonds in the unit due to these N T tetrahedra is thus 4xT1 "N T, each T tetrahedron contributing 4xT1 to such links. In the cordierite Al¢Si s unit there is just a single T16 tetrahedron which is connected solely to T26 tetrahedra (Table 1). Thus 4 x~ 6 = N (A1 ~6 _ Si 26) q_ N (A116 - AI 26). The other T~ tetrahedron T~ 1 occurs twice in every A14Si s unit, and each is connected to two T21 and two T 23 tetrahedra (Table 1). Thus 8 x ~ = N(A1 ~1 -A12~) + N(A1 ~ - Si 2 ~) + N(A11~ _Ala3) + N(A1 x~ _ sia3). If these equations are summed and rearranged we obtain using equation (1): N (A1 T~ - A1 T2) = 12 - 4 XsTi I - - N (A1T~ - Si T~)

where N(A1T~ -A1T~) = N ( A P ~ - A 1 a ~) + N(A11~ -A123) + N ( A P 6 - A 1 a6) Eq. (3) may be used to eliminate N(SiT~-A1T1) ¢

N(Si T~--AlTo)= 5 y~ n[Si(nA1)3 T~ + N (A1T~ - A1T2) - 12 + 4N T'.

4

N(A1-al)= 8 - } ~ n[Si(nA1)] T1 - } ~ n[Si(nA1)] Ta.

(4)

Each tetrahedron within the 6-membered ring (i.e. each T2) has two links to other T 2 tetrahedra, and two to T 1 tetrahedra. Thus, for T 21 8x~t = N ( a l 2' -A123) + N(A12~ -A126) + N(A12~ - A 1 1 ' ) + N(A121 _ S i 23) + N(AI 2~ - Si 26) + N(A12~ - Si' 1). Two similar equations hold for 3"23 and Ta6; if these three equations are summed and the terms grouped

n= 1

n= 1

This derivation of N(A1-A1) is independent of the structural state of the cordierite, that is, whether it possesses the symmetry of the hexagonal (P6/mcc) or orthorhombic (Cccm) space groups. Table 2 lists the number of A 1 - O - A 1 linkages per structural unit A14Si 5 for all the samples investigated by N M R .

References Carpenter MA (1985) Order/disorder transformations in mineral solid solutions. In: Reviews in Mineralogy Vol 14 Ribbe PH (Ed). Mineralogical Society of America Carpenter MA, Putnis A, Navrotsky A, McConnell JDC (1983) Enthalpy effects associated with A1, Si ordering in anhydrous Mg-cordierite. Geochim Cosmochim Acta 47: 899-906 Carpenter MA, McConnell JDC, Navrotsky A (1985) Enthalpies of ordering in the plagioclase feldspar solid solution. Geochim Cosmochim Acta 49:947-966 De Jong BHWB, Brown GE (1980) Polymerization of silicate and aluminate tetrahedra in glasses, melts and aqueous solutions I. Electronic structure of H6Si207, H6A1SiO ~ and H6A120~-. Geochim Cosmochim Acta 44:491-511 Fyfe CA, Thomas JM, Klinowski J, Gobbi GC (1983) Magic angle spinning NMR spectroscopy and the structure of zeolites. Angew Chem 22:259-336 Holm JL, Kleppa OJ (1968) Thermodynamics of the disordering process in albite. Am Mineral 53:123-133 McConnell JDC (1985) Symmetry aspects of order-disorder and the applications of Landau theory. In: Reviews in Mineralogy Vol 14 Ribbe PH (Ed). Mineralogical Society of America McMillan P, Putnis A, Carpenter MA (1984) A Raman spectroscopic study of A1, Si ordering in synthetic magnesium cordierite. Phys Chem Minerals 10:256-260 Navrotsky A, Kleppa OJ (1973) Estimate of enthalpies of transformation and fusion in cordierite. J Am Ceram Soc 56:198-199 Navrotsky A, Peraudeau G, McMillan P, Coutures JP (1982) A thermodynamical study of glasses and crystals along the joins silica-calcium aluminate and silica-sodium aluminate. Geochim Cosmochim Acta 46:2039-2047 Newton RC, Charlu TV, Kleppa OJ (1980) Thermochemistry of the high structural state plagioclases. Geochim Cosmochim Acta 44:933-941 Putnis A (1980a) The distortion index in anhydrous Mgcordierite. Contrib Mineral Petrol 74:135-141 Putnis A (1980b) Order-modulated structures and the thermodynamics of cordierite reactions. Nature 287:128-131 Putnis A, Bish DL (1983) The mechanism and kinetics of A1, Si ordering in Mg cordierite. Am Mineral 68:60-65 Putnis A, Fyfe CA, Gobbi GC (1985) A1, Si ordering in cordierite using "magicangle spinning" NMR. I: Si 29 spectra of synthetic cordierites. Phys Chem Minerals 12:211-216 Received October 10, 1984