J. Phys. Chem. B 2002, 106, 57-62
57
Linear Framework Defects in Zeolite Mordenite B. J. Campbell*,† and A. K. Cheetham Materials Research Laboratory, UniVersity of California, Santa Barbara, California 93106 ReceiVed: July 24, 2001; In Final Form: October 19, 2001
Framework defect correlations are identified in zeolite mordenite that explain both X-ray powder diffraction anomalies and diffuse intensity features in previously reported electron diffraction data. The c/2 displacement of linear “chains” of framework material parallel to the main [001] channel are shown to selectively suppress and broaden the l ) 2n + 1 Bragg reflections, thereby linking these framework defects to observed trends in a number of diffracted intensity ratios. At the same time, these defects are shown to produce thin sheets of diffuse scattering that are restricted to the l ) 2n + 1 planes of reciprocal space, features that have been reported in electron diffraction studies of mordenite. An empirical relationship between the defect concentration and the framework aluminum content is also established.
Introduction Framework defects have been reported to occur in both natural and synthetic varieties of zeolite mordenite, a widely used industrial isomerization catalyst.1,2 Because framework defects alter the local structure (pore size, shape, and connectivity, etc.), they can also influence application-critical properties such as diffusion rate, adsorption preference, and catalytic activity and selectivity.3,4 Many zeolite frameworks are prone to extensive stacking faults, such as zeolite beta5,6 and the ABCD6R type zeolites (PHI, CHA, OFF, etc).7,8 Several zeolites of the pentasil family are also reported to experience faulting, including mordenite, ferrierite, and dachiardite.9 The presence of defects often introduces uncertainty into the relationship between materials processing variables and final material properties. When thoroughly characterized, however, defects can provide additional controlled variables for use in tuning material properties for a given application. The present work attempts to complete our knowledge of the local structure of framework defects in zeolite mordenite. Naturally occurring mordenite is an aluminum-disordered aluminosilicate mineral with a Si/Al ratio (SAR) of about 5. Synthetic varieties can be synthesized with SARs in the range from 5 to 10, and higher SARs can be obtained by subsequent high-temperature dealumination. Mordenite’s large orthorhombic unit cell (a ) 18.1 Å, b ) 20.5 Å, c ) 7.5 Å) hosts a onedimensional system of 6.5 × 7.0 Å 12-membered ring (12MR) channels parallel to [001] that are laden with smaller 8MR side pockets extending along the [010]. The aluminosilicate framework has topological space-group symmetry Cmcm and includes four crystallographically unique tetrahedral sites (Tsites), a total of 48 T-site atoms, and 96 bridging oxygens per unit cell.10 Soon after the mordenite framework structure was determined,11 the discovery that some mordenites had adsorption properties consistent with 12MR channels, while others did not12,13 (large vs small port mordenite), led Meier to propose stacking faults in the (001) planes that either block or restrict the pores.14 † Current address: Materials Science Laboratory, Argonne National Laboratory, Argonne, IL 60439.
Gard and Bennett14-15 soon thereafter performed reciprocal space TEM studies but found no evidence of the diffuse rodshaped scattering features parallel to c* associated with stacking faults. They did, however, observe streaks of diffuse intensity within discrete planes perpendicular to c* and propose the existence of linear c-axis “chains” of material displaced by c/2. Sanders9 conducted a detailed electron diffraction study of faulting in mordenite and reported that its diffuse scattering is confined to thin sheets perpendicular to the c* at odd integer values of l, which appeared as streaks in a*-c* and b*-c* sections, further confirming the conclusion that mordenite has linear, rather than planar, faulting. Single-crystal X-ray diffraction studies of H+-exchanged mordenite16,17 reported difference Fourier map features that looked like low-occupancy 4MR rings located precisely halfway between two proper 4MR pillars and refined the occupancies of the c/2-shifted 4MR atoms to be ∼2%. Itabashi et al.18 of Tosoh Corp., synthesized a series of mordenite powder samples with SAR values between 5 and 10 and found a distinct linear correlation between the SAR and the (111)/(130) and (241)/ (002) diffracted intensity ratios. In 1989, Shiokawa et al.19 performed Rietveld refinements of mordenite using laboratory X-ray diffraction data from a series of Tosoh mordenite powder samples with a range of SARs. By extracting extraframework (EF) atoms from difference Fourier maps, they studied the evolution of the network of EF waters and Na+ cations vs SAR and suggested that EF water and cations might also account for previous reports of c/2-shifted difference Fourier peaks and between diffracted intensity trends. Rudolf and Garce´s20 subsequently carried out additional powder diffraction studies, also using hydrated Tosoh Na mordenite, in which they examined cation/water ordering and c/2 displacements. The structure of a low-aluminum-content sample (SAR ) 9.5) refined nicely using the normal mordenite framework model, whereas the refinement from a sample with higher-aluminum content (SAR ) 6) was greatly improved by incorporating a c/2-shifted copy of the entire structure into the model at the relative 20% occupancy level. They concluded that faulting is related to the trends in the diffracted intensities and that it appears to occur more readily in higher Al containing samples.
10.1021/jp0128372 CCC: $22.00 © 2002 American Chemical Society Published on Web 12/04/2001
58 J. Phys. Chem. B, Vol. 106, No. 1, 2002
Figure 1. Mordenite framework viewed as a series of puckered sheets in the planes perpendicular to [100] that are held apart by 4MR pillars. The white circles represent T-atoms, and the gray circles represent oxygens. The four pillars within the unit cell are labeled A-D after the manner of Sherman and Bennett.24 The bonds between pillar and sheet are omitted to clarify the identity of the distinct subunits.
Figure 2. Puckered sheet structure within the mordenite framework consisting of a “graphite-like” arrangement of six-membered rings or distorted hexagons perpendicular to [100]. The 4MR pillar unit that holds the sheets apart is illustrated at the top. A copy of this pillar is shaded black and properly oriented above the sheet, revealing the hexagonal sheet atoms to which it bonds. Both the 4MR pillars and the O5 and O6 oxygens serve to hold the sheet together along [001].
Shifted-Pillar Defects As a member of the pentasil family of zeolites, the canonical description of the mordenite framework involves the 5MR.10 An alternative description of the mordenite framework that will be especially helpful here is that of a series of puckered sheets aligned normal to the [100] direction, which are separated from one another by a/2 (see Figure 1). Because these sheets consist only of 6MRs, they have a hexagonal or honeycomb-like appearance. Adjacent sheets are related by a mirror plane between them and look identical in projection along [100], though their local curvatures oppose one another. Each pair of adjacent sheets is then connected by a system of pillars, each pillar consisting of a 4MR with eight oxygen legs. It is not unreasonable to suppose that 4MR pillars might occasionally occupy alternative sites displaced from the normal sites by c/2. From Figure 2, one can see that the 4MR pillars share a structural function in common with oxygens O5 and O6 of the hexagonal sheets, in that each acts as a link to hold
Campbell and Cheetham the hexagonal sheets together along the [001] direction; pillars serve as links external to the sheets, while O5 and O6 serve as links within the sheets. At the position displaced by c/2 from any 4MR pillar, the hexagonal sheets above and below the pillar contain a total of four O5 and O6 oxygens. This oxygen quartet has the same formal ionic charge of 8- as the pillar itself, so that the 4MR pillar and O5/O6 quartet can be viewed as interchangeable linkage units that can be used to construct a variety of hypothetical structures that are different from mordenite. Two such alternative pillar arrangements yield the structures of zeolites dachiardite and epistilbite, and the use of different types of pillars to link the hexagonal sheets together leads to the structures of zeolites ferrierite, bikitaite, and Li-ABW.21 The natural mordenite pillar arrangement is illustrated in Figure 3a, where half the pillars shown are above the sheet and half are below it. The pillars above the sheet can be differentiated by the fact that they block the view of the lightly shaded O5 or O6 oxygens immediately below them. An isolated defect involving these linkages might consist of replacing the O5/O6 quartet between two pillars with another pillar (“pillar insertion”), replacing a pillar with an O5/O6 quartet (“pillar omission”), or the exchange of a pillar with one of its adjacent O5/O6 quartets (“pillar shift”). These local configurations may not be energetically favorable, but they are viable framework connectivities and leave no dangling bonds. However, an isolated shifted-pillar defect places a pair of pillars uncomfortably close together along [001] or leaves them uncomfortably far apart, as illustrated in Figure 3b, motivating the idea that the shifts of nearby pillars might be correlated in order to minimize the accompanying strain in the surrounding framework material. The correlated shift of a c-axis chain of these pillars, for example, only requires extraordinary interpillar spacings at the end points of the chain, and when such a chain traverses the entire crystal, no extraordinary spacings need exist at all. The term “chain” here does not indicate that the pillars are directly attached to one another but only infers that they are arrayed in regularly spaced lines along the [001] direction, as seen in Figure 1. The framework connectivity that results from the cooperative c/2 shift of a c-axis chain of 4MR pillars is illustrated in Figure 3c. Selective Bragg Peak Suppression Some definitions and notation need to be clarified prior to exploring the connection between framework defects, diffracted intensities, and diffuse scattering. Direct and reciprocal space positions will be described by unitless vector quantities x and h on the direct and reciprocal crystal bases, {a, b, c} and {a*, b*, c*}, respectively. LD will represent the direct vector lattice, LR will represent the reciprocal vector lattice, and U will represent the set of ideal atomic position vectors within the unit cell. The scattering density n(x) and scattering amplitude n(h) are related to the structure function I(h) in the usual way:
n(x) )
∑
fsδ(x - (S + s))
S∈LD,s∈C
n(h) ) I(h) )
∫e2πih‚xn(x) d3x
(1)
∫e2πih‚(x -x )〈n(x1) n(x2)〉 d3x1 d3x2 ) 〈n(h) n(-h)〉 1
2
(2)
where the brackets indicate an average over any disordered atomic positions in the structure, and atoms are treated as static isotropic point scatterers so as to avoid thermal factors and
Linear Framework Defects in Zeolite Mordenite
J. Phys. Chem. B, Vol. 106, No. 1, 2002 59
Figure 3. Single hexagonal sheet from the mordenite framework appears in each panel, along with the 4MR pillars attached above and below it. Black pillars are in their standard positions, while lightly shaded pillars are shifted by c/2. A pillar lying above the sheet can be distinguished from one lying below it because it blocks the view of one the oxygen atoms within the sheet. The framework segments contain three unit cells along [001] and therefore include only three 4MR pillars from each “chain”. (a) The normal mordenite pillar arrangement. (b) An isolated shifted-pillar defect. Note the effect upon the connectivity within the hexagonal sheet. (c) The cooperative shift of a c-axis chain of 4MR pillars. The affected chain is indicated by the arrow.
angle-dependent scattering factors, which are not of interest here. The structure function, Ip(h), of a perfect crystal (i.e., a crystal free of defects and disorder) can then be expressed as
Ip(h) )
fsfs′e2πih‚(S+s-S′-s′) ) ∑ δ(h - G)|Fh|2 ∑ S,S′∈L ;s,s′∈U G∈L D
R
Fh ≡
∑fse
2πih‚s
(3)
s∈U
The symbols G, S, and s, will hereafter be implicitly assumed to represent vectors contained within spaces LR, LD, and U, respectively. In an imperfect crystal, such that an atom located at x may be displaced from its ideal position by u(x), the structure function becomes
I(h) )
fsfs′e2πih‚(S+s-S′-s′)ΩS+s,S′+s′ ∑ S,S′,s,s′
ΩS+s,S′+s′ ≡ 〈e2πih‚(u(S+s)-u(S′+s′))〉
(4)
The effect of atomic displacements by c/2 on diffracted intensities can be readily calculated using eq 4. While positional correlations will be critical in the discussion in the next section, it is instructive to ignore them initially. Furthermore, it will be assumed that each framework atom in the crystal has an equal probability of being displaced. This is equivalent to giving each atom a fractional occupancy of p0 at the c/2 shifted position and an occupancy of (1 - p0) at the original position, as in the structure refinements of Rudolf and Garce´s.20 While it must be pointed out that only the atoms contained with the pillars themselves and the O5/O6 quartets are actually displaced by c/2 from their normal positions, all of the other framework atoms have symmetry equivalent sites at or very close to their c/2 shifted positions. This contributed to the success of their approach. With these assumptions in mind, ΩS+s)S′+s′ ) 1 and
ΩS+s*S′+s′ ) 〈e2πih‚(u-u′)〉 ) 〈e2πih‚u〉〈e-2πih‚u′〉 ) |〈e2πih‚c/2〉|2 ) |(1 - p0) + p0e-πil|2 ) 1- 4p0(1 - p0) sin2(πl/2) ≡ Ω0(h)
(5)
The structure factor in eq 4 can then be simplified in terms of Ω0 to have the form
I(h) ) Ω0Ip + N(1 - Ω0)
∑s fs2,
(6)
where N is the number of unit cells contained within the crystal volume. The first term, Ω0Ip, represents the defect-modified Bragg reflection intensities and yields an integrated intensity for the (hkl) reflection of
Ihkl ) N|Fhkl|2‚
(
(1 - 2p0)2 if l ) 2n + 1 1 if l ) 2n
)
(7)
The striking feature here is the selective suppression of the odd-l intensities (i.e., Bragg reflections for which the l index is an odd integer) by a quadratic function of the defect concentration. Figure 4 illustrates several calculated powder diffraction patterns that correspond to defect concentrations in the range from p0 ) 0.0 to 0.5. The factor of (1 - 2p0)2 results in a 36% intensity loss at p0 ) 0.1, a 75% intensity loss at p0 ) 0.25, and complete odd-l peak suppression at p0 ) 0.5. The case of p0 ) 1.0 corresponds to a fully c/2-shifted crystal that is once again defect-free. The second term in eq 6 represents diffuse scattering contribution, which is continuous throughout reciprocal space. The factors Ω0 and (1 - Ω0) simply indicate that the diffusely scattered intensity arises at the expense of the Bragg reflection intensities. The implications of eq 7 for experimental powder diffraction data will now be examined. Tosoh HSZ610NAA (SAR ) 5.4) and HSZ640NAA (SAR ) 9.5) Na-mordenite samples, referred to hereafter as T640 and T610, were selected for comparison with previous studies18-20 that utilized similar Tosoh samples. X-ray powder diffraction data were collected using a laboratory Cu KR powder diffractometer and are shown in Figure 5. In this figure, each odd-l peak in the data from the T610 sample is indeed suppressed relative to the same peak from the T640 sample. The low-angle peaks, such as (111), are also strongly affected by EF water and cation content, which varies with the Al content of the sample; but these effects do not systematically differentiate odd-l and even-l intensities, as observed in Figure 5. These trends show the T610 sample to possess a significant c/2 shift-defect concentration relative to T640, while it also has approximately twice as much framework aluminum as T640. A careful examination of the diffraction patterns of Rudolf and Garce´s20 (see their Figure 4a; but note that some peaks are incorrectly labeled) also reveals the same odd-l peak suppression in their SAR ) 5.3 sample relative their SAR ) 9.5 sample, consistent with their conclusion that the amount of defect-shifted
60 J. Phys. Chem. B, Vol. 106, No. 1, 2002
Figure 4. Effect of c/2-shift defects on calculated X-ray powder diffraction intensities over a range of defect concentrations. The odd-l intensities are rapidly extinguished as the defect concentration increases, while the even-l intensities are relatively unaffected.
Campbell and Cheetham
Figure 6. Le Bail profile fit to the Cu KR X-ray powder diffraction from hydrated Na-mordenite (T610). Crosses represent the data, while the upper and lower solid lines are the profile fit and the difference curves, respectively. The (111) and (021) peaks are noticeably broadened relative to the other peaks.
Figure 7. Empirical relationship between the c/2-shift defect concentration and the framework aluminum content (see text). Rs refers to the unknown saturation value of the (111)/(130) intensity ratio at the highly siliceous extreme. Several possible values of Rs (evenly spaced between 3.5 and 6.0) are represented.
Figure 5. Laboratory Cu KR X-ray powder diffraction data from the T640 (top) and T610 (bottom) samples illustrating the selective suppression of odd-l peaks in the lower powder pattern. The strong odd-l peaks are labeled. Overlapping peaks are only labeled if the primary contributor is an odd-l peak.
“normal” peaks in the pattern. The GSAS software package for Rietveld refinement does include a facility that allows one to apply different peak widths to distinct reciprocal sublattices (the odd-l and even-l sublattices in this case), permitting an improved fit to the profile. This unusual broadening pattern provides additional evidence that the c/2-shift defects are correlated so as to form faults. Stacking faults are well-known to produce similar effects in metals.23 Defect Concentration and Aluminum Content
material increases with Al content. We further observe that the odd-l peaks are precisely those that were reported to correlate strongly to the SAR in the studies of Musa et al.22 and Itabashi et al.,18 where each intensity ratio examined consisted of an odd-l intensity divided by an even-l intensity, such that the odd-l intensity decreased with increasing Al content while the even-l intensity varied only slightly. The structural mechanism responsible for these diffracted intensity trends is then resolved. They illustrate a selective suppression of the odd-l reciprocal sublattice intensities due to c/2-shift defects. We also observed the odd-l peaks to be anomalously broadened relative to the even-l peaks in powder diffraction data from mordenite samples from a variety of sources, Tosoh640 being the exception, making it difficult to obtain good Le Bail profile fits to their powder diffraction patterns. The laboratory X-ray diffraction data in Figure 6, from a heavily faulted mordenite sample, reveals the (111) and (201) peaks to have broad Lorentzian tails that distinguish them from the
The well-defined linear relationship between the SAR and the (111)/(130) ratio established by Itabashi et al.18 for hydrated Na mordenite can be used to estimate the relationship between the shifted-pillar defect concentration and the framework Al concentration. This estimate involves some uncertainty, as possible contributions from the extraframework constituents will be ignored. Also, the saturation value of the R ) (111)/(130) ratio at the purely siliceous limit in which the sample is defectfree is unknown. This maximum value (RS) is not known, though it is not likely to be much greater than the value observed in the T640 sample (R ) 3.5), which is already highly siliceous. Combining the linear relationship from Itabashi et al.18 with the factor of (1 - 2p0)2 from eq 7, we obtain the relation RS (1 - 2p0)2 ) (0.56/CAl - 2.56), which is plotted in Figure 7 for several RS values evenly distributed from 3.5 to 6.0. Note that the curves are only valid in the region between CAl ) 0.091 (SAR ) 10) and CAl ) 0.167 (SAR ) 5), where the intensity vs SAR data were initially judged to be linear.
Linear Framework Defects in Zeolite Mordenite
J. Phys. Chem. B, Vol. 106, No. 1, 2002 61
The origin of the relationship in Figure 7 may lie in the local framework distortions induced by a framework Al in an otherwise siliceous region of the crystal. Al maintains a significantly greater distance from bonding framework oxygens (Al-O ) 1.74 Å) than Si (Si-O ) 1.61 Å). During crystal formation, the longer Al-O bonds may cause a new pillar to bond more favorably at the c/2-shifted position. And once initiated, such a defect could propagate along [001] during subsequent crystal growth in order to avoid the disruptive pillar spacings associated with its termination. The fact that the fraction of c/2-shifted 4MR units was reported to be lower in an acid-treated mordenite single crystal17 than in a similar crystal that was not acid-treated16 (1% vs 2%), further suggests that the c/2-shift defects are associated with framework Al sites. Thin Sheets of Diffuse Scattering Perpendicular to c* Next, we take the positional correlations among displaced atoms into account, including details about the types of correlations to be investigated. We begin here with a look at cooperatively shifted chains of 4MR pillars, followed by a discussion of interchain correlations. Because diffusely scattered intensity is observed only in thin sheets of reciprocal space at integral values9,15 of l, we assume that 4MR pillars shift cooperatively as long chains along [001]. Under the assumption that they effectively traverse the entire crystal, all atom pairs within the same chain undergo perfectly correlated displacements (i.e., Ω f 1). It then becomes convenient to separate S into components parallel (S|) and perpendicular (S⊥) to [001], so that eq 4 can now be reexpressed as
I(h) )
∑
fsfse2πih‚(S|+S⊥+s-S′|-S′⊥-s′)ΩS|+S⊥+s-S′|-S′⊥-s′
S|,S⊥s,S′|,S′⊥s′
(8)
Next, taking into account the fact that ΩS+s,S′+s′ ) 1 for the self-correlated pairs and Ω0 for the uncorrelated pairs, straightforward manipulations of the summation lead to the expression
I(h) ) Ω0IP + ID
(9)
where IP(h) is defined as before and the diffuse scattering contribution is
immediately explains one of the most striking features of the experimental diffuse scattering data of Saunders:9 the thin sheets of diffusely scattered intensity appear only in the odd-l planes of reciprocal space, not the even-l planes. The term I⊥(h) in eq 12 contains information about interchain shift correlations and gives rise to nonuniform intensity distributions within the diffuse sheets. Gard and Bennett,15 as well as Sanders,9 reported structure within the diffuse sheets, and Sherman and Bennett24 speculated that the nonuniformities might point to local intergrowths of alternative mordenite pillar arrangements. They proposed three such arrangements, which correspond to the three symmetry-unique ways to shift two of the four pillars within the unit cell (A-B, A-C, and A-D in Figure 1). Equation 12 can be further developed for examining such interchain correlations by reducing the unit cell contents to four pseudoatoms, labeled A-D, situated near the centers of the corresponding four 4MR pillars: (0, y ≈ 1/4, 1/4), (0, - y, 3/ ), (1/ , y + 1/ , 1/ ), (1/ , 1/ - y, 3/ ), each possessing a form 4 2 2 4 2 2 4 factor, fp, which represents the combined scattering strength of that quarter of the unit cell with which is it uniquely associated. A shifted-chain defect is thus approximated by the shift of its representative chain of pseudoatoms, yielding the expression
(p0(1 - p0) - pV xor V′)e2πih‚∆V,V′ ∑ V,V′
I⊥(h) ) fp2
where V′ is summed over the four chains that intersect the unit cell at the origin, V is summed over all chains, and ∆V,V′ is the vector separating representative S| ) 0 pseudoatoms in chains V and V′. For uncorrelated chains, pV xor V′ is simply equal to p0(1 - p0), which is the probability that one chain is shifted multiplied by the probability that the other is not, thereby contributing nothing to I⊥(h). When V ) V′, pV xor V′ must obviously be zero. Nearest-neighbor correlations provide a relatively simple example that can be computed explicitly. When neighboring chains shift together, pV xor V′ ) p0(1 - p0)/2, where the extra factor of 2 arises from the fact that, on average, there is only one unshifted site adjacent to each shifted site rather than two. The resulting expression for I⊥(h) is then
I⊥(h) ) 8fp2p0(1 - p0)M(h)
ID(h) ) (ΩS +s,S′ +s′ - Ω0)fsfs′e2πih‚(S +s-S′ -s′) ∑n δ(l - n)S ∑ S′ s,s′ ⊥
⊥
⊥
⊥
⊥
(10)
⊥
The delta function within ID(h) restricts the diffuse intensity to discrete sheets in the l ) integer planes. Define pV and V′ as the probability that two atoms, V and V′, will shift, pV nor V′ as the probability that neither V nor V′ will shift, and pV xor V′ as the probability that only one of them will shift. Recalling the definition of Ω in eq 4:
ΩV,V′ ) pV and V′ + pV nor V′ + pV xor V′eπih‚c + pV xor V′e-πih‚c ) (1 - 2pV xor V′) + 2pV xor V′ cos(πl) ) 1 - 4pV xor V′ sin (πl/2) (11) which is quite similar to the expression for Ω0. ID(h) can then be reformulated accordingly: 2
ID(h) ) (sin2(πl/2)
∑n δ(l - n))I⊥(h)
(12)
Here, the factor of sin2(πl/2) combines with the delta function so as to be nonzero only when l is an odd integer. This
(13)
(14)
where M(h) depends on the direction of the correlations: sin2(πh/2) for AB-type chain correlations, sin2(πk/2) for ACtype correlations, and cos2(π[hπ ( k]/2) for AD-type correlations. While these simple functions only include the coarsest features that can be identified with each type of nearest-neighbor correlation, the modulations that they impose on the sheets of diffuse scattering persist when the approximations that led to eqs 13 and 14 are relaxed and provide clues to the nature of the interchain correlations that will be resolved by more complete diffuse scattering data. Conclusions Previous reports of correlations between diffracted intensities and aluminum content in as-synthesized Na mordenites (Si/Al ) 5-10) are shown to be explained as a selective suppression of the l ) 2n + 1 Bragg reflections. The suppression of these odd-l reflections is shown to be a consequence of the displacement vector, c/2, by which the affected regions of framework material are shifted. An empirical relationship between Al content and defect concentration is established, indicating that the presence of framework Al may drive the formation of the shifted-chain defects during crystal growth. We demonstrate that
62 J. Phys. Chem. B, Vol. 106, No. 1, 2002 chains of fault-shifted 4MR pillars parallel to the main [001] channel axis produce thin sheets of diffuse intensity that are restricted to the odd-l planes of reciprocal space. The odd-l sheets grow in intensity with increasing defect concentration by a factor of 4p0(1 - p0), while the odd-l diffracted intensities diminish by the same factor: 1 - 4p0(1 - p0) ) (1 - 2p0)2. These quantitative relationships provide a straightforward means of understanding, controlling, and monitoring the presence of shifted-pillar framework defects in zeolite mordenite. Acknowledgment. This work was supported, in part, by the MRSEC Program of the National Science Foundation, Contract No. DMR00-80034. References and Notes (1) Sie, S. T. Stud. Surf. Sci. Catal. 1994, 85, 587-631. (2) Maxwell, I. E.; Williams, C.; Muller, F.; Krutzen, B. Zeolite Catalysis - For the Fuels of Today and Tomorrow; Selected Papers Series; Shell International Chemical Co. Ltd.: 1992. (3) Millward, G. R.; Thomas, J. M.; Terasaki, O.; Watanabe, D. Zeolites 1986, 6, 91. (4) Eckert, J.; Stucky, G. D.; Cheetham, A. K. MRS Bull. 1999, 31. (5) Newsam, J. M.; Treacy, M. M.; Koetsier, W. T.; DeGruyter, C. B. Proc. R. Soc. London A 1988, 420, 375. (6) Higgins, J. B.; LaPierre, R. B.; Schlenker, J. L.; Rohrman, A. C.; Wood, J. D.; Kerr, G. T.; Rohrbaugh, W. J. Zeolites 1988, 8, 446. (7) Breck, D. W. Zeolite Molecular SieVes, Structure, Chemistry, and Use; Wiley & Sons: New York, 1974.
Campbell and Cheetham (8) Skeels, G. W.; Sears, M.; Bateman, C. A.; McGuire, N. K.; Flanigen, E. M.; Kumar, M.; Kirchner, R. M. Micro. Meso. Mater. 1999, 30, 335. (9) Sanders, J. V. Zeolites 1985, 5, 81. (10) Meier, W. M.; Olsen, D. H. Atlas of Zeolite Structure Types, 3rd ed.; published on behalf of the Structure Commission of the International Zeolite Association; Butterworth-Heinemann: London, 1992. (11) Meier, W. M. Z. Kristallografiya 1961, 115, 439. (12) Keough, A. H.; Sand, L. B. J. Am. Chem. Soc. 1961, 83, 3536. (13) Sand, L. B. In Molecular SieVes; The Society of Chemical Industry: London, 1968; pp 71-77. (14) See the special comments at the end of ref 13. (15) Bennett, J. M., Ph.D. Thesis, University of Aberdeen, Scotland, 1966, as cited in ref 24. (16) Mortier, W. J.; Pluth, J. J.; Smith, J. V. Mater. Res. Bull. 1975, 10, 1319. (17) Schlenker, J. L.; Pluth, J. J.; Smith, J. V. Mater. Res. Bull. 1979, 14, 849. (18) Itabashi, K.; Fukushima, T.; Igawa, K. Zeolites 1986, 6, 30. (19) Shiokawa, K.; Ito, M.; Itabashi, K. Zeolites 1989, 9, 170. (20) Rudolf, P. R.; Garce´s, J. M. Zeolites 1994, 14, 137. (21) Barrer, R. M. Zeolites and Clay Minerals as Sorbents and Molecular SieVes; Academic Press: New York, 1978. (22) Musa, M.; Tarina, V.; Stoica, A. D.; Ivanov, E.; Plostinaru, D.; Pop, E.; Pop, Gr.; Ganea, R.; Birjega, R.; Musca, G.; Paukshtis, E. A. Zeolites 1987, 7, 427. (23) Warren, B. E. X-ray Diffraction; Dover: New York, 1990. (24) Sherman, J. D.; Bennett, J. M. In Molecular SieVes; Meier, W. M., Uytterhoeven, J. B., Eds.; American Chemical Society: Washington, DC, 1973; pp 53-65.
