The X-ray diffraction pattern of mullite is very similar to that of sillimanite. Sillimanite is a commonly occurring aluminosilicate mineral stable at high pressures with the chemical formula Al4Si2O10, a 1:1 ratio of silica to alumina.
Roughly speaking, the sillimanite and mullite structures consist of chains of distorted edge-sharing Al-O octahedra at the corners and center of each unit cell running parallel to the c-axis. The chains are cross-linked by Si-O and Al-O corner-sharing tetrahedra . Mullite is a solid solution compound with stoichiometries ranging from relatively silica-rich 3Al2O32SiO2 (3:2 mullite) to alumina-rich 2Al2O3 SiO2 (2:1 mullite). The structure of mullite is summarized in Table 1. Some authors use the Al/ Si ionic ratio when referring to mullite stoichiometry. In this case, 3:2 mullite would have an aluminum/silicon ionic ratio of 3:1. To avoid further confusion and follow the convention most commonly used in the literature, mullite stoichiometry will be based on the alumina/silica molecular ratio. The chemical formula for mullite is often given by Al2(Al2+^Si2_^)O10_x, where x = 0 corresponds to sillimanite, x = 0.25 corresponds
Table 1 Wyckoff positions and coordinates of atom sites for the orthorhombic mullite structure with space group Pbam (No. 55)
The chemical formula is Al2(Al2+2xSi2-2x)O10_x, where x = 0.33 and the calculated density is 3.16 g cm-3. From 
to 3:2 mullite, and x = 0.4 corresponds to 2:1 mullite. Diffusion studies  have shown that the following chemical formula is more appropriate even though it is not commonly seen in the literature:
The symbol denotes an oxygen vacancy. The superscripts VI and IV indicate octahedral and tetrahedral coordination sites, respectively.
With increasing alumina content, Si4+ is replaced by Al3+ and anion (oxygen) vacancies are created to maintain charge neutrality. Accommodating the structural defects causes significant distortions of the aluminum and silicon polyhedra. In mullite (as opposed to sillimanite), there are three (as opposed to four) tetrahedral “chains” in the unit cell, with a somewhat random distribution for silica and alumina tetrahedra . This results in the necessity for distorted alumina tetrahedra to be arranged in an oxygen-deficient tricluster (three tetrahedra sharing single corner-bridging oxygen). These clusters constitute a distinctive element of mullite’s crystal structure [2,5].
Unlike sillimanite, X-ray diffraction patterns of mullite exhibit significant diffuse scattering and possible superlattice reflections. Authors have proposed various models to account for mullite’s anomalous scattering using superlattice refinement, atomic site occupancy factor calculation, and correlated vacancy mapping [2,6,7]. Most work suggests that defects tend to cluster or correlate with short-range order along specific crystallographic directions. Lower alumina concentrations result in less directional correlation of oxygen vacancies or more random vacancy distributions. According to Freimann and Rahman , oxygen vacancies tend to correlate parallel with the lattice parameter a, and to a lesser extent with b. The authors suggest their correlation results could be used to interpret thermal expansion behavior of mullites. As a practical matter, the lattice parameter a correlates linearly with
Fig. 1 Structure of mullite. (a) Average structure and (b) atomic displacements around an oxygen vacancy. From 
Fig. 2 X-ray powder diffraction patterns showing the crystallization of mullite from amorphous precursors as a function of temperature. M denotes mullite peaks, and Sp markers denote the intermediate y-Al2O3 spinel peaks. From 
Al2O3 content. Figure 1 depicts the mullite unit. Atom positions for an intermediate composition of mullite, Al2(Al2+2xSi2_2x)O10_x, where x = 0.33 are provided in Table 1. X-ray powder diffraction patterns demonstrating mullite crystallization from amorphous precursors are shown in Fig. 2 .
It should be noted that there is no convincing evidence of mullite formation in regions of the phase diagram with compositions between 3:2 mullite and sillimanite. In other words, the chemical formula for mullite cannot accommodate x values such that 0 < x < 0.25. Although the presence of a cubic spinel with the stoichiometry and structure similar to that of 2:1 mullite had been reported [9,10], its existence is likely of academic rather than practical significance. What was originally reported as a tetragonal phase of 3:1 mullite  formed by rapid quenching of the melt could be attributed to severe microtwinning of the usual orthorhombic structure . On the other hand, workers have recently reported mullite phases with Al2O3/SiO2 ratios up to and greater than 9:1 [13-15]. These specialty compounds are potentially useful in specific refractory applications due to their high Al2O3 content. Unfortunately, it has proved difficult to produce these ultra-high alumina mullites in sufficient quantity and purity. Further research is required before practical applications for these materials can be envisioned.