Tetrahedral rotation

Figure 6. Stability of illite layers formed by WD mechanism percentage of change in percentage illite layers between 1 and 3 Sr-exchanges is plotted against Of, the angle of tetrahedral rotation. Data from Table V.

Figure 5,44 Sketch of tetrahedral rotational angle a, for two limiting conditions, = 0° and = 12°. From Hazen and Wones (1972). Reprinted with permission of The Mineral-ogical Society of America.

They explained the fit of the dioctahedral and trioctahedral sheets by the thinning of the dioctahedral sheet (2.05 A) and the thickening of the trioctahedral sheet (2.15 A), giving a mean lateral octahedral edge of 3.02 in both sheets. To compensate for the thinning of the dioctahedral sheet the tetrahedral sheet thickened slightly. The tetrahedral rotation is in the same direction as for the other chlorites, but the angle is somewhat less than expected. 

Although the octahedral sheet of chamosite is composed predominantly of Fe2+ (radius = 0.75) rather than the smaller Mg (radius = 0.65), the A1 content (x > 1.20) is sufficiently large so that the tetrahedral rotation is necessary to adjust the size differential (Radoslovich, 1963). As Mg increases relative to Fe2+, less A1 is required to afford a strain-free structure. There appears to be no reason that there cannot be a continuous isomorphous series between chamosite and serpentine. 

Fig.32. Plot of calculated degree of tetrahedral rotation versus calculated b oct./b tet. values for the various clay minerals. Linear relations are based on data from Radoslovich and Norrish (1962). Triangles show range of calculated tetrahedral rotation for the various clay groups (see Fig.31).

The data for the expanded and contracted minerals plot as two separate linear relations with contracted clays having larger tetrahedral rotation values for given oct.Atet. values than the expanded clays. This is presumably due to the K which aids the tetrahedral rotation in the contracted clays. 

Using the structural formulas that were used to plot Fig.29, the oct./ tet. ratio was calculated for a number of clays and the amount of tetrahedral rotation estimated from the graphs in Fig.32. Some of these values are shown in Fig.31. The montmoril-lonites with a low tetrahedral A1 and low octahedral R3+ have high ratio values and presumably a low degree of tetrahedral rotation (0° —1.5°). As the amount of octahedral R3+ increases, Mg decreases, the octahedral sheet becomes smaller, and the amount of tetrahedral rotation increases (6.5°). 

The amount of rotation systematically increases (attaining a maximum value of approximately 10°) concomitantly with an increase in the amount of tetrahedral A1 and an increase in size by the tetrahedral sheet. When much of the octahedral A1 is replaced by the larger Fe3+ (nontronite), the amount of rotation decreases. As the amount of tetrahedral A1 increases, the amount of octahedral R3+ remains relatively constant and the tetrahedral rotation increases from 0°—3° to 7.5°. 

Present data indicate that Fe3+-rich low-charge clays increase their layer charge by increasing the Mg and Fe2+ content of the octahedral sheet at the expense of Fe3 + more so than of Al. The average Al content of glauconite and celadonite is similar to that of nontronite, but the Fe3+ values are lower. With increased octahedral charge there is an increase anion-anion repulsion and the octahedral sheet increases relatively more in the c direction than the 6 direction, which also favors the large cations. Thus, relatively less tetrahedral Al is required to afford the sheet size differential to allow sufficient tetrahedral rotation to lock the K into place. 

The calculated tetrahedral rotation for glauconites with high octahedral R3 + values ranges from 8 to 10°. There appears to be no overlap of the illite values (12°—13.5°). As the amount of octahedral R3+ decreases, the octahedral sheet increases in size and charge and rotation values decrease. As the amount of tetrahedral Al decreases, the sheets become similar in size and the amount of rotation approaches zero (K cannot be locked in position to provide sufficient layer separation). As octa-... 

Using the methods discussed in the tutorial (Sect. 2), the group of symmetry operations of Hso can be constructed explicitly. As is shown in Appendix B, the symmetry group of Hso of (53) is T, the spin double group of the tetrahedral rotation group, is of order 48. T also is the symmetry group of the total Hamiltonian H = Hes + Hso-... 

It may happen that for certain values of / more than one totally symmetric combination (10) may exist. In that case, the / index should be understood as a composite label. As an example, we list in Table I, for values of l up to 10, the tetrahedral rotation functions that transform according to the totally symmetric representation of the tetrahedral group T. These functions are normalized such that... 

Tetrahedral Rotation Functions Adapted to the Site Group O [See Eq. (33)] ... 

The 3-r symbols of the group 0 can be taken as 3-F symbols also for the isomorphic group Ta, and these symbols may again be used for the tetrahedral rotation group T when a real basis is sufficient. 

---