Thermoelastic stability of multiple growth twins in quartz and general geobarothermometric implications

Abstract

Al~tract. The thermoelastic analysis of the possible stable stress-free equilibrium states of polycrystals provides a theoretical way of distinguishing "true" twins from random intergrowths. We analyze in detail a four-fold cyclic twin in quartz, and we show that this gives also the basis for a new technique of geobarothermometry. To obtain the harothermometric function, the simultaneous dependence upon the external pressure and temperature of the lattice-parameter has been estimated; an explicit five-parameter, temperature-corrected version of the Thurston isothermal constitutive equations is presented.Rias~tmto. L'analisi degli stati naturali possibili nei policristalli termoelastici fornisce una base teorica per la distinzione tra "veri" geminati di crescita e formazioni efistalline casuali. Si analizza qui in dettaglio il caso di un gcminato cielico di ordine quattro del quarzo, e si mostra come ei6 offra anche un nuovo metodo geoharotermometrieo. Per rendere esplieita la funzione barotermometrica si ~ stimato l'effetto della pressione e della temperatura ambientali sui parametri reticolari. Si presenta una correzione a cinque parametri dipendenti dalla temperatura delle equazioni costitutive ottenute da Thurston nel caso isotermo.

On the Theory of Cyclic Growth Twins

“…As to how mineralogists interpret "parts", I will analyze two examples of trilling, suggesting three parts, in configurations involving six sections1 opposite sections have the same orientation, so just three orientations are involved. The 90 1 crosses in quartz that Zanzotto [6,7] and I [8] analyzed do allow for some different arrangements of shifts: we did not find observations of these. I shall discuss one of the solutions that qualifies as a cyclic twin, partly to introduce another possible interpretation of the analysis.…”

“…After going through these examples, it occurred to me that the analyses that Zanzotto [6,7] and I [8] did of the 90 1 crosses in 6-quartz, sometimes called low quartz, might be interpreted in a different way. In terms of the usual orthonormal basis, the lattice vectors are of the form Clearly, e 1 6 e 2 bisects the angle between these, thus forming 45 1 angles with both.…”

“…This seems not to happen, indicating that we are dealing with metastable equilibria, unless the environment is quite special. Given suitable measurements of thermal expansion etc, one could calculate pressure-temperature curves along which such configurations might have grown, as Zanzotto [6,7] did for rare 90 1 crosses in quartz. This could be useful in helping to understand the history of the environment in particular regions.…”

On the X-Ray Theory of Twinning

“…Previously, Zanzotto [24,25] had done interesting studies of the effects of pressure and temperature on rare quartz crosses involving these, my studies adding a little to his theory. He found that, in general, changes in pressure and/ or temperature induce shear stresses in such configurations unless their values lie on certain curves.…”

“…Here, (22) 7 requires that, for (9) to be satisfied, ghw T @ ghw p @ 4 and, for simplicity, I will only consider the possibility ghw T @ ghw p @ 4> (25) except where there is a note to the contrary. Then, the work of Pitteri [19] shows that, with U y denoting the rotation with axis y and angle # , either…”