Method for constructing theoretical phase diagrams for tetramethylammonium family crystals
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ICE DYNAMICS AND PHASE TRANSITIONS
Method for Constructing Theoretical Phase Diagrams for Tetramethylammonium Family Crystals D. G. Sannikov Shubnikov Institute of Crystallography, Russian Academy of Sciences, Leninskii pr. 59, Moscow, 119333 Russia email: [email protected] Received February 15, 2012
Abstract—A method for constructing theoretical phase diagrams for tetramethylammonium family crystals is proposed. Temperature–pressure phase diagrams are constructed for different crystals of this family and compared with experimental diagrams. The assumptions and approximations of the method are discussed. DOI: 10.1134/S1063774512060107
SOFT OPTICAL BRANCH IN THE SPECTRUM OF NORMAL VIBRATIONS OF A CRYSTAL Many crystals exhibit a complex sequence of phase transitions (involving an incommensurate phase) with a change in temperature Т, pressure Р, or electric field Е. For clarity, we will use the terminology of lat tice dynamics theory. We naturally assume that the soft optical branch of normal vibrations of a crystal is responsible for the entire set of observed phase transi tions. The dependence of the elasticity coefficient for the soft optical branch on the dimensionless wave number q (q = qa∗, b∗, c∗), α(q), for many experi mentally studied crystals is described by three sim ple relations (1)–(3). α(q) = α – σq + δq2,
σ > 0,
δ > 0.
(1)
The presence of a term linear in q in (1) indicates that the corresponding thermodynamic potential con tains a Lifshitz invariant [1]. Examples of crystals of this type are potassium selenate K2SeO4 and ammo nium fluoroberyllate (NH4)2BeF4. These crystals will not be considered here. They exhibit a sequence of two phase transitions, both with a change in Т and with a change in Р: C – IC – Cm/l, where С is the initial phase, IC is an incommensurate phase, and Cm/l is a commen surate phase (m/l = 1/3 or 1/2). α(q) = α – δq2+ κq4,
δ > 0,
κ > 0.
(2)
Examples of these crystals are thiourea SC(NH2)2 and BCCD (betaine calcium chloride dihydrate). Т–Р, Т–Е, and Т–Р phase diagrams have been observed for these crystals at Е ≠ 0. The proposed method was applied to these crystals, and the corre sponding phase diagrams were constructed (see [2–4] and references therein). These crystals will not be con sidered here either. α(q) = α – δq2 – κq4 + τq6, κ > 0, τ > 0. (3)
This case covers a large family of wellstudied tet ramethylammonium (ТМА) crystals: ТМА–МХ, where ТМА stands for [N(CH3)4]2, М is a metal (Mn, Fe, Co, Ni, Cu, or Zn), and Х is a halogen (Cl, Br, or I) [5, 6]. Below we will construct temperature–pressure (Т–Р) phase diagrams for three such crystals. The method for constructing theoretical Т–Р diagrams is based on a phenomenological approach to the devil’s staircase problem [7]. The optical branch (3) for ТМА crystals has two minima in a certain range of values of the coefficient δ (–κ2/3τ < δ < 0). One minimum is at an arbitrary point of the Brillouin zone; it determines, as in cases (1) and (2), the phase transitions from the initial phase С to an incommen
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