Physics and Chemistry of Minerals

, Volume 35, Issue 1, pp 11–16 | Cite as

Order parameter coupling in leucite: a calorimetric study

  • Helen Newton
  • Stuart A. Hayward
  • Simon A. T. Redfern
Original Paper

Abstract

The thermal anomalies associated with the Ia3dI41/acdI41/a transition sequence of phase transitions in leucite have been studied by differential scanning calorimetry and interpreted with Landau theory. Both transitions are close to the tricritical point. The coupling between the two transitions is biquadratic, and reduces the stability of the I41/a phase.

Keywords

Leucite Calorimetry Phase transitions 

Introduction

Leucite is a framework silicate mineral displaying a rich range of structural behaviour. The fundamental structural units are AlO4 and SiO4 tetrahedra, linked at their vertices to form a network. This network is rather complex; the key features are channels parallel to <111> directions, as shown in Fig. 1.
Fig. 1

Crystal structure of cubic leucite (Peacor 1968), projected down the [111] axis. The AlO4 and SiO4 tetrahedra are disordered, and the K+ cations are shown as isolated spheres

Silicate frameworks of this type tend to be rather flexible (Dove 1997). These degrees of freedom mean that framework silicates frequently undergo displacive phase transitions. They also tend to have good tolerance for chemical substitution. Whilst natural leucites depart only slightly from the ideal composition KAlSi2O6 (Deer et al. 2004), leucite analogues with a wide range of compositions have been synthesised (Bayer 1973; Taylor and Henderson 1968; Galli et al. 1978).

The precise details of the transformation behaviour of leucite have proved difficult to pin down. The most important complication is that there are two phase transitions in leucite, separated by a rather small temperature interval. The first evidence for this was from thermal analysis (Faust 1963) indicating transitions at 938 and 918 K; subsequent measurements (Lange et al. 1986) indicated that the behaviour was more complex and sample-dependent; in particular, annealing a natural leucite reduced the transition temperatures by 24 K. Meanwhile, crystallographic experiments indicated that the space group of the high temperature phase was Ia3d (Peacor 1968) and that of the room temperature phase was I41/a (Mazzi et al. 1976). Later studies (Grögel et al. 1984; Boysen 1990) found the intermediate phase, indicated by the calorimetric data, to have space group I41/acd.

Palmer and co-workers studied leucite using a range of experimental techniques (Palmer et al. 1989, 1990, 1997). Combining these data, Palmer concluded that the Ia3d → I41/acd transition was due to an acoustic shear distortion (Palmer et al. 1990) and that the I41/acd → 41/a transition was associated with the freezing of the K+ substructure combined with an additional framework distortion (Palmer 1990).

Despite the progress in understanding the mechanisms of the transformations in leucite, a complete quantitative description of the transformations has proved more elusive. Approaches based on Landau theory have proved to be very effective in describing phase transitions in framework silicates (Salje et al. 2005), particularly in situations where several transitions interact. In this paper, we combine new calorimetric results for leucite with existing structural data to determine a complete Landau potential for the two phase transitions in leucite and the interaction between them.

Experimental methods

Sample characterisation

A sample of natural leucite from the Roman volcanic province was obtained from the University of Cambridge collection. The lattice parameters and composition of the leucite sample studied here were checked for consistency with other experiments. Lattice parameters were determined using powder X-ray diffraction at room temperature, using a Bruker D8 diffractometer. The diffraction pattern refined as a tetragonal structure, with a = 13.0547(8) Å, c = 13.7555(11) Å. These values are similar to the natural leucite from near Rome, Italy studied by Palmer et al. (1997). They are also close to the values obtained for natural leucite L999 of Palmer et al. (1989, 1990), which were a = 13.059(3) Å, c = 13.756(2) Å.

The composition was determined with a Cameca SX100 electron microprobe. The analysed formula (K0.999Na0.012)(Al1.005Fe0.012Ca0.001)Si1.986O6, is close to the ideal formula of leucite, KAlSi2O6; this is common in natural leucites (Deer et al. 2004). The differences between this sample and natural leucite L999 (Palmer et al. 1989, 1990), which has been analysed as K0.97(Al0.99Fe0.01)Si2.01O6 do not seem significant.

Calorimetric experiment

A large euhedral single crystal was prepared for calorimetry by being cut into thin slices with a diamond saw. It was then polished into a thin disc, with 45 mg mass. This sample was heated in a Perkin-Elmer “Diamond” DSC from 325 to 973 K at a rate of 20 K min−1. Comparison of the heat flow data for this sample with results for an empty calorimeter and a sapphire standard were used to determine the temperature dependence of the specific heat.

Results

The specific heat of leucite as a function of temperature is shown in Fig. 2. Specific heat data are also given in Supplementary Table 1 of Electronic supplementry materials. The baseline specific heat (phonon contribution) is essentially independent of temperature above 800 K, as shown by the broken line. The two peaks in CP are associated with the two phase transitions. The observation that each transition appears as a peak in CP implies that both transitions are Landau tricritical or first order, rather than second order (which would give a step change in CP). The transition temperatures obtained in this study are 919 and 900 K. These temperatures are approximately 20 K below those obtained in earlier studies; the reasons for this discrepancy are unclear, though they are most likely related to differences in the composition and thermal history of the different samples, which may not be detected from simple structural analysis. Another possibility is that there are differences in temperature calibration between different experiments. Integrating the excess in CP over the baseline leads to the excess entropy associated with the transition, as shown in Fig. 3.
Fig. 2

Temperature dependence of the specific heat in leucite, showing the two transitions as peaks in CP

Table 1

Landau coefficients for phase transitions in leucite

Coefficients for Ia3d → I41/acd transition

A1

14.1 (2)

J K−1 mol−1

B1

−836 (9)

J mol−1

C1

13,780 (140)

J mol−1

TC1

917.9 (1)

K

Coefficients for I41/acd → I41/a transition

A2

5.3 (2)

J K−1 mol−1

B2

0

J mol−1

C2

4,830(50)

J mol−1

TC2

907 (1)

K

Coupling coefficients

λ1

0

J mol−1

λ2

3,400(100)

J mol−1

Fig. 3

Temperature dependence of the excess entropy associated with the phase transitions in leucite

Data analysis and discussion

Landau theory

Description of the two phase transitions in leucite requires the use of two coupled order parameters. The overall free energy for the transitions then consists of the energy of each order parameter, and some sort of coupling energy;
$$ G{\left( {Q_{1} ,Q_{2} } \right)} = G{\left( {Q_{1} } \right)} + G{\left( {Q_{2} } \right)} + G{\left( {\rm {coupling}} \right)}, $$
(1)
where Q1 is the order parameter for the Ia3d → I41/acd (higher temperature) transition, and Q2 is the order parameter for the I41/acd → I41/a (lower temperature) transition.
The form of G(coupling) depends on the symmetries of Q1 and Q2. Unusually, two low-order couplings are compatible with observed symmetry changes in leucite. Thus Eq. 1 above expands as
$$ G{\left( {Q_{1} ,Q_{2} } \right)} = \frac{{A_{1} }} {2}{\left( {T - T_{{\rm C1}} } \right)}Q^{2}_{1} + \frac{{B_{1} }} {4}Q^{4}_{1} + \frac{{C_{1} }} {6}Q^{6}_{1} + \frac{{A_{2} }} {2}{\left( {T - T_{{\rm C2}} } \right)}Q^{2}_{2} + \frac{{B_{2} }} {4}Q^{4}_{2} + \frac{{C_{2} }} {6}Q^{6}_{2} + \lambda _{1} Q_{1} Q^{2}_{2} + \lambda _{2} Q^{2}_{1} Q^{2}_{2} . $$
(2)
In the Ia3d phase, both Q1 and Q2 are zero. In the I41/acd phase, Q1 is non-zero and Q2 is zero, and in the I41/a phase both Q1and Q2 are non-zero. In the I41/acd phase, the coupling between Q1and Q2 causes deflection of the order parameter vector. The form of this deflection depends on the nature of the coupling term. It is likely that either λ1 or λ2 is dominant. If λ1 is large λ2 is negligible, the coupling is linear-quadratic. From the equilibrium condition ∂G/∂Qi = 0,
$$ Q_{1} = A_{1} {\left( {T - T_{{\rm C1}} } \right)} + B_{1} Q^{2}_{1} + C_{1} Q^{4}_{1} + \lambda _{1} Q^{{ - 1}}_{1} Q^{2}_{2} $$
(3)
$$ Q_{2} = A_{2} {\left( {T - T_{{\rm C2}} } \right)} + B_{2} Q^{2}_{2} + C_{2} Q^{4}_{2} + 2\lambda _{1} Q_{1} $$
(4)
If, on the other hand, the dominant coupling is biqudratic (negligible λ1, large λ2), the same equilibrium condition leads to
$$ Q_{1} = A_{1} {\left( {T - T_{{\rm C1}} } \right)} + B_{1} Q^{2}_{1} + C_{1} Q^{4}_{1} + 2\lambda _{2} Q^{2}_{2} $$
(5)
$$ Q_{2} = A_{2} {\left( {T - T_{{\rm C2}} } \right)} + B_{2} Q^{2}_{2} + C_{2} Q^{4}_{2} + 2\lambda _{2} Q^{2}_{1} . $$
(6)

Using Eqs. 3 and 5, we may compare the (actual) coupled behaviour of Q1 with its uncoupled behaviour (Q1,0) for each coupling model. If we make the approximation that Q1,0 is nearly tricriticial, then

linear-quadratic coupling:
$$ Q^{4}_{1} = Q^{4}_{{1,0}} - \frac{{\lambda _{1} Q^{2}_{2} }} {{C_{1} Q_{1} }}, $$
(7)
biquadratic coupling:
$$ Q^{4}_{1} = Q^{4}_{{1,0}} - \lambda _{2} Q^{2}_{2} . $$
(8)

The problem of determining the parameters in Eq. 2 therefore needs to be tackled in several discrete steps. Experimental data for the I41/acd phase permits the fitting of the parameters A1, B1, C1 and TC1. These parameters then determine what the behaviour of Q1 in the I41/a phase would be in the absence of order parameter coupling. The deviation of Q1 from this behaviour constrains the form and magnitude of the coupling, which then allows the bare free energy parameters for Q2 (A2 etc.) to be determined.

Use of spontaneous strains to measure order parameters

In the study of Palmer et al. (1989), experimental spontaneous strain data are used to quantify the order parameters. Two strains are observed in leucite; the symmetry breaking strain esb and the volume strain ensb, as defined by Palmer et al. (1989). The temperature dependencies of these strains are shown in Fig. 4. The transition temperatures observed in this study (938 and 918 K) are approximately 20 K higher than obtained here.
Fig. 4

Temperature dependencies of the symmetry-breaking strain (esb, solid circles) and non-symmetry-breaking strain (ensb, open circles) in leucite, from the X-ray diffraction data of Palmer et al. (1989)

Carpenter et al. (1998) review the use of spontaneous strains to measure order parameters associated with phase transitions in minerals. The lowest order coupling (for example e ∝ Q or e ∝ Q2) allowed is constrained by group theoretical principles. Higher order couplings not commonly significant, but are allowed.

The symmetry breaking strain esb vanished at the upper transition temperature. This suggests that it can be associated with Q1. From symmetry arguments, we expect esb ∝ Q. A volume strain, ensb is allowed in any structural phase transition, with ensb ∝ Q2.

In the case of leucite, the volume strain associated with Q1 is rather small; the major contribution to the volume strain only occurs below the lower transition temperature. This point is made most clearly by plotting values of esb and ensb against each other, as shown in Fig. 5. Interestingly, the relationship between the two strains in the I41/acd phase field shows the two strains being proportional. Since volume strains are always proportional to Q2, this implies that the actual coupling between esb and the order parameter is of the form esb ∝ Q2.
Fig. 5

Mutual variation of the strains esb and ensb at various temperatures in leucite; data from Palmer et al. (1989). Temperatures in K are indicated next to data points. For 20 K below the upper transition temperature, esb and ensb are proportional (broken line); with further cooling, ensb increases more rapidly than esb

Characterisation of the Ia3→ I41/acd transition

In the I41/acd phase, Eq. 2 simplifies to
$$ G{\left( {Q_{1} ,0} \right)} = \frac{{A_{1} }} {2}{\left( {T - T_{{\rm C1}} } \right)}Q^{2}_{1} + \frac{{B_{1} }} {4}Q^{4}_{1} + \frac{{C_{1} }} {6}Q^{6}_{1} $$
(9)
and so the specific heat anomaly is given by
$$ {\left( {\frac{T} {{\Delta C_{\rm P} }}} \right)}^{2} = \frac{{4B^{2}_{1} + 16A_{1} C_{1} {\left( {T_{\rm C} - T} \right)}}} {{A^{4}_{1} }}. $$
(10)
The temperature dependence of (TCP)2 is shown in Fig. 6; the linear behaviour seen in the vicintiy of the upper transition temperature is consistent with a Landau-like first order transition. If the transition temperature is taken to be the temperature of the CP maximum, we obtain TTR = 919.18 K. The upper stability limit of the tetragonal phase is the temperatue where ΔCP extrapolates to infinity, giving T2 = 918.78 K. Hence the transition is only slightly first order. Given TTR, T2 and the gradient of the line in Fig. 6, the proceedure outined in Hayward et al. (2000) was used to determine the Landau coefficients, giving the results in the first part of Table 1.
Fig. 6

Landau analysis of specific heat data for leucite in the vicinity of the Ia3d → I41/acd phase transition in leucite

Coupling between Q1 and Q2

In Table 1, The coefficient B1 is negative, but very small relative to C1. Therefore, the Ia3d → I41/acd transition is first order, but only marginally so. Using the coefficients A1, B1, C1 and TC1 to calculate the temperature dependence of Q1, we find that the behaviour of Q1 is extremely close to tricritical (Q14 ∝ ∣TC − T∣). As noted above, the non symmetry-breaking strain is proportional to Q2, so the square of ensb should be linear with temperature. This behaviour is seen in the I41/acd stability field (Fig. 7), but not in the I41/a field. This deviation is due to the effect of the second order parameter Q2 coupling to the first order parameter Q1. As shown in Fig. 8, the form of this deviation is Δ(esb2) ∝ Δ(Q14) ∝ ∣TC2 − T∣. Comparing this behaviour with the two coupling models described by Eqs. 7 and 8 indicates that the behaviour of Q2 is essentially tricritical, and the coupling between Q1 and Q2 is biquadratic.
Fig. 7

Temperature dependence of the square of the tetragonal strain in leucite from Palmer et al. (1989). The solid line shows the behaviour expected for an exactly tricritical Ia3d → I41/acd transition

Fig. 8

Excess in the square of the tetragonal strain in leucite. The fit line is to a model Δ(esb2) ∝ ∣TC2 − T∣, which is consistent with biquadratic coupling

The sign of the coupling constant λ2 is positive (that is, the coupling increases the overall free energy of the system). This is manifest by the observation that the lower temperature peak in the dsc data occurs at 900 K, whereas the lower “bare” transition temperature is 907 K.

Bare behaviour of Q2

As noted above, the deviations seen in Q1 below the lower transition temperature, together with the form of the specific heat anomaly, imply that the behaviour of Q2 is essentially tricritical. No latent heat was observed around TC2. Whilst a small positive B2 is possible, none of the results of this study require it, or allow its numerical value to be determined. As a result of the coupling between Q1 and Q2, the lower temperature transition happens around 10 K below TC2; this is because the coupling reduces the stability of the I41/a phase.

Assuming that the lower temperature transition is exactly tricritical, there are then two independent parameters describing the thermodynamics of this transition. The value of A2 was found from the magnitude of the entropy anomaly at the lower transition, and TC2 from the observed transition temperature (taking account of the effect of the coupling with Q1). For a tricritical transition, B2 = 0, and C2 = A2 TC2, both by definition.

Final remarks

From Eq. 2, the excess entropy associated with the two phase transitions is given by
$$ S{\left( {Q_{1} ,Q_{2} } \right)} = - {\left( {\frac{{A_{1} }} {2}Q^{2}_{1} + \frac{{A_{2} }} {2}Q^{2}_{2} } \right)} $$
(11)
As a final test of the model described in this study, we compare the experimental measurements of the total excess entropy with the predictions of Eq. 11. To deal with the different TC values observed in various studies, we applied a constant shift of 20 K to the Q1, Q2 (T) data from Palmer et al. (1989); the resulting comparison in shown in Fig. 9, and shows good agreement over a range of 80 K. At lower temperatures, uncertainty in the phonon baseline contribution to CP (see Fig. 2) makes the experimental determination of the total excess entropy increasingly problematic.
Fig. 9

Comparison of the observed excess entropy (solid line) with model derived from experimental measurements of the order parameters (data from Palmer et al. (1989) with rescaled temperatures)

Our new measurements of the heat capacity of leucite demonstrate that the underlying thermodynamic character of both transitions is near-tricritical. Landau theory successfully describes the individual transitions, and the coupling between them. The data indicate that the dominant interaction between the two order parameters is biquadratic; whilst a linear-quadratic coupling is allowed by symmetry, in practice it is either undetectably weak or absent. The likely reason for this is that coupling via the spontaneous strain is a physically plausible mechanism for the interaction between the two order parameters (Salje and Devarajan 1986), and that coupling is biquadratic in the case of leucite.

References

  1. Bayer G (1973) Thermal expansion of new leucite-type compounds. Naturwissenshaften 60:102–103Google Scholar
  2. Boysen H (1990) Neutron scattering and phase transitions in leucite. In: Salje EKH (ed) Phase transitions in ferroelastic and co-elastic materials. Cambridge University Press, Cambridge, pp 334–349Google Scholar
  3. Carpenter MA, Salje EKH, Graeme-Barber A (1998) Spontaneous strain as a determinant of thermodynamic properties for phase transitions in minerals. Eur J Mineral 10:621–691Google Scholar
  4. Deer WA, Howie RA, Wise WS, Zussman J (2004) Rock-forming minerals: volume 4B. Framework silicates, silica minerals, Feldspathoids and zeolites, 2nd edn. Geological Society, London, 958 ppGoogle Scholar
  5. Dove MT (1997) Theory of displacive phase transitions in minerals. Am Mineral 82:213–244Google Scholar
  6. Faust GT (1963) Phase transition in synthetic and natural leucite. Schweiz Mineral Petrograph Mitteil 43:165–195Google Scholar
  7. Galli E, Gottardi G, Mazzi E (1978) The natural and synthetic phases with the leucite framework. Mineral Petrograph Acta 22:185–193Google Scholar
  8. Grögel T, Boysen H, Frey F (1984) Neutron powder investigation of I41/aIa3d in leucite. Coll Abst 13th Int Cong Cryst C256–C257Google Scholar
  9. Hayward SA, Romero FJ, Gallardo MC, del Cerro J, Gibaud A, Salje EKH (2000) Cubic–tetragonal phase transition in KMnF3: excess entropy and spontaneous strain. J Phys: Condens Matt 12:1133–1142CrossRefGoogle Scholar
  10. Lange RA, Carmichael ISE, Stebbins JF (1986) Phase transitions in leucite (KAlSi2O6), orthorhombic KAlSiO4, and their iron analogues (KFeSi2O6, KFeSiO4). Am Mineral 71:937–945Google Scholar
  11. Mazzi F, Galli E, Gottardi G (1976) The structure of tetragonal leucite. Am Mineral 61:108–115Google Scholar
  12. Palmer DC (1990) Phase transitions in leucite. PhD thesis, University of CambridgeGoogle Scholar
  13. Palmer DC, Salje EKH, Schmahl WW (1989) Phase transitions in leucite: X-ray diffraction studies. Phys Chem Minerals 16:714–719CrossRefGoogle Scholar
  14. Palmer DC, Bismayer U, Salje EKH (1990) Phase transitions in leucite: order parameter behaviour and the Landau potential deduced from Raman spectroscopy and birefringence studies. Phys Chem Minerals 17:259–265Google Scholar
  15. Palmer DC, Dove MT, Ibberson RI, Powell BM (1997) Structural behavior, crystal chemistry, and phase transitions in substituted leucite: high-resolution neutron powder diffraction studies. Am Mineral 82:16–29Google Scholar
  16. Peacor DR (1968) A high temperature single crystal diffractometer study of leucite (K,Na)AlSi2O6. Zeit Krist 127:213–224CrossRefGoogle Scholar
  17. Salje E, Devarajan V (1986) Phase transitions in systems with strain-induced coupling between two order parameters. Phase Trans 6:235–248CrossRefGoogle Scholar
  18. Salje EKH, Hayward SA, Lee WT (2005) Phase transitions: structure and microstructure. Acta Cryst 61:3–18Google Scholar
  19. Taylor D, Henderson CMB (1968) The thermal expansion of the leucite group of minerals. Am Mineral 53:1476–1489Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Helen Newton
    • 1
  • Stuart A. Hayward
    • 1
  • Simon A. T. Redfern
    • 1
  1. 1.Department of Earth SciencesUniversity of CambridgeCambridgeUK

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