# Order parameter coupling in leucite: a calorimetric study

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## Abstract

The thermal anomalies associated with the *Ia*3*d* → *I*4_{1}/*acd* → *I*4_{1}/*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 *I*4_{1}/*a* phase.

### Keywords

Leucite Calorimetry Phase transitions## Introduction

_{4}and SiO

_{4}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.

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 KAlSi_{2}O_{6} (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 *Ia*3*d* (Peacor 1968) and that of the room temperature phase was *I*4_{1}/*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 *I*4_{1}/*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 *Ia*3*d* →* I*4_{1}/*acd* transition was due to an acoustic shear distortion (Palmer et al. 1990) and that the *I*4_{1}/*acd* → 4_{1}/*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 (K_{0.999}Na_{0.012})(Al_{1.005}Fe_{0.012}Ca_{0.001})Si_{1.986}O_{6}, is close to the ideal formula of leucite, KAlSi_{2}O_{6}; 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 K_{0.97}(Al_{0.99}Fe_{0.01})Si_{2.01}O_{6} 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

*C*

_{P}are associated with the two phase transitions. The observation that each transition appears as a peak in

*C*

_{P}implies that both transitions are Landau tricritical or first order, rather than second order (which would give a step change in

*C*

_{P}). 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

*C*

_{P}over the baseline leads to the excess entropy associated with the transition, as shown in Fig. 3.

Landau coefficients for phase transitions in leucite

Coefficients for | ||

| 14.1 (2) | J K |

| −836 (9) | J mol |

| 13,780 (140) | J mol |

| 917.9 (1) | K |

Coefficients for | ||

| 5.3 (2) | J K |

| 0 | J mol |

| 4,830(50) | J mol |

| 907 (1) | K |

Coupling coefficients | ||

λ | 0 | J mol |

λ | 3,400(100) | J mol |

## Data analysis and discussion

### Landau theory

*Q*

_{1}is the order parameter for the

*Ia*3

*d*→

*I*4

_{1}/

*acd*(higher temperature) transition, and

*Q*

_{2}is the order parameter for the

*I*4

_{1}/

*acd*→

*I*4

_{1}/

*a*(lower temperature) transition.

*G*(coupling) depends on the symmetries of

*Q*

_{1}and

*Q*

_{2}. Unusually, two low-order couplings are compatible with observed symmetry changes in leucite. Thus Eq. 1 above expands as

*Ia*3

*d*phase, both

*Q*

_{1}and

*Q*

_{2}are zero. In the

*I*4

_{1}/

*acd*phase,

*Q*

_{1}is non-zero and

*Q*

_{2}is zero, and in the

*I*4

_{1}/

*a*phase both

*Q*

_{1}and

*Q*

_{2}are non-zero. In the

*I*4

_{1}/

*acd*phase, the coupling between

*Q*

_{1}and

*Q*

_{2}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 λ

_{1}or λ

_{2}is dominant. If λ

_{1}is large λ

_{2}is negligible, the coupling is linear-quadratic. From the equilibrium condition ∂

*G*/∂

*Q*

_{i}= 0,

_{1}, large λ

_{2}), the same equilibrium condition leads to

Using Eqs. 3 and 5, we may compare the (actual) coupled behaviour of *Q*_{1} with its uncoupled behaviour (*Q*_{1,0}) for each coupling model. If we make the approximation that *Q*_{1,0} is nearly tricriticial, then

The problem of determining the parameters in Eq. 2 therefore needs to be tackled in several discrete steps. Experimental data for the *I*4_{1}/*acd* phase permits the fitting of the parameters *A*_{1}, *B*_{1}, *C*_{1} and *T*_{C1}. These parameters then determine what the behaviour of *Q*_{1} in the *I*4_{1}/*a* phase would be in the absence of order parameter coupling. The deviation of *Q*_{1} from this behaviour constrains the form and magnitude of the coupling, which then allows the bare free energy parameters for *Q*_{2} (*A*_{2} etc.) to be determined.

### Use of spontaneous strains to measure order parameters

*e*

_{sb}and the volume strain

*e*

_{nsb}, 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.

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* ∝ *Q*^{2}) allowed is constrained by group theoretical principles. Higher order couplings not commonly significant, but are allowed.

The symmetry breaking strain *e*_{sb} vanished at the upper transition temperature. This suggests that it can be associated with *Q*_{1}. From symmetry arguments, we expect *e*_{sb} ∝ *Q*. A volume strain, *e*_{nsb} is allowed in any structural phase transition, with *e*_{nsb} ∝ *Q*^{2}.

*Q*

_{1}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

*e*

_{sb}and

*e*

_{nsb}against each other, as shown in Fig. 5. Interestingly, the relationship between the two strains in the

*I*4

_{1}/

*acd*phase field shows the two strains being proportional. Since volume strains are always proportional to

*Q*

^{2}, this implies that the actual coupling between

*e*

_{sb}and the order parameter is of the form

*e*

_{sb}∝

*Q*

^{2}.

### Characterisation of the *Ia*3*d *→ *I*4_{1}/*acd* transition

*I*4

_{1}/

*acd*phase, Eq. 2 simplifies to

*T*/Δ

*C*

_{P})

^{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

*C*

_{P}maximum, we obtain

*T*

_{TR}= 919.18 K. The upper stability limit of the tetragonal phase is the temperatue where Δ

*C*

_{P}extrapolates to infinity, giving

*T*

_{2}= 918.78 K. Hence the transition is only slightly first order. Given

*T*

_{TR},

*T*

_{2}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.

### Coupling between *Q*_{1} and *Q*_{2}

*B*

_{1}is negative, but very small relative to

*C*

_{1}. Therefore, the

*Ia*3

*d*→

*I*4

_{1}/

*acd*transition is first order, but only marginally so. Using the coefficients

*A*

_{1},

*B*

_{1},

*C*

_{1}and

*T*

_{C1}to calculate the temperature dependence of

*Q*

_{1}, we find that the behaviour of

*Q*

_{1}is extremely close to tricritical (

*Q*

_{1}

^{4}∝ ∣

*T*

_{C}−

*T*∣). As noted above, the non symmetry-breaking strain is proportional to

*Q*

^{2}, so the square of

*e*

_{nsb}should be linear with temperature. This behaviour is seen in the

*I*4

_{1}/

*acd*stability field (Fig. 7), but not in the

*I*4

_{1}/

*a*field. This deviation is due to the effect of the second order parameter

*Q*

_{2}coupling to the first order parameter

*Q*

_{1}. As shown in Fig. 8, the form of this deviation is Δ(

*e*

_{sb}

^{2}) ∝ Δ(

*Q*

_{1}

^{4}) ∝ ∣

*T*

_{C2}−

*T*∣. Comparing this behaviour with the two coupling models described by Eqs. 7 and 8 indicates that the behaviour of

*Q*

_{2}is essentially tricritical, and the coupling between

*Q*

_{1}and

*Q*

_{2}is biquadratic.

The sign of the coupling constant λ_{2} 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 *Q*_{2}

As noted above, the deviations seen in *Q*_{1} below the lower transition temperature, together with the form of the specific heat anomaly, imply that the behaviour of *Q*_{2} is essentially tricritical. No latent heat was observed around *T*_{C2}. Whilst a small positive *B*_{2} 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 *Q*_{1} and *Q*_{2}, the lower temperature transition happens around 10 K below *T*_{C2}; this is because the coupling reduces the stability of the *I*4_{1}/*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 *A*_{2} was found from the magnitude of the entropy anomaly at the lower transition, and *T*_{C2} from the observed transition temperature (taking account of the effect of the coupling with *Q*_{1}). For a tricritical transition, *B*_{2} = 0, and *C*_{2} = *A*_{2}* T*_{C2}, both by definition.

## Final remarks

*T*

_{C}values observed in various studies, we applied a constant shift of 20 K to the

*Q*

_{1},

*Q*

_{2}(

*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

*C*

_{P}(see Fig. 2) makes the experimental determination of the total excess entropy increasingly problematic.

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.

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