ORIGINAL PAPER

Order parameter coupling in leucite: a calorimetric study

Helen Newton Æ Stuart A. Hayward ÆSimon A. T. Redfern

Received: 2 August 2006 / Accepted: 19 September 2007 / Published online: 31 October 2007

� Springer-Verlag 2007

Abstract The thermal anomalies associated with the Ia3d

? I41/acd ? I41/a transition sequence of phase transitions

in leucite have been studied by differential scanning cal-

orimetry 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.

Silicate frameworks of this type tend to be rather flex-

ible (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 transi-

tions 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 sam-

ple-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 (Grogel 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 elu-

sive. Approaches based on Landau theory have proved to

be very effective in describing phase transitions in

framework silicates (Salje et al. 2005), particularly in sit-

uations 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.

H. Newton � S. A. Hayward (&) � S. A. T. Redfern

Department of Earth Sciences,

University of Cambridge, Downing Street,

Cambridge CB2 3EQ, UK

e-mail: satr@esc.cam.ac.uk

123

Phys Chem Minerals (2008) 35:11–16

DOI 10.1007/s00269-007-0193-3

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 deter-

mined 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) A,

c = 13.7555(11) A. 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) A, c = 13.756(2) A.

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 for-

mula 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 calorim-

etry 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 Sup-

plementary Table 1 of Electronic supplementry materials.

The baseline specific heat (phonon contribution) is essen-

tially 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 tran-

sition 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.

0

50

100

150

200

250

300

CP / J K

-1 mol -1

300 400 500 600 700 800 9000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

CP /

J K

-1 g

-1

T / K

Fig. 2 Temperature dependence of the specific heat in leucite,

showing the two transitions as peaks in CP

a

b

c

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

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

k1 0 J mol–1

k2 3,400(100) J mol–1

12 Phys Chem Minerals (2008) 35:11–16

123

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 differ-

ences 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.

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 Q1;Q2ð Þ ¼ G Q1ð Þ þ G Q2ð Þ þ G couplingð Þ; ð1Þ

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 com-

patible with observed symmetry changes in leucite. Thus

Eq. 1 above expands as

G Q1;Q2ð Þ¼A1

2T�TC1ð ÞQ2

1þB1

4Q4

1þC1

6Q6

1þA2

2T�TC2ð Þ

�Q22þ

B2

4Q4

2þC2

6Q6

2þk1Q1Q22þk2Q2

1Q22:

ð2Þ

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 k1 or k2 is

dominant. If k1 is large k2 is negligible, the coupling is linear-

quadratic. From the equilibrium condition qG/qQi = 0,

Q1 ¼ A1 T � TC1ð Þ þ B1Q21 þ C1Q4

1 þ k1Q�11 Q2

2 ð3Þ

Q2 ¼ A2 T � TC2ð Þ þ B2Q22 þ C2Q4

2 þ 2k1Q1 ð4Þ

If, on the other hand, the dominant coupling is

biqudratic (negligible k1, large k2), the same equilibrium

condition leads to

Q1 ¼ A1 T � T1ð Þ þ B1Q21 þ C1Q4

1 þ 2k2Q22 ð5Þ

Q2 ¼ A2 T � TC2ð Þ þ B2Q22 þ C2Q4

2 þ 2k2Q21: ð6Þ

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:

Q41 ¼ Q4

1;0 �k1Q2

2

C1Q1

; ð7Þ

biquadratic coupling:

Q41 ¼ Q4

1;0 � k2Q22: ð8Þ

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 sponta-

neous strain data are used to quantify the order parameters.

0840 860 880 900 920 940

1

2

3

4

5

-∆S

/ J

K-1

mol

-1

T / K

Fig. 3 Temperature dependence of the excess entropy associated

with the phase transitions in leucite

Phys Chem Minerals (2008) 35:11–16 13

123

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.

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 com-

monly significant, but are allowed.

The symmetry breaking strain esb vanished at the upper

transition temperature. This suggests that it can be asso-

ciated 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. Interest-

ingly, 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.

Characterisation of the Ia3d ? I41/acd transition

In the I41/acd phase, Eq. 2 simplifies to

G Q1; 0ð Þ ¼ A1

2T � TC1ð ÞQ2

1 þB1

4Q4

1 þC1

6Q6

1 ð9Þ

and so the specific heat anomaly is given by

T

DCP

� �2

¼ 4B21 þ 16A1C1 TC � Tð Þ

A41

: ð10Þ

The temperature dependence of (T/DCP)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 DCP extrapolates to infinity,

800 820 840 860 880 900 920 940 9600.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

e

T / K

Fig. 4 Temperature dependencies of the symmetry-breaking strain

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

(1989)

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.0160.000

0.002

0.004

0.006

0.008

0.010

0.012

813835

846857

867867

878889

900

911

913922

922928

933938

943

e ns

b

e sb

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

915 916 917 918 9190

50

100

150

200

250

300

350

transition temperature:∆CP maximumTTR = 918.18 K

upper critical temperature:∆CP extrapolates to infintiy

T2 = 918.78 K

lower critical temperature:TC = 917.90 K

(T/ ∆

CP)2 /

K4 J

-2 m

ol2

T / K

Fig. 6 Landau analysis of specific heat data for leucite in the vicinity

of the Ia3d ? I41/acd phase transition in leucite

14 Phys Chem Minerals (2008) 35:11–16

123

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.

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 � jTC – Tj). 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 devia-

tion is D(esb2 ) � D(Q1

4) � jTC2 – Tj. Comparing this

behaviour with the two coupling models described by

Eqs. 7 and 8 indicates that the behaviour of Q2 is essen-

tially tricritical, and the coupling between Q1 and Q2 is

biquadratic.

The sign of the coupling constant k2 is positive (that is,

the coupling increases the overall free energy of the sys-

tem). 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 spe-

cific 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 parame-

ters 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 Q1;Q2ð Þ ¼ � A1

2Q2

1 þA2

2Q2

2

� �ð11Þ

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

0.00000

0.00005

0.00010

0.00015

0.00020

I41/a I41/acd Ia3d

(e s

b)2

T / K840 860 880 900 920 940 960820

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/acdtransition

-0.00002

0.00000

0.00002

0.00004

0.00006

0.00008

0.00010

(e s

b)2 -

(e

sb0)

2

T / K840 860 880 900 920 940 960800 820

Fig. 8 Excess in the square of the tetragonal strain in leucite. The fit

line is to a model D(esb2 ) � jTC2 – Tj, which is consistent with

biquadratic coupling

Phys Chem Minerals (2008) 35:11–16 15

123

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.

Our new measurements of the heat capacity of leucite

demonstrate that the underlying thermodynamic character

of both transitions is near-tricritical. Landau theory suc-

cessfully describes the individual transitions, and the

coupling between them. The data indicate that the domi-

nant 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

Bayer G (1973) Thermal expansion of new leucite-type compounds.

Naturwissenshaften 60:102–103

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 334–349

Carpenter MA, Salje EKH, Graeme-Barber A (1998) Spontaneous

strain as a determinant of thermodynamic properties for phase

transitions in minerals. Eur J Mineral 10:621–691

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, Lon-

don, 958 pp

Dove MT (1997) Theory of displacive phase transitions in minerals.

Am Mineral 82:213–244

Faust GT (1963) Phase transition in synthetic and natural leucite.

Schweiz Mineral Petrograph Mitteil 43:165–195

Galli E, Gottardi G, Mazzi E (1978) The natural and synthetic phases

with the leucite framework. Mineral Petrograph Acta 22:185–

193

Grogel T, Boysen H, Frey F (1984) Neutron powder investigation of

I41/a–Ia3d in leucite. Coll Abst 13th Int Cong Cryst C256–C257

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–

1142

Lange RA, Carmichael ISE, Stebbins JF (1986) Phase transitions in

leucite (KAlSi2O6), orthorhombic KAlSiO4, and their iron

analogues (KFeSi2O6, KFeSiO4). Am Mineral 71:937–945

Mazzi F, Galli E, Gottardi G (1976) The structure of tetragonal

leucite. Am Mineral 61:108–115

Palmer DC (1990) Phase transitions in leucite. PhD thesis, University

of Cambridge

Palmer DC, Salje EKH, Schmahl WW (1989) Phase transitions in

leucite: X-ray diffraction studies. Phys Chem Minerals 16:714–

719

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 17:259–265

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 82:16–29

Peacor DR (1968) A high temperature single crystal diffractometer

study of leucite (K,Na)AlSi2O6. Zeit Krist 127:213–224

Salje E, Devarajan V (1986) Phase transitions in systems with strain-

induced coupling between two order parameters. Phase Trans

6:235–248

Salje EKH, Hayward SA, Lee WT (2005) Phase transitions: structure

and microstructure. Acta Cryst 61:3–18

Taylor D, Henderson CMB (1968) The thermal expansion of the

leucite group of minerals. Am Mineral 53:1476–1489

0820 840 860 880 900 920

1

2

3

4

5

calculated from strain data

experimental data from calorimeter

- ∆S

/ J

K-1

mol

-1

T / K

Fig. 9 Comparison of the observed excess entropy (solid line) with

model derived from experimental measurements of the order param-

eters (data from Palmer et al. (1989) with rescaled temperatures)

16 Phys Chem Minerals (2008) 35:11–16

123