ORIGINAL PAPER

Laboratory parallel-beam transmission X-ray powder diffractioninvestigation of the thermal behavior of nitratine NaNO3:spontaneous strain and structure evolution

Paolo Ballirano

Received: 4 November 2010 / Accepted: 14 March 2011 / Published online: 29 March 2011

� Springer-Verlag 2011

Abstract Present work provides in-situ structural data at

a fine temperature scale from RT to the melting point of

nitratine, NaNO3. From the analysis of log e33 versus log t

plots, it is possible to prove that an univocal indication on

the R 3c (low temperature, LT) ? R 3m (high temperature,

HT) transition mechanism cannot be obtained because of

the relevant role played by the arbitrary assumptions

required for defining the c0 dependence from temperature

of the HT phase. This is due to the occurrence of excess

thermal expansion for the HT phase. A significantly better

fit for an Ising-spin structural model over a non-Ising rigid-

body one has been obtained for the LT phase. Moreover,

the Ising model led to a smooth variation of the oxygen site

x fractional coordinate throughout the transition. The

structure of the HT polymorph has been successfully

refined considering an oxygen site at x, 0, �, with 50%

occupancy. Such model was the only acceptable one from

the crystal chemical point of view as the alternative model

(oxygen site at x, y, z with 25% occupancy) led to unre-

alistically aplanar NO�3 groups.

Keywords Nitratine � NaNO3 � X-ray powder

diffraction � Rietveld method � Spontaneous strain �Phase transition

Introduction

A relevant number of papers dealing with the temperature

dependence of the structure of nitratine, NaNO3, have

been published so far (for an extended bibliographic list

see: Harris 1999; Antao et al. 2008). The interest is due to

the fact that such a relatively simple material undergoes

an order/disorder transition that is related to an orienta-

tional disorder of the NO�3 group leading to a symmetry

change R 3c (low temperature, LT phase) ? R 3m (high

temperature, HT phase) resulting in a c axis halving. Such

transition is marked by the disappearance of superlattice

reflections in diffraction experiments. Nitratine is isotypic

with calcite, CaCO3, which shows a similar transition.

However, experimental difficulties (CO2 overpressure of

at least 2–3 atm is required to prevent calcite decompo-

sition; Jacobs et al. 1981; Dove and Powell 1989; Dove

et al. 2005) render the study of the thermal behavior of

nitratine much more simple. Surprisingly, modeling of

both transitions using rigorous thermodynamic treatments

has been largely unsatisfactory (Harris 1999). Attempts to

interpret such transition have been undertaken using data

from very different experimental techniques including

birefringence measurements (Poon and Salje 1988), X-ray

powder diffractometry (Reeder et al. 1988; Antao et al.

2008), X-ray single-crystal diffractometry (Schmahl and

Salje 1989), calorimetry (Reinsborough and Whetmore

1967; Jriri et al. 1995), dilatometry (Takeuchi and Sasaki

1992), and IR spectroscopy (Brehat and Wyncke 1985;

Harris et al. 1990) among the others. A comprehensive re-

analysis of reference experimental data, carried out by

Harris (1999), has produced a convergence to a possible

transition model of two-dimensional XY type (Bramwell

and Holdsworth 1993a, b).

Electronic supplementary material The online version of thisarticle (doi:10.1007/s00269-011-0425-4) contains supplementarymaterial, which is available to authorized users.

P. Ballirano (&)

Dipartimento di Scienze della Terra, Sapienza Universita

degli Studi di Roma, P.le Aldo Moro 5, 00185 Rome, Italy

e-mail: paolo.ballirano@uniroma1.it

123

Phys Chem Minerals (2011) 38:531–541

DOI 10.1007/s00269-011-0425-4

Among the different properties used for thermodynamic

analysis of phase transitions, lattice parameters variation as

a function of T has received special attention in particular

via the concept of spontaneous strain e. This is a second-

rank tensor, constrained by symmetry, which has been

largely used as a determinant of thermodynamic properties

for phase transitions (Carpenter et al. 1998). In the case of

nitratine, the e33 strain component, acting along the c axis,

is the only relevant. Clearly, a very stringent request for a

modeling of the spontaneous strain data is the availability of

very accurate cell parameters for both LT and HT poly-

morphs. A central role is represented by the extrapolation of

the c0 cell parameter of the HT phase below Tc. A part of the

small accessible thermal range between Tc and the melting

point, an excess thermal expansion is shown by the HT

phase. In fact, the ‘‘normal’’ thermal expansion is expected

to be of the order of approximately ac = 10-5 K-1 instead

of a measured value of more than one order of magnitude

greater. Reeder et al. (1988) attributed to local fluctuations

or precursoring effects such unexpected behavior.

Nitratine has a very anisotropic dependence of cell

parameters from T, the c axis being the softest against

heating. Comparison between the data of Reeder et al.

(1988) and Antao et al. (2008) indicates significant dif-

ferences, especially near Tc leading to relevant differences

on the calculated spontaneous strain. In particular, Antao

et al. (2008) report an anomalous break on the temperature

dependence of the a cell parameter at Tc. whose occurrence

seems to be difficult to conciliate with classical thermo-

dynamic models.

Reeder et al. (1988) used a slightly decreasing depen-

dence of c0 from temperature for the HT phase for the

analysis of the spontaneous strain related to the transition.

Dependence was established making a few arbitrary

assumptions: they fixed c0 at a ‘‘reasonable’’ value, at an

arbitrary temperature exceeding the melting point. It was

reported that, at a given arbitrary T, different combinations

of c0 and ac produced similar critical exponents b provid-

ing the temperature dependence of the macroscopic order

parameter Q. Such exponents were derived from the slope,

expected to be equal to 2b, of the log e33 versus log t plot

[t = (Tc - T)/Tc] The selection of such dependence of c0

from temperature had the consequence, among the others,

to produce a non-zero strain at Tc and conspicuous ‘‘tails’’

above Tc. The temperature dependence of e233-t showed a

linear region from ca. 185 to ca. 435 K leading to an

extrapolated Tc of 597 K. After substitution of the extrap-

olated for the effective Tc, a critical exponent b = 0.25

compatible with a tricritical behavior was obtained.

Instead, the value of the critical exponent near Tc was of ca.

0.22 depending on the choice of c0 and ac indicating a

crossover between tricritical to a three-state Potts model.

Harris (1999) strongly argued, on theoretical grounds,

against this interpretation showing that an equally satis-

factory fit (no statistical indicators were reported) can be

obtained using a single b value of 0.22(1) over the

200 K \ T \ 550 K range. This value is consistent with

both three-state Potts model (predicted 0.21) and with a

two-dimensional XY model (predicted ca. 0.231).

According to the different results obtained by those

authors from different data treatment procedures used, it is

legitimate to wonder which is the influence of the various

arbitrary assumptions used on the derived value of the

critical exponent. This is particularly crucial, in the present

case of nitratine, because b takes values very similar for

different proposed models.

A few structural investigations of nitratine at non-

ambient conditions have been performed in the past. The

structure of NaNO3 consists of alternating layers of nitrate

groups and sodium cations stacked along [001]. The nitrate

group has been definitely proved to be perfectly planar

from a very accurate (wR = 0.007) room temperature (RT)

single-crystal synchrotron radiation structural analysis

(Gonschorek et al. 1995). Each Na is octahedrally coordi-

nated to three oxygen atoms pertaining to the upper layer

and three pertaining to the lower one. At RT the in-sheet

nitrate groups have a ‘‘ferro’’ order whereas an ‘‘antiferro’’

order exists between successive layers. Such relationship

implies a relative 60� rotation of the NO�3 groups lying in

successive layers.

Cherin et al. (1967), from the measurement of hkl,

l = 2n ? 1 reflections only (i.e. those for which only the

oxygen atoms contribute), determined the position and

atomic displacement parameters (ADPs) of the oxygen

atoms up to 473 K. They reported a regular decrease of the

N–O bond distance attributed to increasing libration of the

nitrate groups about the threefold axis. Subsequently, Paul

and Pryor (1972), from single-crystal neutron diffraction,

analyzed in more detail, but a very large temperature scale,

the structural evolution up to 563 K. This investigation

reported the occurrence, at temperature exceeding 503 K,

of a partial disorder of the nitrate groups that converged

toward a complete disorder at the transition temperature Tc

of 548 K. However, the authors refined the HT phase in R

3c instead of R 3m considering the oxygen atoms distrib-

uted between two sites approximately at ±x, 0, �, each one

with 50% occupancy, but allowing refinement of separate

positions and displacement parameters. Gonschorek et al.

(2000) refined the structure of NaNO3 at 100, 120, and

563 K. The structure of the HT polymorph, converging to a

very respectable wR of 0.017 from single-crystal neutron

diffraction, was refined in R 3m with a single oxygen site at

x, 0, � with 50% occupancy. A strong anharmonic motion

was observed at HT.

532 Phys Chem Minerals (2011) 38:531–541

123

Very recently Antao et al. (2008), by synchrotron radia-

tion X-ray powder diffraction, published a thorough analysis

of the thermal dependence of the nitratine structure from

300 K up to its melting point at a very narrow temperature

scale. Two models were investigated for the LT phase:

Model 1: No oxygen positional disorder; corresponding

to the RT structure of Gonschorek et al. (1995).

Model 2: Ising model; two oxygen sites at ±x, 0, �. This

model correspond to that used by Paul and Pryor (1972)

for T [ 503 K.

However, Antao et al. (2008) stated that ‘‘The present IP

(Image Plate) XRD data does not allow us to choose one

structural model over the other model, but the gradual

structural changes would tend to favour model-2’’. Disor-

der was found to start at T [ 327 K. Moreover, it is worth

noting that it was statistically significant (at the 3r level:

r = 0.004) at T [ 400 K.

Moreover, Antao et al. (2008) refined the structure of the

HT phase in R 3m with the planar nitrate groups occurring

in two different orientations, rotated by 180�, each one with

50% occupancy, and located at z = 0.47 and 0.53. This

corresponds to oxygen atoms lying at site x, y, z (x ca. 0.24,

y ca. 0, z ca. 0.47) with 25% occupancy. As a result, a

strongly aplanar NO�3 group was observed differently from

Gonschorek et al. (2000).

It should be pointed out that the structural investigation

of Antao et al. (2008) was based on a relatively limited

angular range, extending up to sinh/k = 0.40 A-1. In fact,

this reduced data set forced those authors to use con-

strained isotropic displacement parameters for the carbon

and the oxygen atoms (C Uiso = O Uiso). The same

approach has been used for a more recent paper on calcite

(Antao et al. 2009). However, Paul and Pryor (1972) and

Gonschorek et al. (1995) indicated a severe anisotropy of

the oxygen thermal ellipsoid at each analyzed temperature,

a fact confirmed by Antao et al. (2008) itself from a RT

refinement based on synchrotron high-resolution powder

X-ray diffraction (HRPXRD) data extending up to sinh/

k = 1.05 A-1. Moreover, Gonschorek et al. (1995, 2000)

reported a significant reduction of the wR agreement

indices after inclusion of anharmonic motion to the

refinements. Therefore, it is expected that the ability

to properly model the thermal motion could produce

significant improvements on the detailed analysis of the

R 3c ? R 3m transition as it has been very recently proved

for calcite (Ballirano 2011a). Ballirano (2011a) has dem-

onstrated that the improvement in fast detectors coupled

with a laboratory parallel-beam transmission experimental

set-up renders possible to extract high-quality non-ambient

structural data from Rietveld refinements. Exploiting such

opportunity, present work is expected to provide accurate

in-situ structural data at a fine temperature scale from RT

to the melting point. In fact, non-ambient structural data of

such quality, reporting individual ADPs, are available at a

few temperatures only (Paul and Pryor 1972; Gonschorek

et al. 1995, 2000). In particular, present work is aimed to:

(a) check, in the present case, the feasibility of thermody-

namic modeling of the transition from spontaneous strain

analysis; (b) define the best way of describing the electron

density map around the nitrogen atom for the LT phase; (c)

determine unequivocally the structure of the HT phase,

recently revised, and its temperature dependence.

Experimental

Powder of synthetic analytical-grade NaNO3 (Merck, prod-

uct 6546) was loaded in a 0.7 mm diameter SiO2-glass

capillary that was glued to a 1.2 mm inner diameter Al2O3

tube by means of an high-purity alumina ceramic (Resbond

989). The capillary/tube assembly was subsequently aligned

on a standard goniometer head. Data were collected, using

Cu Ka radiation, on a parallel-beam Bruker AXS D8

Advance automated diffractometer operating in h-h geome-

try. It is fitted with diffracted-beam radial Soller slits and a

PSD VANTEC-1 detector set to a 6� 2h aperture and a

prototype of capillary heating chamber (Ballirano and Melis

2007). Thermal calibration of the chamber was carried out

using MgO (periclase) as standard (Reeber et al. 1995). Error

in temperature measurements is estimated to be of ±1 K.

Diffraction data were collected in the angular range 20–140�2h (sinh/k = 0.61 A-1) with a step size of 0.0214� 2h, and

1.8 s of counting time. The 25� \ 2h\ 27� angular range

was excluded from the refinement because of the occurrence

of a very small Cu Kb component of the strong 103 reflection

arising from a non perfect monochromatization of the inci-

dent beam by the Gobel mirror. Within this angular range, no

reflections of both LT and HT polymorphs occur.

Thermal behavior of nitratine was investigated in the

303–503 K temperature range at steps of 5 K and in

the 505–583 K temperature range at steps of 2 K. The

R 3c ? R 3m transition was observed at 551 K and was

easily monitored from the complete disappearance of the

hkl, l = 2n ? 1 superstructure reflections. For kinetic

reasons, melting started at 577 K and was completed at

581 K. A magnified view of the complete data set is

reported in Fig. 1 as a 3D-plot.

Diffraction data were evaluated by the Rietveld method

(Rietveld 1969) using TOPAS v. 4.2 (Bruker AXS 2009)

operating in launch-mode. This program implements the

Fundamental Parameters Approach FPA (Cheary and

Coelho 1992). FPA is a convolution approach in which the

peak-shape is synthesized from a priori known features of

Phys Chem Minerals (2011) 38:531–541 533

123

the diffractometer (i.e. the emission profile of the source,

the width of the slits, the angle of divergence of the inci-

dent beam) and the microstructural features of the speci-

men. This approach is believed to improve the stability and

the quality of the refinement, especially with respect to the

extraction of microstructural parameters. Peak shape was

modeled through FPA imposing a simple axial model

(10.4 mm) and the size of the divergence slit (0.2 mm).

Peak broadening was assumed to follow a Lorentzian (size)

and a Gaussian (strain) behavior (Delhez et al. 1993).

Anisotropic peak broadening was modeled using spherical

harmonics (nine parameters up to the 8th order) coupled

with the exp_conv_const macro. Peaks position was cor-

rected for sample displacement from the focusing circle.

The background was fitted with a 33-term Chebyshev

polynomial of the first kind. Such a large number of terms

are required for a proper fit of the amorphous contribution

of the glass-capillary. Absorption was refined at 303 K

considering the contribution of the aluminum windows of

the heating chamber following the formalism of Sabine

et al. (1998) for a cylindrical sample. Such value was kept

fixed for all the non-ambient refinements.

Preferred orientation was modeled by means of spherical

harmonics (nine refinable parameters up to the 8th order).

A first series of refinements was carried out allowing opti-

mization of the spherical harmonics terms that were found to

be extremely small (as expected for a capillary mount) and

constant throughout the analyzed thermal range. The final

structural data set was obtained keeping the spherical har-

monics terms fixed to the corresponding averaged values.

Three different models for the LT (R 3c) polymorph

were tested:

Model 1: No oxygen positional disorder: corresponding

to the RT structure of Gonschorek et al. (1995); ADPs

were refined for Na and O whereas N was refined

isotropically.

Model 2: Ising model: two oxygen sites at ±x, 0, � were

used to model NO�3 disorder. This model correspond to

that proposed by Paul and Pryor (1972) and Antao et al.

(2008); ADPs for Na and O, N refined isotropically.

Model 3: No oxygen positional disorder and rigid body

constraints; same as model 1 except for the nitrate group

being treated as a rigid body, following the same

approach of Markgraf and Reeder (1985) and Dove et al.

(2005) for calcite. According to the 32 point symmetry

of the nitrate group in nitratine, T11 (=T22), T33, L11

(=L22), L33, SAA (=S33 - S11) and SBB (=S11 - S33)

tensor elements are possible refinable parameters. How-

ever, as reported by Dove et al. (2005) for calcite, screw

components SAA, and SBB (=-SAA) were unstable and

therefore were not refined. In this case, the GSAS

crystallographic suite of programs (Larson and Von

Dreele 2000) with the EXPGUI graphical interface

(Toby 2001) was used for evaluating the diffraction data.

Model 4: Ising model and rigid-body constraints; the

same approach used by Swainson and Brown (1997) for

the refinement of the structure of ammonium perrhenate.

Neutral scattering factors were used. The choice to refine N

isotropically is supported by experimental evidences indicat-

ing limited anisotropy of the thermal ellipsoid at each tem-

perature (Paul and Pryor 1972; Gonschorek et al. 1995, 2000)

and by the request to avoid possible correlations at HT with the

Uij tensors of the oxygen atom that were observed in a test run.

Similarly, two different models for the HT (R 3m)

polymorph were tested:

Model 1: Oxygen atom at x, 0, � (site occupancy 50%)

as reported by Gonschorek et al. (2000); ADPs for Na

and O, N refined isotropically.

Model 2: Oxygen atom at x, y, z (site occupancy 25%) as

reported by Antao et al. (2008); ADPs for Na and O, N

refined isotropically.

Fig. 1 Magnified view of the

full experimental data set

reported as a 3D-plot. Transition

can be monitored from the

disappearance of the 113

superlattice reflection

534 Phys Chem Minerals (2011) 38:531–541

123

The peak shape modeling procedure was the same

adopted for the LT polymorph.

Full structural data are deposited under the form of CIF

files as Supplemental Material. Examples of Rietveld plots

of data collected at selected temperatures are shown in

Figure S1.

Discussion

Thermal expansion and spontaneous strain

The dependence of cell parameters and volume from

temperature are reported in Fig. 2. No differences at the 1rlevel were observed among parameters obtained from the

various tested models. In order to allow a direct compari-

son between the two polymorphs, c and V of the HT phase

are plotted multiplied by two. Data are in substantial

agreement with those of Reeder et al. (1988), Gonschorek

et al. (1995, 2000), as well as those of Antao et al. (2008).

However, differently from Antao et al. (2008) no break of

the a cell parameter at Tc has been observed. Moreover, a

regular increase of the anisotropic peak broadening, due to

strain, has been observed to start at ca. 450 K. It becomes

relevant at 510 K, increases up to 547 K, and then it sig-

nificantly decreases until Tc is reached. It is reasonable to

hypothesize that the latter behavior could be, at least par-

tially, influenced by the occurrence of some longitudinal

thermal gradient along the sample implying mixing of both

LT and HT phases in the 549 K \ T \ 551 K range.

Above Tc, the anisotropic peak broadening is suddenly

released becoming small and characterized by slightly

fluctuating spherical harmonics terms signaling the pres-

ence of correlations among them. Therefore, the reported

irregular behavior of the a cell parameter near Tc, reported

by Antao et al. (2008), is reasonably due to an imperfect

modeling of the peak shape. This hypothesis has been

confirmed owing to the fact that removal from the refine-

ments of the anisotropic peak broadening produced a fairly

irregular a cell parameter near Tc.

The variation of the unit cell parameters and volume

with temperature below Tc have been empirically fitted

with a second-order polynomial of type p = a0 ?

a1T ? a2T2. Results of non-linear fitting are reported in

Table 1. As reported in reference data, thermal expansion

is significantly anisotropic. In fact, the a cell parameter

increases by 0.3% from RT to Tc, whereas the c cell

parameter increases by 4.8%. The expansion of the a cell

parameter perfectly agrees with that calculated by Liu et al.

(2001) from molecular dynamics differently from the

expansion of the c cell parameter that is significantly larger

than the 2.9% calculated by the same authors. Above Tc,

and up to the melting point, i.e. within the small

553 K \ T \ 581 K thermal range, both cell parameters

expand linearly; the c parameter expands at a rate of one

order of magnitude greater than a. Data for the a cell

parameter are fairly scattered due to the very small

expansion coefficient coupled with the effects of the

slightly fluctuating spherical harmonics terms describing

the anisotropic peak broadening. Results of the linear fit-

ting of the variation of cell parameters and volume at

temperatures above Tc are reported in Table 2. Linear

extrapolation of the cell parameters and volume of the HT

polymorph below Tc indicates the presence of significant

Fig. 2 Dependence of cell parameters and volume from temperature:

a a and c cell parameter; b cell volume. Data from Gonschorek et al.

(1995, 2000), labeled as G95 and G00, respectively, are reported for

comparison

Table 1 Results from data-fitting procedures using the polynomial

p = a0 ? a1T ? a2T2 for thermal expansion data below Tc

a c V

R2 0.999 0.994 0.995

a0 5.0760(9) 17.39(9) 388.4(2)

a1 -5.9(4) 9 10-5 -4.2(4) 9 10-3 -1.1(1) 9 10-1

a2 1.47(5) 9 10-7 8.2(5) 9 10-6 2.1(1) 9 10-4

Phys Chem Minerals (2011) 38:531–541 535

123

spontaneous strain as previously reported by many authors.

Moreover, crossing of the linear extrapolations with the

corresponding LT phase cell parameters and volume con-

firms the occurrence of significant excess thermal expan-

sion for the HT phase as indicated by Reeder et al. (1988).

In the LT ? HT phase transition, the spontaneous strain

is not symmetry-breaking, as both polymorphs belong to the

trigonal point group 3m, and transforms as Cidentity showing

a co-elastic behavior. Analysis of the present data set clearly

indicates the strong dependence of both the extension of

e233-t linearity region and the value of the critical exponent

b from the selection of the dependence of c0 from temper-

ature. Different models were tested and evaluated from a

purely statistical point of view. A remarkably good

(R2 = 0.9996) linear e233-t dependence has been obtained

for the present analyzed thermal range keeping fixed c0 to

the value determined at Tc. However, such strategy is not

free from criticism. In fact, it has been considered in the past

as acceptable for large strains albeit significant errors are

expected to occur (Carpenter et al. 1998). Minor deviation

from linearity became detectable at t = ca. 0.06 (T ca.

520 K). Attempts to use the same c0 dependence from

temperature of Reeder et al. (1988) produced a reduction of

the region of linearity by increasing ac passing from t [ ca.

0.16 for ac = 0.5 9 10-5 K-1 to t [ ca. 0.23 for ac =

2 9 10-5 K-1. Putting the data on a log e33 versus log t plot

produced extended linearity regions well above 520 K.

Considering data for T \ 537 K, critical exponents in the

0.1860(9)–0.1904(9) range were obtained using the Reeder

et al. (1988) approach, whereas keeping fixed c0 to the value

determined at Tc, provided a value of 0.2849(9). Those

values are not consistent with the generally accepted theo-

retical models for the LT ? HT phase transition (tricritical,

two-dimensional XY, three-dimensional XY, three-

dimensional 3-state Potts, three-dimensional Ising models;

see Harris 1999).

An alternative c0 dependence from temperature has been

subsequently tested, hypothesizing as ‘‘observed’’ value for

the HT paraphase, that measured just before melting and

investigating the effect of different values of ac on the bcritical exponent. This hypothesis was explored because it

seems reasonable that precursoring effects could last

at melting. Similarly to Reeder et al. (1988) it was

checked the 0.5 9 10-5 K-1 \ ac \ 1.5 9 10-5 K-1

range of thermal expansion. The effect of increasing ac was

a reduction of b from 0.2242(4) to 0.2217(3) with R2 values

regularly increasing from 0.9996 to 0.9999 (Fig. 3). It is

worth noting that such c0 dependence from temperature

lead to a good fit of the low temperature log e33 versus log

t values re-calculated from the original data of Reeder et al.

(1988), at least before saturation that has been reported to

occur at 70 K. Such agreement is surprising because of the

large difference in accuracy between the two data sets. In

fact, the standard deviation is one order of magnitude

smaller for the present data set as compared with reference

data. Apparently, such c0 dependence from temperature

seems to have a more sound justification with respect to

those proposed in reference data. Moreover, it has the

advantage to fit with a single critical exponent log e33

versus log t data spanning over a large reduced temperature

range. Unfortunately, such values of the critical exponent

are consistent with both two-dimensional XY (expected bca. 0.231) and three-dimensional 3-state Potts (expected

b = 0.21) models.

Therefore, no clear indication on the transition mecha-

nism can be extracted from an analysis of the spontaneous

strain alone despite the availability of accurate cell

parameters for the LT phase (standard deviation of the

order of a few parts per million). This is because of the

crucial role played by the arbitrary assumptions required

for defining the c0 dependence from temperature for the

derivation of the critical exponent value. This result may be

extended to reference data.

Structure evolution

Regarding the LT polymorph, the Ising-model (model 2)

produced, by far, the best agreement parameters set as

compared with those obtained from the remaining models.

It is not confirmed the statement by Lefebvre et al. (1984)

that above 523 K a free-rotational model provides a better

Table 2 Results from data-fitting procedures using the polynomial

p = a0 ? a1T for thermal expansion data above Tc

a c V

R2 0.870 0.998 0.997

a0 5.058(3) 15.54(2) 344.8(9)

a1 5.4(6) 9 10-5 3.84(4) 9 10-3 9.5(2) 9 10-2

Fig. 3 Log e33 versus log t plot

536 Phys Chem Minerals (2011) 38:531–541

123

fit of the data. Differences become relevant as the disordered

orientation has an occupancy exceeding 3% (Figure S2).

However, differently from Antao et al. (2008), despite

significant differences between the x site fractional coor-

dinates, N–O bond distances corrected for rigid-body

motion (Downs et al. 1992) show comparable values for

all analyzed models. Moreover, only the Ising model

provides a smooth, continuous variation of the x site

fractional coordinate throughout the transition. Such dis-

crepancy with reference data may be reasonably due to

the higher resolution of the present data set and to the

corresponding ability to refine anisotropically the crystal

structure. In fact, anisotropic structure refinements were

perfectly stable at all temperatures. Model 1 (no disorder,

ADPs) performed significantly better than model 3 (no

disorder, TLS) because of the inability to refine the screw

components of the TLS model. Insertion of the corre-

sponding SAA = -SBB values recalculated from the ADPs

of model 1 led to nearly identical agreement indices

between model 1 and model 3 and irrelevant structural

differences between the two models as previously repor-

ted for calcite, from X-ray single-crystal analysis, by

Markgraf and Reeder (1985). Moreover, a difficulty to

reach convergence for model 3 was consistently observed.

Significant differences were observed between refined

values of both T and L tensors determined with or without

inclusion of the screw components. This fact could have

influenced the structural results reported by Swainson

et al. (1998) and Dove et al. (2005). Similarly, model 4

(Ising model, TLS) produced nearly the same agreement

indices and structural results of model 2 after the inclu-

sion of the screw components. Therefore, results from the

anisotropic structure refinements of the Ising model will

be reported and discussed in the present work because

they provide a far better way to describe the electron

density map of the LT phase.

The two different models tested for the HT polymorph

(R 3m) produced similar agreement parameters sets. Model

2 (oxygen atom at x, y, z, site occupancy 25%; ADPs for

Na and O, N refined isotropically) performed slightly better

than model 1 (oxygen atom at x, 0, �, site occupancy 50%;

ADPs for Na and O, N refined isotropically). However,

model 2 led to unrealistically aplanar NO�3 groups with

values of the distance between the nitrogen atom and the

plane of the three oxygen atoms well above those consid-

ered as ‘‘definitely proved’’ by Jarosch and Zemann (1983).

Therefore, structural data from model 1 will be reported

and discussed in the present work owing to the fact that

such model was the only to provide a reasonable structure

from the crystal chemical point of view.

Evolution of the oxygen site x fractional coordinate is

reported in Fig. 4, oxygen site occupancy in Fig. 5, and

anisotropic (for Na and O) and isotropic thermal dis-

placement parameters (for N) in Figure S3.

The general behavior of the oxygen site x fractional

coordinate dependence from T follows closely that of Paul

and Pryor (1972), Cherin et al. (1967), and Gonschorek

et al. (1995, 2000), a part for a slight systematic dis-

placement. However, the behavior is different with respect

to that recently reported by Antao et al. (2008), in partic-

ular near Tc. This may be possibly due to the effect of the

constraint C Uiso = O Uiso adopted by those authors. Bal-

lirano (2011a) has reported a similar discrepancy for calcite

that was analyzed by Antao et al. (2009) using the same

constraint. The x coordinate has a peculiar behavior as it

decreases linearly up to ca. 470 K and subsequently shows

a steeper decrease up to Tc. Above Tc, the x coordinate

decreases at a nonlinear reduced rate until melting. This

Fig. 4 Temperature evolution of the oxygen site x fractional coor-

dinate. Data from Paul and Pryor (1972) and Gonschorek et al. (1995,

2000), labeled as P&P72, G95, and G00, respectively, are reported for

comparison

Fig. 5 Temperature evolution of the oxygen site occupancy. Data

from Paul and Pryor (1972) and Antao et al. (2008), labeled as P&P72

and A08, respectively, are reported for comparison

Phys Chem Minerals (2011) 38:531–541 537

123

general behavior, coupled with the anisotropic thermal

expansion, results in an increase in the Na–O bond distance

and a strong, apparent, reduction of the N–O bond distance.

However, after correction for rigid-body motion the N–O

bond distance was found to increase from ca. 1.265 to

1.275 A at 547 K and subsequently to decrease to ca.

1.255 A at Tc. Above Tc, the value remains approximately

constant (Fig. 6). It is worth noting that Markgraf and

Reeder (1985) and Ballirano (2011a) have observed an

increase approximately of the same magnitude of the C–O

bond distance of calcite. Considering as correct the

assumption of Gonschorek et al. (2000) that the N–O bond

distance is substantially independent from T, we may

possibly attribute such behavior to the combined effect of

minor longitudinal thermal gradient and to the inability to

refine anisotropically the nitrogen atom.

Moreover, the plot of the temperature dependence of the

x coordinate bears strong similarities with that of the

temperature dependence of the birefringence reported by

Poon and Salje (1988) and to the temperature evolution of

the observed frequencies of the m2 and P modes reported by

Harris et al. (1990) indicating a coupling between the

oxygen atom positions and those parameters.

The evolution of the oxygen site occupancy is similar to

that reported by Antao et al. (2008). Disorder has been

found to start at 353 K but it becomes statistically signif-

icant (at the 3r level: r = 0.001) at T [ 370 K (Fig. 5).

Therefore, as indicated by Gonschorek et al. (1995), the

Ising-spin variable saturates well above RT and the varia-

tion of the intensity of superlattice hkl, l = 2n ? 1

reflections below RT is only a measure of the reduced

mean-square amplitude of vibration of the oxygen atoms

i.e. those contributing to such reflections.

Analysis of the anisotropic and isotropic thermal dis-

placement parameters of nitratine reveals a good agreement

with single-crystal data of Paul and Pryor (1972) and

Gonschorek et al. (1995, 2000) for both LT and HT

polymorphs (Figure S3). In fact, it should be reminded that

Paul and Pryor (1972) attributed the anomalously low Uijs

for Na observed at 541 K to anomalous absorption arising

from intense diffuse scattering. Finally, Paul and Pryor

(1972) refined the HT polymorph in space group R 3c. This

fact implies a different orientation of the oxygen atom

thermal ellipsoid resulting in significantly different U23

tensors. The spectacular increase of the oxygen atom

vibration with T, reported by Cherin et al. (1967), was not

observed in this work similarly to Paul and Pryor (1972).

Above Tc, there is a reasonable continuity of the dis-

placement parameters with those of the LT phase. The

more relevant difference is related to the significant

decrease of the U11 tensor for Na in the HT phase.

The combined Ising model-TLS refinements of the

present powder data led to the general relationships

L33 [ L11 and T33 [ T11 that are consistent with those

reported by Markgraf and Reeder (1985) for calcite. As

example, values of ca. 10� forffiffiffiffiffiffiffi

L33

pat 453 K, ca. 12� at

523 K, and ca. 14� at Tc (Fig. 7) were obtained.

Evaluation of the O–O contacts dependence from

T indicates a generalized expansion a part of the apparent

contraction of the O1-O1 intra nitrate contact (Fig. 8). The

O1-O2 in-sheet distance increases from ca. 2.57 to ca.

2.68 A at Tc, reaching a value of ca. 2.78 A at the melting

point, indicating that in a static model it is impossible a

mixing of both orientations of nitrate groups within the

same sheet. In fact, the shortest known non-bonding O–O

contacts between two oxygen atoms outside a coordination

polyhedron, excluding HP phases, are of ca. 2.85 A, in the

case of magnesite and smithsonite (Effenberger et al. 1981).

Fig. 6 Temperature dependence of the N–O bond distance corrected

for rigid-body motion. Data from Gonschorek et al. (1995, 2000),

labeled as G95 and G00, respectively, are reported for comparison.

Dashed line is a guide for the eye

Fig. 7 Temperature evolution of Ljj and Tjj tensors coefficients for

model 4 refinements (Ising model, TLS)

538 Phys Chem Minerals (2011) 38:531–541

123

This value is close to 2.80 A, the sum of two oxygen atoms

van der Waals radii.

An O1-O2 in sheet distance of ca. 2.80 A can be obtained

at RT exclusively by a synchronous opposite rotation of

neighboring nitrate groups by ±20� around the c axis and by

a few degrees less at higher temperature as a combined

result of the small thermal expansion of the a cell parameter

and of the evolution of the oxygen site x fractional coor-

dinate. Therefore, coupling of the libration around the c axis

with that, relevant albeit smaller, around the a axis could

lead to ‘‘dynamical’’ O–O contacts compatible with the

mixing of both orientations of nitrate groups within the

same sheet, at least at a temperature exceeding 500 K.

Moreover, the O1-O2 sheet-to-sheet distance increases

from ca. 2.90 to ca. 3.08 A indicating a full compatibility of

a ‘‘ferro’’ order between successive layers.

According to those results, it seems clear that the dis-

ordering mechanism acts by segregation of the competing

orientation by synchronous switch of layers. Moreover, it

seems possible, at least in principle and at high tempera-

tures, the mixing of both orientation within the same sheet

because of the coupled effect of the limited thermal

expansion of the a axis and the increasing librations.

Therefore, it can be hypothesized, above a certain tem-

perature, the occurrence of those competing mechanisms

eventually leading to a change of regime in the critical

exponent reported by different researchers. However, no

such change in regime has been observed from the present

log e33 versus log t data near Tc.

Evaluation of the dependence from T of the Na site

coordination provides interesting results. In the attempt to

understand the possible role of Na in the disorder process,

different polyhedral parameters were calculated using the

IVTON2 software (last version of the IVTON program

of Balic-Zunic and Vickovic 1996). In particular, apart

of the observed polyhedral volume Vp, we have analyzed

the dependence from T of the volume distortion,

m = (Vi - Vp)/Vi, where Vi is the ideal polyhedral volume.

The first sphere of coordination of Na expands significantly

from a polyhedral volume Vp of ca. 18.40, at RT, to ca.

21.00 A3 at the melting point (Figure S4). The largest rate

of expansion is observed within the 543 \ T \ Tc thermal

range. The volume distortion m is very small and regularly

increases from 0.2 at RT to 1.4 % at Tc. Above the tran-

sition the polyhedron starts to regularize and m decreases to

0.8 % at the melting point. Six further oxygen atoms are

coordinated at ca. 3.50 A giving rise to a moderately dis-

torted polyhedron. Such contact distance remains approx-

imately constant (3.48–3.52 A) throughout the entire

thermal range differently from the first sphere of coordi-

nation. As can be seen from Figure S5, the second-sphere

polyhedral volume regularly expands up to 543 K but such

expansion is counterbalanced by a regularization of the

polyhedron. After such temperature, a small contraction

occurs followed by an almost constant value after Tc. On

the contrary, distortion reduces at a faster rate in the

543 K \ T \ Tc range, that in which the largest rate of

expansion of the first sphere coordination occurs, and

subsequently decreases with the previous rate.

Bond valence analysis, carried out using the Brese and

O’Keeffe (1991) parameters, indicates that the Na site is

significantly overbonded at RT as a result of a bond

valence sum of ca. 1.18 valence units (v.u.). Similar values

are obtained from reference Na–O bond distances (Figure

S6). The bond valence sum decreases as temperature

increases and near Tc crosses the value of one. The pres-

ence of very large over/under bonding of Na sites has been

observed in natrite Na2CO3 too (Dusek et al. 2003). Within

the natrite structure, there are three different Na sites, two

of which, Na(1) and Na(2), are sixfold coordinated whereas

the third, Na(3), is ninefold. Both Na(1) and Na(2) are

strongly overbonded at RT (ca. 1.35 v.u.), whereas Na(3) is

underbonded (ca. 0.90 v.u.). Analysis of the dependence

from T of the valence sum over the three sites (Ballirano

2011b) reveals a generalized reduction with increasing

temperature. Natrite undergoes three d C2/m(a0c) (com-

mensurately modulated) ? c C2/m(a0c) (incommensurately

modulated) ? b C2/m ? a P63/mmc phase transitions. An

unsaturated bonding potential at the ninefold coordinated

site has been claimed in the past to be the driving force of

the various polymorphic phase transitions (Dusek et al.

2003). However, in the case on both nitratine and natrite,

the common feature is the strong overbonding of the six-

fold coordinated Na sites that is regularly released passing

from the LT to the HT phases.

Fig. 8 Temperature evolution of O–O contacts

Phys Chem Minerals (2011) 38:531–541 539

123

Conclusions

Present data show that no univocal indication on the tran-

sition mechanism can be extracted from the analysis of the

spontaneous strain alone despite the availability of accurate

cell parameters for the LT phase. The reason is that the

arbitrary assumptions required for defining the dependence

from temperature of c0 for the HT phase play a crucial role

for the derivation of the critical exponent value. This result

has to be extended to reference work. The experimental

data provide a significantly better fit for an Ising-spin

structural model over a non-Ising TLS one. Moreover,

the Ising model is consistent with a smooth variation of

the oxygen site x fractional coordinate throughout the

R 3c ? R 3m transition. Analysis of the structural data

indicates that the ‘‘antiferro’’ order between successive

layers is always possible. Therefore, the disordering

mechanism should act by segregation of the competing

orientation by synchronous switch of layers. However, with

increasing temperature, ‘‘dynamic’’ O–O contacts fitting an

in-sheet ‘‘antiferro’’ order of the nitrate groups (deriving

from rapidly growing displacement parameters) may occur.

Therefore, the occurrence, at temperatures near Tc, of two

competing mechanisms of disordering could be, at least in

principle, responsible of the change in regime of the critical

exponent b reported by a few authors (see for example

Harris 1999). However, no change in regime near Tc has

been observed from the present log e33 versus log t data.

Despite the fact that the two different structural models

tested for the HT polymorph produced similar agreement

parameters sets, model 1 (oxygen atom at x, 0, �, site

occupancy 50%; ADPs for Na and O, N refined isotropi-

cally) was the only to provide a reasonable structure from

the crystal chemical point of view. In fact, model 2,

recently proposed by Antao et al. (2008) led to unrealisti-

cally aplanar NO�3 groups with values of the distance

between the nitrogen atom and the plane of the three

oxygen atoms significantly exceeding those considered as

‘‘definitely proved’’ by Jarosch and Zemann (1983).

Acknowledgments The manuscript benefited from the constructive

review of an anonymous referee and Editor A. Kavner. Financial

support from Sapienza Universita di Roma is acknowledged.

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