ORIGINAL PAPER

In situ high-temperature X-ray and neutron powder diffractionstudy of cation partitioning in synthetic Mg(Fe0.5Al0.5)2O4 spinel

Nicoletta Marinoni • Davide Levy • Monica Dapiaggi •

Alessandro Pavese • Ronald I. Smith

Received: 4 February 2009 / Accepted: 9 May 2010 / Published online: 26 May 2010

� Springer-Verlag 2010

Abstract The intra-crystalline cation partitioning over

T- and M-sites in a synthetic Mg(Fe,Al)2O4 spinel sample has

been determined as a function of temperature by Rietveld

structure refinements from powder diffraction data, com-

bining in situ high-temperature neutron powder diffraction

(NPD; POLARIS diffractometer, at ISIS, Rutherford

Appleton Laboratory, UK), to determine the Mg and Al

occupancy factors, with in situ high-temperature X-ray

powder diffraction, to fix the Fe3? distribution. The results

obtained agree with a two-stage reaction, in which an ini-

tial exchange between Fe3? and Mg, the former leaving

and the latter entering tetrahedral sites, is successively

followed by a rearrangement involving also Al. The mea-

sured cation distribution has then been compared and dis-

cussed with that calculated by the Maximum Configuration

Entropy principle, for which only NPD patterns have been

used. The cation partitioning has finally been interpreted in

the light of the configuration model of O’Neill and

Navrotsky.

Keywords Synthetic spinel � X-ray powder diffraction �Neutron powder diffraction � Cation ordering

versus temperature

Introduction

Spinels are ternary oxides (SG: Fd�3m) with an elementary

cell containing 32 oxygen atoms in cubic close packing

arrangement, 16 octahedrally (M) and 8 tetrahedrally (T)

coordinated sites which can host several bi-, tri- and tetra-

valent cations of the first group (typically in natural phases:

Fe3?, Fe2?, Al, Mg, Cr, Zn and Ti). AB2O4 spinel phases

can exhibit two possible extreme cation distributions,

usually addressed to as ‘‘normal’’ or ‘‘inverse’’ configura-

tions. In the former, A2?- and B3?-cations are tetrahedrally

and octahedrally coordinated, respectively, whereas in the

latter the tetrahedral sites are occupied by half of the B3?

cations with the displaced A2? cations octahedrally coor-

dinated in the vacated M-sites. In addition, intermediate

cation arrangements are often observed in nature. Several

studies have shown that spinel cation partitioning changes

as a function of pressure and temperature, indicating that

both energy and configuration entropy play a key role in

determining the atomic distribution over the two different

crystallographic sites (Antao et al. 2005; Lavina et al.

2002; Pavese et al. 1999).

Many investigations, using a variety of complementary

techniques, have looked at intra-crystalline cation order–

disorder in AB2O4 spinels formed in binary systems (see

Lavina et al. 2004 for a review). However, just a modest

number of studies have hitherto been carried out on atomic

partitioning in spinels involving three cation chemical

species. In particular, Mg–Al–Fe3? bearing spinels are

found in a wide range of geological environments, from

N. Marinoni (&) � M. Dapiaggi � A. Pavese

Dipartimento di Scienze della Terra ‘‘Ardito Desio’’, Universita

degli Studi di Milano, Via Botticelli 23, 20133 Milan, Italy

e-mail: nicoletta.marinoni@unimi.it

D. Levy

Dipartimento di Scienze Mineralogiche e Petrologiche,

Universita degli Studi di Torino, Via Valperga Caluso 35,

10125 Turin, Italy

A. Pavese

Section of Milan, National Research Council,

IDPA, Via Botticelli 23, 20133 Milan, Italy

R. I. Smith

ISIS Facility, Science and Technology Facilities Council,

Rutherford Appleton Laboratory, Harwell Science

and Innovation Campus, Didcot OX11 0QX, UK

123

Phys Chem Minerals (2011) 38:11–19

DOI 10.1007/s00269-010-0377-0

upper mantle to crust, and crystallize under diverse phys-

ical–chemical conditions. Their importance as petrogenetic

indicators has motivated a relevant interest (Sack and

Ghiorso 1991; Gasparik and Newton 1984).

Relatively few studies can be found in literature on Mg–

Al–Fe3? spinels by in situ measurements (Harrison et al.

1999; Pavese et al. 1999), although such experiments are a

key to get an insight into the ‘‘real time’’ behaviour of

phases under non-ambient conditions of temperature and

pressure (Harrison and Putnis 1996; O’Neill et al. 1992;

Wood et al. 1986). The comparatively recent investigation

by Martignago et al. (2003, 2006) reports a thorough in situ

high-temperature X-ray diffraction study on synthetic and

natural Mg–Al–Fe3? spinel specimens. But, foremost, the

atomic partitioning in three cation spinels has usually been

determined exploiting crystal–chemical models that relate

site occupancy factors to bond lengths, rather than provide

the cation distribution by a direct measurement of the site

contents.

In such a light, the present investigation aims at mea-

suring the cation partitioning as a function of T in a synthetic

Mg–Al–Fe3? spinel sample by combining in situ high-

temperature (HT) X-ray powder diffraction (XRPD) and

neutron powder diffraction (NPD). HT-XRPD is effective to

determine the Fe3? distribution with respect to Mg and Al

(whose cations are isoelectronic), while HT-NPD allows

one to reliably discriminate between Mg and Al, exploiting

their neutron scattering length difference (0.5375 and

0.345 9 10-12 cm, respectively). In this view, the present

study represents a novelty, fixing the cation partitioning as a

function of T in Mg–Al–Fe3? spinel through a direct in situ

determination of the site contents, without any crystal–

chemical modelling or a priori assumption. The cation

distribution so determined is also compared with that

derived from the Maximum Configuration Entropy princi-

ple, and then used to calculate the thermodynamic param-

eters of the configuration model (O’Neill and Navrotsky

1984). Moreover, the composition we explore in the present

work, i.e. Fe3?:Al *1, differs from those of Mg–Al–Fe3?

spinels investigated in earlier studies.

Experimental

Spinel synthesis

Mg(Fe,Al)2O4 spinel was synthesised by a high-tempera-

ture solid-state reaction of MgO, Al2O3 and Fe2O3 reagent

grades (Aldrich chemicals). The oxide mixture (MgO,

Al2O3 and Fe2O3 in terms of a mole ratio of 1:0.5:0.5) was

first ground at room temperature by a planetary ball mill for

11 h; then pressed into pellets and treated in a muffle

furnace at 1,100�C, without atmosphere control, for 72 h

with a cooling rate about 20�C/h. The full treatment cycle

was repeated twice. The sample crystallinity was checked

by means of a Bragg–Brentano Panalytical X’pert Pro

MPD diffractometer. The lattice parameters of the syn-

thetic spinel were determined at room temperature by using

Si (NBS SRM 640b) as an internal standard. No trace of the

reagent oxides was found. The spinel refinement yields a

cell edge of 8.2395(7) A, which corresponds to an average

value of the cell parameter between the two end-members,

spinel in sensu stricto [MgAl2O4 with a = 8.0859 A

(Andreozzi et al. 2000, 2001)] and hercinite [MgFe2O4

with a = 8.3600 A (Nakatsuka et al. 2004)].

Mossbauer spectroscopy measurements, here not reported

for the sake of brevity, confirmed iron to occur in the sample

as Fe3?. The Mg(Fe,Al)2O4 composition was determined by

the ICP-AES technique, using a model JV24 of JOBIN–

YVON, and results in Mg0.99(2)(Al0.48(2)Fe0.49(2))2O4-x,

where x is such as to provide electro-neutrality. Taking into

account the uncertainties reported above, we assume the

nominal composition of Mg(Fe0.5Al0.5)2O4, in all the struc-

ture refinements discussed henceforth.

Data collection

High-temperature X-ray diffraction data ([298 K) were

collected with a Philips X’Pert h-h diffractometer (CuKaradiation monochromatised by a flat graphite crystal),

equipped with a furnace (AHT PAP1600), calibrated for

sample position and temperature, from 5� to 100� 2h, with

a 2h-step of 0.02� and counting time of 3 s/step. 20%wt Si

powder (NBS SRM 640b) was added to spinel for goni-

ometer zero calibration at room temperature. High-tem-

perature measurements were recorded at room temperature

and every 100th degree from 373 to 1,273 K, in air.

High-temperature time-of-flight powder neutron dif-

fraction data were collected using the Polaris medium

resolution diffractometer at the ISIS pulsed spallation

source (RAL-Rutherford Appleton Laboratory, UK).

Approximately 3 g of sample was loaded into a thin-walled

cylindrical vanadium sample can and placed in a RAL

furnace, which operates under high vacuum. Diffraction

patterns were collected at the same temperatures as the

XRPD measurements, with the furnace temperature being

monitored and controlled using a type K thermocouple

placed in contact with the outside of the sample can. Data

collected in the backscattering (average 2h = 145�) and 2h*90� detector banks over the d-spacing range 0.4–4.2 A

were used in Rietveld refinements.

Data treatment

All the diffraction patterns were analysed by the full profile

Rietveld method implemented in the GSAS software (Larson

12 Phys Chem Minerals (2011) 38:11–19

123

and Von Dreele 2004) using the EXPGUI graphical interface

(Toby 2001). In the case of the XRPD patterns, the back-

ground was modelled by Chebyshev polynomials with ten

terms; the Bragg diffraction peaks were described by

pseudo-Voigt-like profiles, whose Gaussian and Lorentzian

components of the FWHM were modelled as a function of 2hby FWHMGAUSS

2 = g1tan2h ? g2tanh ? g3 (Caglioti et al.

1958) and FWHMLOR = l1 tanh ? l2/cosh (McCusker et al.

1999), respectively. The coefficients g1, g3, l1 and l2 of the

FWHM were refined. The peak asymmetry was treated

according to Finger et al. (1994).

The lattice parameters, isotropic atomic displacement

parameters and oxygen coordinate (u,u,u) were refined,

along with the site occupancy factors of Fe. Only the Fe

occupancy can be reliably determined by XRPD, since iron

scatters X-rays much more strongly than magnesium and

aluminium which, in turn, are quasi-isoelectronic species. In

this analysis, we have treated Al and Mg like one chemical

species, adopting the X-ray scattering curve of the former.

Such a simplification does not skew our results, as proven by

repeating the calculation using the scattering curve of Mg in

place of Al’s. Note that tests, performed to assess the effect of

the starting cation distribution on the final results, showed

that essentially identical structure parameters were produced

by the least squares regardless of the starting values.

With the NPD patterns, the background was modelled

using a Chebyschev polynomial of the first kind with 14

terms. For structure refinement with Polaris data GSAS peak

function 3 (two back-to-back exponential functions convo-

luted with a pseudo-Voigt function) is used. The exponential

functions describe the moderator effects on the diffraction

peak shape, while the pseudo-Voigt function (which is a sum

of Gaussian and Lorentzian functions) describes the instru-

mental and sample contributions to the peak shape (Larson

and Von Dreele 2004). The starting peak shape and width

parameters, which are contained in the instrument parameter

file, were obtained from analysis of a silicon powder sample

(NBS SRM 640b). The oxygen coordinate (u), the isotropic

atomic displacement parameters, the tetrahedral site neutron

scattering length (btet), constraining that of the M-site (boct)

to the chemical composition, were also refined. The site

occupancy factors of Mg and Al were then determined by

means of btet and boct, keeping the site contents of Fe fixed at

the values from XRPD. In Fig. 1a, b the XRPD and NPD

profiles at ambient conditions are shown.

Results and discussion

Thermal expansion

In Fig. 2 and Table 1 the cell edge (a) as a function of

T measured by XRPD and NPD is shown; the a-linear

thermal expansion coefficient has been calculated accord-

ing to the usual definition, i.e. aa(T) = 1/a(da/dT).

A good agreement between the a-values from XRPD

and NPD is observable up to 973 K, whereas a mismatch

appears above 1,073 K, which could be due to either a

sample misalignment or a temperature measurement

problem (Dapiaggi et al. 2002). Because of the larger

sample size and fixed scattering geometry used in time-of-

flight diffraction we believe that this technique provides a

more representative measurement for the spinel cell edge,

even taking into account that the cubic symmetry of spinel

does not require a high d-resolution. Furthermore, in the

neutron diffraction measurement the thermocouple is able

to be placed closer to the sample, separated from it only by

the thin (thermally conducting) metal sample can instead of

the insulating alumina holder of the furnace used for the

X-ray data collections. For all these reasons, we have

chosen to focus on the a-values measured from NPD

only. a(T) can be split into two parts, either exhibiting a

quasi-linear trend: the one that ranges from ambient

2θ (deg)

inte

nsity

-500

0

500

1000

1500

2000

2500

Time-of-light (ms)

20 30 40 50 60 70 80 90

4 6 8 10 12 14 16 18

inte

nist

y

-20

0

20

40

60

80

100

120

(a)

(b)

* *

Fig. 1 Observed, calculated and residual patterns of a XRPD and

b NPD data collections at ambient conditions. Asterisk refers to the

Si (NBS SRM 640b) main diffraction peaks in the XRPD pattern

Phys Chem Minerals (2011) 38:11–19 13

123

conditions to 773 K (haai & 9.03(2) 10-6 K-1), and that

from 773 K to the highest temperature achieved

(haai & 14.5(1) 10-6 K-1). These two curves intersect

one another at *760 K, which provides an estimation of

the temperature whereat occurs the slope discontinuity.

Note that the slope discontinuity is likely related to the

onset of a steady octahedral/tetrahedral cation diffusion

underlying an atomic rearrangement that will be discussed

in the following sections.

Cation partitioning

In Fig. 3a the Fe3? content in the tetrahedral site versus

temperature determined by XRPD is plotted. Below 773 K

no well-defined partitioning rearrangement is observed,

save a weak indication of a slight Fe-adjustment at 573 K,

a possible token of thermal re-equilibration. At higher

temperature, Fe3? starts apparently moving from tetrahe-

dral to octahedral sites, in keeping with the slope change of

a as a function of T, discussed above. The variation with T

of the tetrahedral site neutron scattering length (Fig. 4)

follows a trend qualitatively similar to that of Fe3? shown

in Fig. 3a, and ‘‘not incompatible’’ with a possible though

very modest and non-quantifiable re-equilibration cation

diffusion onset at about 573 K. Combining the Fe3? par-

titioning from XRPD with the neutron scattering lengths of

the T- and M-sites after NPD refinements, the Mg and Al

distribution has been inferred and plotted in Fig. 3a–c.

Three temperature ranges can be observed and discussed,

using the results and T-points of Table 2:

1. room temperature \ T \ 773 K: no solid evidence of

steady and relevant cation diffusion is observed, within

the limits of the present experimental resolution and of

the order–disorder reaction kinetics;

2. 773 B T \ 973 K: mainly Fe3? and Mg exchanges are

involved, as proven by the regular increase of Fe and

Mg in M-and T-sites, respectively;

3. T C 973 K: Al, too, participates positively in the diffu-

sion process. In particular, Mg-atoms tend to prefer the

T-sites, whereas [IV]Al decreases in favour of a sixfold

coordination. Fe3? keeps diffusing from T- to M-sites.

Note that earlier works, by Martignago et al. (2006) and

Antao et al. (2005), claim to observe signals of cation

rearrangement at higher Ts than ours, i.e. over the 820–

850 K range. Such discrepancies have to be considered

taking into account that: (1) different chemical composi-

tions are involved; (2) synthetic samples often bear such a

variety of deviations from ideal as makes it difficult to

compare one work with another, in particular as far as

reactivity and the related phenomenology are concerned.

A further possible cation partitioning scheme based on

the notion of ‘‘Maximum Configuration Entropy’’ (MCE)

has been used to derive an Mg–Al–Fe3? distribution

accordingly. The present case can be envisaged as an

extension of the MEM technique, developed for electron

Fig. 2 Lattice parameter a as a function of temperature. Filled and

open symbols refer to XRPD and NPD refinements, respectively

Table 1 Structural parameters as a function of temperature: lattice

parameter (A), oxygen positional parameter (u), isotropic thermal

parameters (9100; A2) of oxygen (UO), tetrahedral (UT) and octa-

hedral (UM) cation, determined by XRPD and NPD structural

refinements

T (K) a u UO UT UM Rwp%

XRPD

298 8.2395(2) 0.2581(5) 3.31(9) 2.60(9) 2.46(9) 1.87

373 8.2439(2) 0.2578(4) 3.38(9) 2.48(8) 2.63(9) 1.85

473 8.2509(2) 0.2571(5) 3.70(8) 2.68(9) 2.19(9) 1.79

573 8.2580(2) 0.2575(5) 3.86(7) 2.63(8) 2.51(8) 1.77

673 8.2660(2) 0.2576(5) 4.25(9) 3.23(9) 2.68(8) 1.78

773 8.2750(2) 0.2573(5) 4.51(9) 3.12(9) 2.92(8) 1.83

873 8.2841(1) 0.2569(5) 4.42(7) 3.24(9) 2.99(9) 1.81

973 8.2940(1) 0.2573(5) 4.73(8) 3.60(8) 2.93(9) 1.79

1,073 8.3045(2) 0.2562(5) 4.85(9) 3.48(9) 3.12(9) 1.74

1,173 8.3152(2) 0.2566(5) 4.99(9) 3.55(8) 3.25(8) 1.75

1,273 8.3258(2) 0.2573(5) 5.02(9) 3.48(9) 3.66(8) 1.74

NPD

298 8.2395(1) 0.25954(6) 1.04(1) 0.46(1) 0.54(1) 1.77

373 8.2439(1) 0.25951(8) 1.06(1) 0.51(1) 0.59(1) 1.73

473 8.2507(2) 0.25953(8) 1.19(1) 0.63(1) 0.68(1) 1.66

573 8.2579(2) 0.25944(8) 1.31(1) 0.73(1) 0.82(1) 1.63

673 8.2653(2) 0.25946(9) 1.46(1) 0.87(1) 0.96(1) 1.61

773 8.2734(2) 0.25946(9) 1.58(1) 0.98(1) 1.07(1) 1.59

873 8.2825(1) 0.25942(9) 1.71(1) 1.15(1) 1.18(1) 1.52

973 8.2935(1) 0.25936(9) 1.85(1) 1.29(1) 1.29(1) 1.49

1,073 8.3078(2) 0.25945(9) 2.03(1) 1.48(1) 1.33(2) 1.46

1,173 8.3205(2) 0.25949(9) 2.20(1) 1.67(2) 1.45(2) 1.41

1,273 8.3297(2) 0.25935(9) 2.39(1) 1.79(2) 1.69(2) 1.52

Rwp% = 100H(P

((Iobs - Ical)2w/Iobs

2 w))

14 Phys Chem Minerals (2011) 38:11–19

123

density (see Merli et al. 2009 for applications of mineral-

ogical interest) and here adapted to the determination of

cation partitioning.

This approach allows one to use the results from NPD

data collections only, though at the cost of introducing the

MCE principle, which relies on seeking such a cation

distribution as:

a. maximises the configuration entropy (Sconfig), namely

Sconfig ¼X

a¼1;3

X

j¼1;n

�/jxaj ln xa

j

� �ð1Þ

where /j is the jth-site multiplicity to Z (formula units

per cell) ratio, and xja corresponds to the occupancy

factor of the a-species in the jth-site;

b. fulfils the crystal–chemical constraints below, i.e.

xatet þ 2xa

oct ¼ Ca ð2aÞ

boct ¼ xFeoct � bFe þ xMg

oct � bMg þ xAloct � bAl ð2bÞ

btet ¼ xFetet � bFe þ xMg

tet � bMg þ xA1tet � bAl ð2cÞ

where Ca is the content of the a-species per formula

unit, boct and btet the measured site neutron scattering

lengths and ba the neutron scattering length of the a-

species.

On an energetic viewpoint, such a principle attributes a

driving role to the -T 9 Sconfig contribution in stabilizing a

given atom distribution through a solid at ambient pressure

and temperature T. By means of Eqs. 2a, 2b, 2c, it is

straightforward to demonstrate that this problem reduces to

searching for a maximum in a one-dimension space. The

cation partitioning obtained, set out in Table 2, shows

some differences with respect to that from combining

XRPD with NPD, discussed above. This fact can be

interpreted as a hint that energy plays so vital a part in

stabilizing an atomic rearrangement as to compensate the

mere configuration entropy contribution, which grows in

importance upon T. For instance, above 1,200 K, the cation

distribution measured by HT-XRPD and HT-NPD is

affected by a ‘‘configuration entropy penalty’’ of &2.4 kJ/

mol with respect to that based on MCE.

Cation–oxygen bond lengths

In Fig. 5 the tetrahedral and octahedral cation–oxygen

bond lengths (dtet and doct) as a function of temperature are

Fig. 3 a Fe3?, b Mg and c Al occupancy factors in T-site versus T,

by Rietveld refinements. Fe3? from XRPD; Mg and Al from NPD,

constraining the Fe3? occupancy factors to the values from XRPD.

Data from Table 2

Fig. 4 btet T-site neutron scattering length (10-12 cm) as a function

of T, determined from NPD structural refinements

Phys Chem Minerals (2011) 38:11–19 15

123

reported. A discrepancy is observed between the dtet- and

doct-values from XRPD and NPD, and reflects X-rays being

sensitive to electron density, whereas neutrons interact with

the atomic nuclei directly (Finney 1995). This might be

consistent with the hypothesis of an oxygen atom polari-

zation, characterized by an electron cloud having its

barycentre shifted towards the octahedral site. In addition,

the scattering power of the O atoms relative to Fe, Mg and

Al is higher with neutrons than with X-rays, meaning that

the oxygen atom positions will be determined more accu-

rately in the NPD results (Finney 1995).

At T \700 K the bond length changes are due to mere

thermal expansion, whereas in the higher temperature

regime, there occurs an additional contribution due to

cation diffusion over the T- and M-sites, affecting dtet and

doct through both the average sizes of the cations involved

and the different response to T of their bonds. The XRPD

points in Fig. 5 seem to be more scattered than the NPD

measurements: for instance, the T–O bond lengths from

XRPD range from 1.887(7) to 1.909(7) A´

during cation

diffusion, whereas the ones from NPD look like less

responsive to heating. In addition, the larger momentum

transfer range (smaller d-spacing) measured on Polaris,

which allows better refinement of site occupancy and

thermal vibration parameters, and the better oxygen scat-

tering power (as mentioned in the previous comment) mean

the NPD bond lengths are probably a more reliable indi-

cation of what is happening as the temperature is increased.

Theoretical cation–oxygen distances have been calcu-

lated following two methods:

1. using the tabulated cation–oxygen bond lengths of

Shannon 1976, that are in good agreement with the

ones reported by Lavina et al. 2002 and O’Neill and

Navrotsky 1983 for Al, Mg and Fe3? at room

temperature, and the thermal expansion coefficients

determined by the Megaw’s formula (Megaw 1938);

2. resorting to the model of Hazen and Yang (1999) and

the parameters therein reported.

Both methods use linear models to describe the effect of

a solid solution on the response of a site to heating. The

theoretical curves exhibit a degree of agreement with the

observed ones in terms of general trend and, in particular,

the former model nears our XRPD data up to about 800 K.

Isotropic thermal parameter (Uiso) and oxygen

coordinate (u)

The isotropic thermal parameters as a function of T are

plotted in Fig. 6 and, as expected, those measured by NPD

Table 2 Cation distribution of Fe3?, Mg and Al in T-site, as a function of T

T (K) Fe3? Mg Al Fe3? (MCE) Mg (MCE) Al (MCE) DSMCE-O [J/(mol K)] TDSMCE-O (kJ/mol)

298 0.517 0.365 0.116 0.5471 0.2740 0.1789 0.302 0.090

373 0.518 0.363 0.115 0.5425 0.2740 0.1835 0.202 0.076

473 0.519 0.365 0.116 0.5480 0.2740 0.1780 0.301 0.142

573 0.509 0.384 0.107 0.5441 0.2751 0.1808 0.428 0.245

673 0.511 0.369 0.119 0.5409 0.2762 0.1829 0.312 0.210

773 0.502 0.386 0.111 0.5372 0.2773 0.1855 0.421 0.325

873 0.491 0.403 0.105 0.5312 0.2773 0.1915 0.543 0.474

973 0.480 0.394 0.125 0.5154 0.2845 0.2001 0.411 0.400

1,073 0.463 0.423 0.113 0.5076 0.2845 0.2079 0.626 0.672

1,173 0.435 0.474 0.090 0.4944 0.2913 0.2143 1.139 1.336

1,273 0.402 0.532 0.065 0.4780 0.2962 0.2258 1.896 2.413

The estimated uncertainties for Fe3?, Mg and Al are *0.01, *0.004 and *0.006, respectively. The uncertainties on the Al/Mg occupancy

factors here reported are obtained setting those of Fe at the values from XRPD and do not take into account the uncertainty on the Fe3?

occupancy factors

X (MCE): tetrahedral occupancy factors of the X-species, determined according to the principle of maximum configuration entropy (ME);

DSMCE-O: configuration entropy difference between MCE-configuration and observed partitioning

Fig. 5 Experimental and calculated (T–O) and (M–O) bond lengths

versus T. The size of the symbols is comparable to their estimated

standard deviations

16 Phys Chem Minerals (2011) 38:11–19

123

are smaller than their counterparts determined by XRPD.

These differences are well known to reflect that neutrons

and X-rays have different interactions with atoms. One has

also to take into account that the thermal parameters are

often affected by spurious effects due to the experimental

setup (absorption phenomena, for instance).

As a whole the Uisos, which increase upon heating

because of the increase of the atomic thermal vibration

amplitude, show behaviours consistent with the masses of

the related atoms: T- and M-sites, characterized by com-

parable masses, have similar thermal parameters, while O,

lighter than the others, exhibits a larger one.

The oxygen positional parameter (u) as a function of T

measured by XRPD and NPD are shown in Fig. 7; the

mismatch between the two data sets is attributable to the

fact that in the NPD the scattering factor of oxygen is

comparatively large, whereas in XRPD O is the weakest

‘‘scatterer’’. In the present study we believe that NPD data

provide a more reliable measure for the oxygen coordi-

nates, which range from 0.25954(6) to 0.25935(9).

Thermodynamic modelling

The basic assumptions of the model of (O’Neill and Nav-

rotsky 1983, 1984) have been used here to derive from the

present data the energy parameters related to cation

ordering. More complex models (see for instance Redfern

et al. 1999; Pavese et. al 2000) would not be more effective

on account of the comparatively small set of available

measurements. The cation distribution of three chemical

species over the M- and T-sites can be parameterised in

terms of x and y, representing Mg and Fe occupancy factors

at the tetrahedral position, respectively. In such a view, the

Gibbs energy difference with respect to a reference con-

figuration (x0 and y0; we use that having Al ordered at the

octahedral position, and Mg and Fe3? fully disordered, i.e.

x0 = y0 = 0.5), results in:

DG ¼ G T; Uð Þ � G T ; U0ð Þ¼ �T � S x; yð Þconf�S x0; y0ð Þconf

� �þ Ax x� x0ð Þ

þ Ay y� y0ð Þþ Bx x� x0ð Þ2þBy y� y0ð Þ2þCxy x� x0ð Þ y� y0ð Þ

ð3Þ

where U means ‘‘configuration’’, and the ordering energy

has been supposed expandable up to a second order in x

and y around U0. From the equation above descend the

equilibrium constraints at a given T, i.e.

oDG=ox ¼ 0

and

oDG=oy ¼ 0

In explicit form, one has

RT � log 1� xð Þ 1� x� yð Þ= x xþ yð Þ½ �f g¼ Ax þ 2Bx x� x0ð Þ þ Cxy y� y0ð Þ ð4aÞ

RT � log 1� yð Þ 1� x� yð Þ= y yþ xð Þ½ �f g¼ Ay þ 2By y� y0ð Þ þ Cxy x� x0ð Þ: ð4bÞ

If one treats Eqs. 4a and 4b by the least squares method

using the cation distributions versus T here reported, it

turns out that the Cxy interference term is less than 3rsignificant; for this reason we have chosen to set it at zero.

In so doing, one obtains Ax = -25.2(3), Ay = -13.6(3),

Bx = -128(3), By = 104(6) kJ/mol. In Table 3 we report

for the sake of completeness the energetic contributions to

the Gibbs energy measured by means of the model above.

Mg and Fe3?, in turn, affect the configuration energy

differently, the former having a Bx \ 0, whereas the latter a

By [ 0. In this light, the configuration energy favours Mg

to enter the T-site, with Fe3? moving to the octahedral

Fig. 6 Isotropic displacement parameters (Uiso) as a function of

temperatureFig. 7 Oxygen positional parameter (u) versus T. Filled and opencircles refer to XRPD and NPD refinements, respectively

Phys Chem Minerals (2011) 38:11–19 17

123

position, in keeping with an experimental cation

partitioning trend (Fig. 3) featured by a degree of

depletion of Fe3? from the T-site, and an enrichment of

Mg therein.

If one searches for that U that minimises DG at a given

temperature, one finds out a cation distribution having a

trend versus T consistent with the observed one, though

bearing different occupancy factor values. Such an aspect

can be interpreted as a token of a partial inadequacy to

model DG, and of the need of a larger data set to better

constraint the thermodynamic parameters.

Conclusions

Structure refinements of X-ray and neutron powder dif-

fraction data on a synthetic Mg(Fe,Al)2O4 spinel have

allowed one to propose an intra-crystalline cation parti-

tioning scheme as a function of temperature. The Fe

occupancy has been fixed by HT-XRPD, whereas HT-NPD

has enabled the Mg and Al partitioning to be determined.

In this light, the present work represents an instance of

synergic use of two ‘‘parallel’’ diffraction experiments

(HT-XRPD and HT-NPD) to fix univocally and directly the

cation partitioning in an Mg–Al–Fe3? spinel.

Cation diffusion onset was qualitatively ‘‘monitored’’ by

observing a discontinuity (*760 K) on the lattice param-

eter thermal expansion coefficient, and some discrepancies

with earlier studies on comparatively similar samples are

revealed.

The sample here studied exhibits a re-adjustment of

Fe3? and Mg, only, entering M- and T-sites, respectively,

up to 973 K, where a rearrangement of the Al partitioning

starts, too. In the high-temperature regime, Fe3? and Al

tend to migrate towards the sixfold coordination site,

whereas Mg shows a preference for the tetrahedral site,

achieving an occupancy factor of 0.532 at 1,273 K. In

particular, the [IV]Fe/[IV]Al ratio, that maintains much lar-

ger than 1 at any T, passes from 4.4, at room temperature,

to 6.2, at 1,273 K, betokening a remarkable propensity of

Al for the octahedral coordination, specially at HT.

Therefore, one in general gathers that: (1) the T-site is

mainly occupied by Fe3? and Mg, in terms of [IV]Fe3?

[[IV]Mg up to 973 K, whereas at higher Ts [IV]Fe3?

\[IV]Mg; (2) Al principally dwells at the octahedral site and

its occupancy factor increases upon heating. This produces

a configuration less and less disordered, as one can realize

by inspecting the trend of the configuration entropy of

Table 3. In this view, this study leads to a cation distri-

bution that differs from those of Martignago et al. (2006),

who observe a tendency towards an increase of disorder as

a function of temperature, proven by an increase of the

configuration entropy in the Fe-richest sample they studied.

However, one has to take into account that the quoted

authors have investigated Mg–Al–Fe spinels with different

compositions with respect to ours, i.e. poorer of Fe. In this

view, the Fe3?:Al ratio seems to be a key to govern the

cation ordering which stabilizes at a given temperature in

Mg–Al–Fe3? bearing spinels. We wish, however, to

underscore once more that synthetic spinels bear deviations

from ideal that might be material to the intra-crystalline

atomic diffusion and related phenomena.

The temperature-dependent cation distribution has also

been inferred by the MCE principle, using NPD data only;

the results suggest that the configuration energy plays a

relevant part in affecting the partitioning, at the expense of

the configuration entropy contribution of the Gibbs energy.

The presented cation distribution as a function of T

has been interpreted using the configuration model of

O’Neill and Navrotsky (1984) [Eq. 3; Ax = -25.2(3), Ay =

-13.6(3), Bx = -128(3), By = 104(6) kJ/mol], which

leads to that [IV]Mg[0.5, [IV]Fe\0.5 and [IV]Fe/[IV]Al �1

promote stability of the Mg–Al–Fe3? solid solution upon

increasing temperature, if Fe3?:Al *1.

Acknowledgments Neutron scattering beam time at ISIS was pro-

vided by the CCLRC (now STFC). The research was financially

supported by Italian MURST and CNR. The authors acknowledge

Prof. Umberto Russo (University of Padova, Italy) for Mossbauer

measurements. The authors are indebted to the reviewers for their

suggestions that have really enhanced the original manuscript.

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