Phys Chem Minerals (2001) 28: 110±118

Ó Springer-Verlag 2001

ORIGINAL ARTICLE

T. Yamanaka
H. Shimazu
K. Ota

Electric conductivity of Fe2 SiO4 ±Fe3O4 spinel solid solutions

Received: 15 March 2000 / Accepted: 4 September 2000

Abstract The spinel solid solution was found to exist in the whole range between Fe3 O4 and c-Fe2 SiO4 at over 10 GPa. The resistivity of Fe3 x Six O4 (0:0 < x < 0:288) was measured in the temperature range of 80  300 K by the AC impedance method. Electron hopping between Fe3‡ and Fe2‡ in the octahedral site of iron-rich phases gives a large electric conductivity at room temperature. The activation energy of the electron hopping becomes larger with increasing c-Fe2 SiO4 component. A nonlinear change in electric conductivity is not simply caused by the statistical probability of Fe3‡ ±Fe2‡ electron hopping with increasing the total Si content. This is probably because a large number of Si4‡ ions occupies the octahedral site and the adjacent Fe2‡ keeping the local electric neutrality around Si4‡ makes a cluster, which generates a local deformation by Si substitution. The temperature dependence of the conductivity of solid solutions indicates the Verwey transition temperature, which decreases from 124(2) K at x ˆ 0 (Fe3 O4 ) to 102(5) K at x ˆ 0:288, and the electric conductivity gap at the transition temperature decreases with Si4‡ substitution. Key words Fe3)xSixO4 á Solid solution á Cation distribution á Electric conductivity á Verwey transition temperature á Nonstoichiometry

T. Yamanaka
H. Shimazu Department of Earth and Space Science, Graduate School of Science, Osaka University, 1-1 Machikaneyama Toyonaka, Osaka 560-0043, Japan e-mail: [email protected] Fax/Tel.: 081-6-6850-5793 K. Ota Institute of Scienti®c and Industrial Researches, Osaka University, 1-1 Yamadaoka Suita, Osaka 565-0871, Japan

Introduction Investigations of magnetite and related phases have been carried out because of their signi®cance for the Earth's mantle. Isothermal compression and high-pressure phase transformation were demonstrated (Mao et al. 1974; Pasternak et al. 1994; Fei et al. 1999). Substitution of Ti4‡ in Fe3 O4 makes titanoferrous magnetite, which is found in basic igneous rocks and forms a continuous solid solution between magnetite Fe3 O4 and ulvospinel Fe2 TiO4 (Verhoogen 1962). Price (1981) reinvestigated the subsolidus phase relations in the titanomagnetite solid solution series. Many silicate spinel and spinelloid phase stabilities and their crystal structures have been investigated, because they may be important phases in pyrolite upper mantle or spinel lherzolite at about the transition zone between 400 and 650 km. The phase diagram of Mg2 SiO4 ±Fe2 SiO4 solid solution at high pressure and temperature (Akimoto et al. 1967; Yagi et al. 1987; Katsura and Ito 1989; Rubie and Brearley 1990) indicates the transformation to spinel structure of c-Fe2 SiO4 at about 6 GPa (Ohtaka et al. 1997). Yagi et al. (1974) and Marumo et al. (1977) reported the crystal structure of c-Fe2 SiO4 . Yamanaka (1986) reported the structure change and its kinetics as functions of temperature and heating duration. The pressure-volume-temperature relation of c-Fe2 SiO4 has been investigated by static compression measurement (Plymate and Stoutt 1994). Si4‡ substitution in magnetite is rarely found in natural minerals. However, the solid solution of Fe3 x Six O4 is one of the signi®cant subjects in the Earth's interior. The recent discovery of siliceous magnetite (Shiga 1988) stimulated the investigation of solid solutions between magnetite and iron-silicate spinel. A phase diagram was ®rst studied at high pressure, 7 and 9 GPa (O'Neill and Canil 1992). We studied the phase diagram of the Fe2 SiO4 ±Fe3 O4 system and found a spinelloid phase in the intermediate region of 0:4 < x < 0:7 of Fe3 x Six O4 in the pressure range from 3 to 9 GPa at 1200  C (Ohtaka

111

et al. 1997). Recently, Woodland and Angel (1998) reinvestigated the spinelloid phase region and reported three spinelloid phases, which have structures analogous to phases II, III and V of nickel-aluminosilicate spinelloids (Ma 1974; Ma and Sahl 1975a; Ma et al. 1975, 1978c; Akaogi and Navrotsky 1984). Ross et al. (1992) analyzed the crystal structure of spinelloid Fe2:2 Si0:8 O4 (Fa40-Mt60) and described a structure similar to phase V. Structure re®nement of phase II was made by Angel and Woodland (1998). We made single-crystal structure re®nements of x ˆ 0:000; 0:090; 0:288; 0:750; 0:920; 1:000 of Fe3 x Six O4 spinels (Yamanaka et al. 1998) and found that their lattice constants of solid solutions show a large deviation from a linear change with the composition of x. The electron hopping between Fe2‡ and Fe3‡ in the octahedral site in Fe3 O4 magnetite changes with temperature. Hopping becomes frozen at temperatures below about 120 K and the electric charge localization causes the cation-ordered distribution and lattice deformation; this is known as the Verwey transition (Verwey 1939; Verwey and Heilman 1947). Charge localization in magnetite induces its lattice transformation from cubic to monoclinic symmetry (Iida 1980). The monoclinic structure has not yet been analyzed, because diculties arise from the complicated polysynthetic twin and domain distortion on passing through the transition temperature (OÈzdemir and Dunlop 1998). The Verwey transition temperature, Tv decreases with increasing Fe cation vacancy in magnetite (Aragon et al. 1985; Kronmuller and Walz 1980; Tamura 1990). In this paper Tv and electric conductivity dependence on the composition x in Fe3 x Six O4 have been measured, and the mechanism is discussed from the viewpoint of the e€ect of silicon substitution on the hopping electric conductivity and on lattice deformation. The magnetic ordering of Fe3 x Six O4 solid solutions is possibly related to the charge transfer. The possible e€ects of magnetic ordering on the electric conductivity will be discussed in Table 1 Structure parameters of Fe3-x SixO4 spinel solid solution as (MT){MO}2O4. u in (uuu) indicates the atomic coordinate of the positional parameter

another paper of this journal (Yamanaka and Okita 2001). Their Curie temperatures decrease gradually, but not linearly, from 851 to 12 K with increasing content of nonmagnetic ions Si4‡ . Magnetic hysteresis becomes more noticeable in the solid solutions having a larger content of Fe2 SiO4 .

Experiment Sample preparation under high pressure In the course of this study we determined the phase diagram of the Fe2 SiO4 ±Fe3 O4 spinel solid solution and clari®ed the stability region of the spinel phase at high pressure and temperature. Powder mixtures of reagent grade Fe3 O4 and well-characterized a-Fe2 SiO4 were used for high-pressure synthesis. The latter was prepared from FeOOH and SiO2 at 1200  C in a CO2 gas ¯ow. These starting materials were loaded in a gold capsule 3 mm long. A cubic-multianvil high-pressure apparatus was used for the synthesis of all samples of Fe3 x Six O4 solid solutions. A platinum or graphite heater, together with an MgO sleeve insulator, was used in the cell assembly of 10-mm edge length. The detailed assembly was reported in our previous study of the phase diagram (Ohtaka et al. 1997). A split-sphere multianvil-type apparatus was also applied using MgO pressure media of 14-mm edge octahedra. Sample synthesis pressures at 1200  C (10  C) were set in the range of 6± 10 GPa, which is in the stability region of spinel solid solutions in accordance with our previous phase diagram. The sample temperature was measured with a Pt/Pt13Rh thermocouple, which was inserted into the center of the cell. The synthesis pressures and temperatures are listed in Table 1. The samples were kept for 5 h and then quenched under pressure. The recovered aggregate samples of about 1:0  0:8  2:0 mm were used for electric conductivity measurements. All recovered samples were con®rmed by powder di€raction study to be spinel single-phase without any spinelloid phase. Their compositional analyses were made with an electron probe microanalyzer (EPMA). The homogeneity of the samples was also con®rmed. Grain sizes and their textures were examined by scanning electron microscope (SEM) image. It was known that magnetite includes such a cation-de®cient nonstoichiometric compound as Fe3 d O4 . Accordingly, the present electric conductivity measurement considered the e€ect of nonstoichiometry. In order to avoid complexity in the cation distribution, preparation of stoichiometric samples is ecient in the present

Sample(x)

0.000

0.090

0.288

0.750

0.920

1.000

Pressure (GPa) Temperature (°C) Lattice constant a (AÊ) Oxygen (uuu) Site occupancy MO (´2) Ai (Fe) Ai (Si) MT (´1) Ai (Fe) Ai (Si) Interatomic distance MT ±O(AÊ) ´4 MO±O(AÊ) ´6 MO±MO(AÊ) ´6 MO±MT(AÊ) ´6 MT ±MO(AÊ) ´12
6 6 10 10 10 9 1200 ‹ 10 1200 ‹ 10 1200 ‹ 10 1200 ‹ 10 1200 ‹ 10 1200 ‹ 10 8.394 (1) 0.3798 (1)

8.392 (2) 0.3792 (3)

8.374 (2) 0.3769 (1)

8.286 (1) 0.3700 (2)

8.256 (1) 0.3666 (1)

8.2374 (9) 0.3656 (2)

1.0 0.0

0.998 0.002

0.975 0.029

0.963 0.037

0.985 0.015

0.987 0.013

0.914 (7) 0.086

0.769 (4) 0.231

0.324 (7) 0.676

0.110 (5) 0.890

0.026 0.974

1.878 2.063 2.967 3.479 3.479 123.9

1.840 2.078 2.961 3.472 3.472 124.6

1.722 2.114 2.963 3.435 3.435 126.9

1.667 2.135 2.919 3.423 3.423 127.9

1.650 2.139 2.911 3.414 3.414 128.1

1.0 0.0 and angle 1.886 (1) 2.059 (1) 2.968 (1) 3.480 (1) 3.480 (1) 123.7

(2) (2) (2) (2) (2)

(1) (1) (2) (2) (2)

(2) (2) (1) (2) (2)

(1) (1) (1) (1) (1)

(1) (1) (1) (1) (1)

112 electric conductivity. For examining the e€ect of nonstoichiometry on the electric conductivity, samples of Fe3 d O4 were prepared under controlled oxygen partial pressure Po2 with controlling CO2 and H2 mixed-gas ¯ow rate. Two samples were synthesized at 1300  C, Po2 ˆ 10 4 and 10 8 at ambient pressure, and a highly stoichiometric Fe3 O4 was also prepared at 6 GPa and 1200  C under reducing conditions using a graphite heater. Electric conductivity measurement The resistivity of Fe3 x Six O4 was measured in the temperature range of 80  300 K by the AC impedance method. The sample temperature was measured by a sheath thermocouple attached to the sample holder. The measurement was conducted for each sample under He dry gas ¯ow. To measure the internal grain resistivity and grain boundary resistivity of samples, respectively, impedance measurement (Z) was performed from the frequency response analysis. The electric conductivity was measured by the four-probe AC complex impedance method using an impedance meter (LCR meter) with increasing frequency in the range of 1 MHz  2:0 Hz. The present experimental system and sample assembly are shown in Fig. 1. The diagram of the electric circuit is illustrated in Fig. 2. The diagram represents the impedance Z on the complex plane. Z is related to resistivity R (real part) and electric capacity X (imaginary part) by Z ˆ R ‡ iX ˆ jZj€h, where R ˆ jZj cos h and X ˆ 1=xC ˆ jZj sin h. 1=Z ˆ 1=…R ‡ jX † ˆ Y ˆ G ‡ iB, where admittance Y, conductance G, and susceptance B are expressed at angular frequency x by: Fig. 1 A, B Electric resistivity measurement system. A Sample temperature-controlling system. B Electrode and sample assembly

G ˆ R=fR2 ‡ …1=xC†2 g: conductance

…1†

B ˆ i…1=xC†=fR2 ‡ …1=xC†2 g: susceptance :

…2†

Then the relation between G and B is: …G

1=2R†2 ‡ B2 ˆ …1=2R†2 :

…3†

Consequently, when B ˆ 0, R can be measured. An example of the impedance plot of magnetite is shown in Fig. 3. Tv was observed from the thermal change of the electric conductivity at elevated temperature and de®ned by the in¯ection point of the continuous change.

Results and discussion 1. Cation distribution and crystal chemistry It was known that Fe3 O4 has an inverse spinel structure, …Fe3‡ †‰Fe3‡ ; Fe2‡ ŠO24 , and that ideal Fe2 SiO4 is a normal spinel structure of …Si4‡ †‰Fe2‡ ; Fe2‡ ŠO24 . All through this paper, cations in parentheses and brackets indicate the tetrahedral and octahedral cations, MT and MO respectively. Our previous phase study found a complete solid solution between Fe3 O4 and c-Fe2 SiO4 at pressures of

113

deviation from the ideally linear change with composition. An X-ray single-crystal structure re®nement of Fe2:712 Si0:288 O4 reveals the cation distribution of 4‡ 4‡ 2 2‡ 3‡ …Fe3‡ 0:78 ; Si0:22 †‰Fe1:288 Fe0:66 2Si0:057 ŠO4 . It is a signi®cant feature that not less than 20% of the total silicon occupies the octahedral sites; Si4‡ substitutes for Fe3‡ in the octahedral site and occupies 3% of the octahedral cations (Table 1). Hence, the spinel solid solution can be expressed, in general, by the following cation distribution using the disordered parameter t: tetr

Fig. 2 A, B AC Complex impedance measurement. A Complex impedance diagram. B R-C parallel circuit

Fig. 3 AC Complex impedance plot of magnetite. The AC complex impedance plot of stoichiometric magnetite (x = 0.0) was performed by LCR meter with a frequency range of 1 MHz  2.0 Hz. The impedance Z (real part) with phase angle h (imaginary part) is plotted on the complex plane

9 GPa and 1200  C (Ohtaka et al. 1997). In the solid solution, the Si4‡ substitution in Fe3 O4 can be realized by the following ideal substitution: tetr

Fe3‡ ‡ oct Fe3‡ ˆ tetr Si4‡ ‡ oct Fe2‡ :

…4†

The ideal solid-solid reaction is based on the following Si4‡ substitution for Fe3‡ at the tetrahedral site: …1

x†…Fe3‡ †‰Fe2‡ ; Fe3‡ ŠO4 ‡ x…Si4‡ †‰Fe2‡ ; Fe2‡ ŠO4 4‡ 2‡ 3‡ ˆ …Fe3‡ 1 x ; Six †‰Fe1‡x ; Fe1 x ŠO4

…5†

Structure parameters signi®cant for the present study were obtained from our single-crystal structure re®nements of spinel solid solutions (Yamanaka et al. 1998). Only information pertinent to the current subject is presented in Table 1. In this paper, sample compositions are expressed by Fe3 x Six O4 , where x is equivalent to the molfraction of Fe2 SiO4 expressed in Eq. (5). Within the experimental precision, the unit-cell volume calculated from the lattice constant shows a small

4‡ ‰Fe3‡ 1 x‡xt ; Six…1

t† Š

octa

3‡ ‰Fe2‡ 1‡x ; Fe1 x

4‡ xt ; Sixt ŠO4

:

…6†

The parameter t relates the nonlinear change of the lattice constants with composition. Marumo et al. (1977) proposed the disorder of the Si4‡ ion in both cation sites of silicate spinels. Recalculations from their results shown by Eq. (6) indicate the cation disorder in c-Fe2 SiO4 …x ˆ 1:0; xt ˆ 0:023  0:01†, c-Co2 SiO4 …x ˆ 1:0; xt ˆ 0:034  0:008†, and c-Ni2 SiO4 …x ˆ 1:0; xt ˆ 0:005  0:012†. The small number of Si4‡ ions intrinsically occupies the octahedral site. Interatomic MT ±O and MO ±O distances and site volumes of MT O4 and MO O6 are listed in Table 1, where MT and MO represent tetrahedral and octahedral cations, respectively. MT ±O distance decreases with c-Fe2 SiO4 content, and this tendency is a change similar to the lattice constant, while MO ±O distance increases with the content. These changes result from the substitution of a smaller Si4‡ cation for a larger Fe3‡ in the tetrahedral site and simultaneously Fe3‡ replacement with a larger Fe2‡ in the octahedral site. Consequently, the volume of the MT O4 tetrahedra is reduced, while that of the MO O6 octahedra is expanded. The cation distributions of Fe3 x Six O4 solid solutions control their electric conductivity, because electron hopping between Fe2‡ and Fe3‡ at the octahedral site is the main electric conductivity mechanism. The site symmetries of the tetrahedral MT and octahedral MO site are 43 m and 3, respectively. The deformation of the octahedral site owing to the variation of oxygen positional u-parameters from 0.3797 to 0.3658 in¯uences the exchange interaction. The ideal octahedral site has a site symmetry of m3m at u ˆ 0:375. It is estimated from the structure analysis that the sample of x ˆ 0:42 shows the ideal structure. 2. Cation-de®cient nonstoichiometry In this paper we examine the e€ect of nonstoichiometry on electric conductivity. Several models of cation distribution can be proposed. 1. Stoichiometric: Si4‡ ion in tetrahedral and octahedral site. 4‡ 4‡ tetr octa 3‡ ‰Fe3‡ ‰Fe2‡ 1 x‡xt ; Six…1 t† Š 1‡x ; Fe1 x xt ; Sixt ŠO4 2. Nonstoichiometric: cation vacancy substituted for Fe2‡ and Fe3‡ ions in octahedral site.

114 tetr

4‡ octa …Fe3‡ ‰Fe2‡ 1 x ; Six † 1‡x

3‡ 3d ; Fe1 x‡2d ; (dŠO4

;

where the symbol ( indicates the vacant site. 3. x…Fe2 SiO4 † ‡ …1 tetr

x†…c-Fe2 Od †

4‡ octa 3‡ …Fe3‡ ‰Fe2‡ 1 x ; Six † 2x ; Fe5=3…1

x† ; (1=3…1 x† ŠO4

:

It was known that a high electric conductivity in magnetite at room temperature is induced mainly from electron hopping between Fe2‡ and Fe3‡ in the octahedral site. A fairly high electric conductivity of Fe3 O4 …2:5  104 X 1 m 1 † and its temperature change indicates a semiconducting character. The conductivity can be detected in the wide range of the solid solution. With increasing Si content, the conductivity gradually decreases and c-Fe2 SiO4 …x ˆ 1:0† is relatively insulating …5:0  10 6 X 1 m 1 †, because almost no iron ions exist in the tetrahedral site but only Fe2‡ ions in the octahedral site. Then c-Fe2 SiO4 cannot expect an electron hopping mechanism but has a di€erent mechanism of electron conductivity, other than the hopping. 3. Electric conductivity change with temperature and Fe2 SiO4 component Conductivity is partly caused by holes and impurities in many oxides. Two di€erent nonstoichiometric magnetite samples were synthesized under di€erent partial pressure of oxygen, Po2 at ambient pressure, and a stoichiometric sample was prepared under high pressure, as mentioned earlier. These three samples were examined for conductivity measurement (Fig. 4). It was found that the nonstoichiometric sample showed slightly lower conductivity than the stoichiometric

Fig. 4 Electric conductivity of Fe3 d O4 magnetite with regard to their stoichiometry. Three samples are di€erent in stiochiometry. The sample prepared by high-pressure synthesis is a stoichiometric magnetite

Fig. 5 Temperature dependence of electric conductivity of Fe3 x Six O4 spinel solid solution …0 < x < 0:288† in the temperature range between 80 and 300 K

Fig. 6 Electric conductivity of Fe3 x SiOx O4 at 80, 150, 200, and 300 K as a function of x

115

one. It is con®rmed that the sample synthesized at 6 GPa under reduced condition showed the highest conductivity among the three samples. This result is in good agreement with that of Aragon et al. (1985). The high conductivity of the stoichiometric sample is probably due to the greater probability of electron hopping. The conductivities of Fe3 x Six O4 solid solutions of x ˆ 0:0, 0.052, 0.085, 0.107, 0.169, and 0.288 were measured at elevating temperature from 80 to 300 K and they are decreased with Si substitution, as seen from Fig. 5. The conductivities of all samples become larger with increasing temperature; the conductivity of each sample at 80, 150, 200, and 300 K is plotted in the logarithmic scale in Fig. 6. r…x† above Tv indicates about 2 orders of magnitude greater than that below Tv where electron hopping is frozen. The electric conduction below Tv is induced from mechanisms other than electron hopping, such as phonon or polaron conduction. In the compositional range of x ˆ 0:0  0:169; r…x† linearly decreases with x, as shown in Fig. 6, but these slopes do not continue to data at x = 0.288. This is caused by a di€erent electric conduction mechanism of electron hopping.

The number of hopping paths is reduced by the substitution of silicon ions in the octahedral site. When x = 0.288, silicon occupies not less than 20% of the octahedral cations. As shown in Table 1, octahedral cation MO has six nearest-neighbor cations MO with a distance of 2:968  2:911 AÊ and six second neighbors of tetrahedral cations MT with 3:480  3:414 AÊ. On the other hand MT has 12 nearest MO within 3:480  3:414 AÊ. Then six short MO ±MO pairs make electron hopping possible, as shown in Fig. 7. In an ideally perfect Fe3 O4 crystal, 6n paths will be expected, where n is the number of iron ions at octahedral site. Electron hopping does not occur in ideal …Si4‡ †‰Fe2‡ 2 ŠO4 spinel. When one silicon ion substitutes for octahedral iron, six hopping paths will be discontinued. Therefore, the number of hopping paths decreases. Q(x) is obtained from the slope of ln r…T † versus 1/T, which is plotted in Fig. 8. The activation energy Q(x) of hopping shown in Fig. 9 is raised by lessening the hopping probability.

Fig. 7 MT and Mo con®guration in spinel structure. Solid lines indicate electron hopping paths between Fe2‡ and Fe3‡ at octahedral sites in Fe3 O4

116

where x is the silicon content, r0 is the observed electric conductivity of stoichiometric magnetite …102 X 1 m 1 †, and a is a lattice perfection parameter, experimentally optimized to a ˆ 2 in the present experiment. 4. Verwey transition temperature

Fig. 8 ln r…T † in the temperature region above Tv is presented as an inverse temperature (1/T )

The Verwey transition temperature, Tv , of magnetite decreases with increasing Fe cation vacancy (Aragon et al. 1985). Further, Tamura (1990) reported that the Tv of magnetite Fe3 d O4 was lowered from about 120 to 90 K with increasing nonstoichiometry, d, and increasing compression pressure. Then the cation vacancies and the charge transfer in¯uence the conductivity. All samples of solid solutions prepared by the present highpressure synthesis are probably good stoichiometric compounds, as mentioned above. Consequently, the cation distribution of the solid solutions can be expressed by the structure in Eq. (6). Fig. 10 shows the thermal change of Tv with x of Fe3 x Six O4 . In the present experiment, Tv was de®ned by the in¯ection point of the thermal change of electric conductivity shown in Fig. 6. As shown in Table 2, Tv was decreasing with c-Fe2 SiO4 content in Fe3 x Six O4 from 124(2) K at x = 0 …Fe3 O4 † to 102…5† K at x = 0.288. Electron hopping between Fe3‡ and Fe2‡ at the octahedral site is perturbed by Si4‡ substitution at the site. Tv is a€ected by both hopping distance and the probability of electric charge transfer. From the crystal structure data in Table 1, the lattice constant decreases with x content and Mo ±Mo distance becomes smaller. Then in the case of large x, a small thermal energy brings electron hopping. The freezing temperature of the elec-

Fig. 9 Apparent activation energy Q(x) of electron hopping

This ®gure proves that the activation energy of electric conductivity depends on the silicon concentration. This is probably because silicon at the octahedral site hinders the hopping. Q(x) is also related to the Fe±Fe interatomic distance, which is de®ned by the lattice constant. r is expressed in terms of Q(x) by the following equation: r ˆ r0 exp‰ Q…x†=akT Š ;

…7†

Fig. 10 Verwey transition temperature Tv change with x in Fe3 x Six O4

117 Table 2 Verwey transition temperature and electric conductivity gap x in Fe3 x Six O4

Tm /K

0.000 0.052 0.085 0.107 0.169 0.288

124 117 115 111 108 102

‹ ‹ ‹ ‹ ‹ ‹

Dr=X 3 5 5 5 5 5

1

8.5(0.4) 8.0(0.4) 1.0(0.5) 2.5(0.5) 1.1(0.7) 1.1(0.7)

m ´ ´ ´ ´ ´ ´

1

10)1 10)1 10)1 10)2 10)4 10)6

tron hopping is lowered with increasing x. It is also clari®ed that the electric conductivity gap …Dr† at the transition temperature becomes smaller with Si4‡ substitution. Dr is shown in Fig. 11. These experimental results can be explained by the probability of charge transfer in Fe2‡ and Fe3‡ cation pairs and by the lattice deformation. The observed relaxation processes are short- and long-range rearrangements of Fe2‡ and Fe3‡ ionic states within domain walls below Tv .

Summary The nonlinear change of electric conductivity with x in Fe3 x Six O4 indicates that the conductivity is not simply caused by the statistical probability of Fe3‡ ±Fe2‡ electron hopping. The conductivity is a€ected by the silicon content (x) and hopping activation energy Q(x) in the octahedral site. The activation energy depends on the silicon contents and is also a€ected by the charge ordering in octahedral site. However, ideally random distribution of Fe3‡ and Fe2‡ is not performed in these solid solutions, because the adjacent Fe2‡ keeping the local electric neutrality around Si4‡ makes a cluster. This inhomogeneity makes the a parameter not unity.

Fig. 11 Electric conductivity gap at the Verwey transition temperature Tv . Triangle is the gap in the nonstoichiometric magnetite

Another reason may be the fact that the lattice deformation is changed by the silicon substitution. The physical properties of Fe2 SiO4 ±Fe3 O4 solid solution system are important in the study of the geophysical processes in the Earth's upper mantle, and they are controlled by cation distribution. In the present study, the site occupancy test by X-ray structure re®nements has suggested that Fe3 x Six O4 solid solutions have a cation distribution expressed by structure Eq. (6). The observed relaxation processes are short- and long-range rearrangements of Fe2‡ and Fe3‡ ionic states within a domain wall below Tv . Tv of Fe3 x Six O4 depends on the composition x, and the electric conductivity gap at the transition temperature becomes smaller with Si4‡ substitution. The physical properties of the Fe2 SiO4 ± Fe3 O4 solid solution system are helpful for studies of geophysical processes in the Earth's upper mantle, and they are in¯uenced by the charge distribution under the Earth's oxidation state. Acknowledgements The present series of works are partly supported by Grants-in-Aid nos.A(2)-10304044 and B(2)-11694076 of the Ministry of Education, Science, Sport, and Culture, Japan.

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