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The effect of the crystal orientation on the rate of formation of cation-excess magnetite
__ +__ lilt2
a
SOLID STATE
ELSEYIER
IONICS
Solid State Ionics 89 (1996) 279-286
The effect of the crystal orientation on the rate of formation of cation-excess magnetite T. Togawa, T. Sano, Y. Wada, T. Yamamoto, Department of Chemistry, Research Center
M. Tsuji, Y. Tamaura”
for Carbon Recycling and Utilization, Tokyo Institute of Technology, 2-12. I, Ookayatnu, Meguro-ku, Tokyo 152, Japan
Received
17 July 1995; accepted
10 February
1996
Abstract Cation-excess magnetite (Fe,O,_s) has been found to form by passing H, gas through magnetite powder at 300°C with its spine1 structure retained. Cation-excess magnetite iS a metastable phase in the transformation of magnetite into a-Fe. The lattice constant of the cation-excess magnetite was enlarged to a maximum value of 0.8407 nm, which is substantially larger than that of stoichiometric magnetite (a, = 0.8396 nm). The formation mechanism of cation-excess magnetite in the spine1 structure was studied using two specimens of magnetite crystals with (I 11) and ( 100) planes developed. They were referred to as (11 I)- and (lOO)-magnetite. The formation of cu-Fe was decreased over the former, where wiistite and cation-excess magnetite were formed while keeping the fee arrangement of the oxide ions in the solid. On the other hand, cr-Fe was more easily formed over the latter. It is considered that the cation excess state can appear when lattice oxygen is removed faster than the formation of Fe*+ ions coming from Fe3+ ions through the H, reduction and that the cation excess state is stabilized due to electron hopping between Fe’+ and Fe’+ Ions in the B site of the spine1 structure. Keywords: Cation-excess
magnetite;
Spine1 structure;
Electron
hopping;
1. Introduction At room temperature magnetite has an inverse spine1 structure, Fe3+(Fe2+Fe3+)04; one third of the cations occupy tetrahedral interstices in a c.c.p. assembly of O*- ions and equal numbers of Fe*+ and Fe 3+ ions occupy octahedral interstices. It is believed that Fe2+ and Fe3+ ions are randomly distributed over the occupied tetrahedral and octahedral sites at high temperature. A slightly oxidized magnetite has a metal deficit relative to the stoichiometric composition and the formula may accordingly *Corresponding
author. Fax:
0167.2738/96/$15.00 Copyright PI1 SOI67-2738(96)00358-X
+ 81-3-5734-3337 01996
Crystal orientation
be written Fe3_c0, [l-3]. The 5 value increases with decreasing temperature at constant partial pressure of oxygen. It is commonly assumed that the metal deficiency constitutes the presence of unoccupied cation sites. Although very few reports for cation-excess spine1 compounds, Fe,O,_,, are available, there are a few studies on the reaction of gaseous oxide with cation-excess metal oxides [4-61. Dieckmann has studied cation-excess magnetite in the temperature range 900- 14OO”C, where it exists in equilibrium with wiistite [4]. Recently, Tamaura and Tabata have reported that there exists a cation-excess magnetite at around 300°C in the course of the H, reduction of magnetite into a-Fe [5]. They confirmed
Elsevier Science B.V. All rights reserved
280
T. Tqawa
et al.
I Solid
the cation-excess state by means of chemical analysis, X-ray diffractometry and the MSssbauer effect [7,8]. Since this cation-excess state can be obtained by removing lattice oxygen from the spine1 structure, cation-excess magnetite is in an oxygen deficient state compared with the stoichiometric composition of magnetite (Fe,O,). So it may be also called H, reduced magnetite in a cation-excess state. Based on the analysis on the Mijssbauer parameters, they concluded that H, reduced magnetite has a deformed spine1 structure with more A sites occupied than stoichiometric magnetite. The cation-excess magnetite with the deformed spine1 structure is unstable and is strongly inclined to restore the stoichiometric compound. This instability gives rise to a high reduction potential; CO, gas is easily decomposed to carbon and oxygen at 300°C [7,8]. In the phase diagram published by Darken and Gurry [9], coexisting stable phases at around 300°C in the Fe-O system are stoichiometric magnetite (Fe,O,) and (Y-Fe. The cation-excess magnetite can be obtained as a metastable intermediate in the transformation reaction of magnetite into a-Fe by means of the H, reduction at around 300°C; this metastable phase of cation-excess magnetite can exist under a nonequilibrium condition [7,8]. Therefore, the formation of the cation-excess state depends on the reducing conditions such as temperature and H, gas flow rate etc.; the cation-excess state will appear when lattice oxygen is removed with a faster rate than that of reconstruction of cations into the metallic iron crystal structure. The removal rate of lattice oxygen can be expected to be accelerated by increasing the H, flow rate at a constant temperature. Reconstruction of the cations into crystalline metallic iron requires cation movement to make a network of Fe0 such as a cluster. Of course these rates will be associated with the electron-transfer mechanism from cations to H, gas. In previous reports [7,8], we have suggested that the high reduction potential of cation-excess magnetite is due to the existence of electron hopping between Fe’+ and Fe3+ in the octahedral sites of the spine1 structure. The objectives of the present work are to clarify the effect of the crystal orientation of magnetite crystallites on the formation rate of the cation-excess state with the spine1 structure retained and to discuss the formation mechanism of the cation-excess state.
State
lonics
89 (1996)
279-286
2. Experimental 2. I. Materials All chemicals used for sample preparation were of analytical grade, and distilled water was used for the preparation of solutions, FeSO, * 7H,O, ZnSO, . 7H,O and NaOH were supplied by Wako Chemical Industries, Ltd. H,, N, and air gases used were research grade. 2.2. Preparation
of Zn(lI) ferrite,
and (lOO)- and
(I I I)-magnetite
Zn(II) ferrite was synthesized by air oxidation of aqueous suspensions of Fe(U) hydroxides and Zn(I1) ions according to the wet method reported previously [lo]. After passing nitrogen gas through distilled water (4.0 dm3) for 30 min, FeSO, .7H,O (300 g) and ZnSO, .7H,O (15 1 g) were added. The pH of the solution was adjusted to 9 by adding a 3.0 mol dmm3 NaOH solution. Air was passed through the alkaline suspension at 80°C for 6 h. The reaction pH was kept constant at 9 during the air oxidation. The product was collected by decantation. After washing with distilled water and acetone successively, the product was dried in a nitrogen gas stream at 300°C. The product was identified by X-ray diffractometry with FeKa radiation (Rigaku, Model RAD2A diffractometer). The X-ray diffraction pattern of the Zn(II) ferrite showed a single phase of the spine1 type compound without any other peaks assigned to a-Fe,O, and iron hydroxides such as cu-FeOOH. The chemical composition of the Zn(I1) ferrite was determined by atomic absorption spectroscopy for Zn(I1) and Fetola, contents, and by calorimetry with 2,2’-bipyridine [l l] for Fe(U) and Fetota, content. Magnetite powders with surface crystal planes of ( 111) and ( 100) were supplied by the Toda Kogyo Company. They were synthesized by air oxidation of Fe(U) ions in aqueous suspension. Hereafter these magnetites will be referred to as (1 1 1)-magnetite and (lOO)-magnetite, respectively. 2.3. H2 reduction The powder samples (l-4 g) were placed in a quartz tube of the reaction cell (diameter: 10 mm,
T. Togawa et al. I Solid State lonics 89 (1996) 279-286
length: 350 mm). The reaction cell was evacuated, and then heated in an electric furnace at 300°C. After passing H, gas through the samples for l- 12 h at 3OO”C, the samples were quenched by quickly placing the reaction cell in a refrigerant of ice/NaCl while passing nitrogen gas through the reaction cell. The samples were taken out under nitrogen atmosphere, and subjected to X-ray diffractometry while preventing oxidation.
b)
281
I
3. Results and discussion 3.1. Crystal orientation magnetite &ace
of the (lOO)- and (111)
X-ray diffractometry showed that the (11 l)- and (lOO)-magnetite powder samples gave only the peaks assigned to those of spine1 type compound, as shown by patterns (a) in Figs. 1 and 2. Fig. 3(a) and (b) are electron-micrographs for ( 11 l)- and ( lOO)-magnetite powders. The particle shapes estimated from the various images projected on the TEM photograph papers are cubes (( lOO)-magnetite) and octahedrons (( 111 )-magnetite). The average particle size estimated from the electron micrographs is 430 nm for (lOO)-magnetite, and 390 nm (a) and 440 nm (b) for (Ill)-magnetite [(a) and (b) are the edges of the octahedron shown in Fig. 31. Fig. 4 shows the electron diffraction patterns for (lOO)-magnetite (a) and (1 1 1)-magnetite (b). These electron diffraction patterns were taken by allowing the electron beam to strike the surface of the single crystal of the (lOO)or (ill)-magnetite perpendicular to its surface. As can be seen from Fig. 4(a), a typical reciprocal lattice for a face-centred lattice, for which the conditions for reflection are taken that h, k and 1 should be either all odd or all even for a particular reflection. Since 020, 200 and 220 appears in the zero layer of the reciprocal lattice (Fig. 4(a)) for a face-centred lattice, the lattice constant a, was evaluated as to be 0.839 nm. This value is nearly equal to the lattice constant of the stoichiometric magnetite (a, = 0.83967 nm) [12]. Thus, the surface of the (lOO)magnetite is oriented toward the (100) direction. Fig. 4(b) shows a typical hexagonal reciprocal lattice. This reciprocal lattice pattern can be projected by taking the diffraction pattern allowing the electron
I
I
I
I
40
50
60
j
28 (Cl.&) Fig. 1. Change in the powder XRD patterns during H, reduction of (11 l)-magnetite for 0 h (a) (non-reduced), 1 h (b), 3 h (c) and 6 h (d) respectively. H, flow rate = 0.1 dm3 min-‘.
beam to strike the single crystal having a facecentred lattice in the (111) direction. The (1 ll)magnetite is the spine1 type compound with a facecentred lattice. Thus, the reciprocal lattice pattern of Fig. 4(b) shows that the surface of the single crystal of (11 I)-magnetite is oriented toward (111) direction, for the electron beam was allowed to strike the surface of the (ill)-magnetite perpendicular to the surface. 3.2. H, reduction of (1 II)- and (100)~magnetite Fig. 1 shows the time variation of the X-ray diffraction pattern of (ill)-magnetite in the H, reduction at 300°C. Small wtistite peaks appeared along with the strong peaks of the spine1 structure at 1 h of H, reduction time (pattern (b) in Fig. 1). During the reduction time from 1 h to 3 h, a small peak for a-Fe appeared along with the small peaks
T. Togawa et al. I Solid State Ionics 89 (1996)
282
I
I
I
I
40
50
60
J
20 (Cu&) Fig. 2. Change in the powder XRD patterns during H, reduction of (lOO)-magnetite for 0 h (a), 0.5 h (b) and 3 h (c) respectively. H, flow rate = 0.1 dm3 min-‘.
of wi.istite and the strong peaks of the spine1 structure. After further reduction for 6 h, the peak intensity for a-Fe became stronger, but that for wtistite rather smaller (Fig. l(d)). Thus, in the early stage of the H, reduction of (Ill)-magnetite at 300°C (patterns (a)-(d)), some part of (11 l)-magnetite is transformed into wiistite; the removal of lattice oxygen from the surface of the ( 111) plane of the spine1 structure results in the formation of wtistite with the NaCl structure [13]. Since the wtistite phase was observed by X-ray diffractometry, as shown by Fig. l(b) and (c), it has at least 2.0 nm to 10 nm range of crystal structure. The spine1 structure and the NaCl structure have the same cubic close packing of oxide ions. Therefore, the wtistite crystal structure can be formed only by moving Fe*+ ions, which are formed from Fe3+ ions by the H, reduction, into interstitial B sites of the spine1 structure while maintaining the cubic close packing of the oxide ions (excess oxide ions should be removed by reaction with H, as H,O). On the other hand, as can be seen from curve A in
279-286
Fig. 5 and the X-ray diffraction patterns in Fig. 1, the lattice constant of the spine1 structure increased with an increase in the peak intensity of the wiistite in the X-ray diffraction pattern. Since the expansion of the cell volume of the spine1 structure is accompanied by the wustite phase formation (Fig. l(b)), it is considered that the expansion of the cell volume comes from the movement of Fe*+ ions, which are formed from Fe3+ ions by H, reduction, into interstitial B sites of the spine1 structure. The lattice constant of the spine1 structure rapidly increased with H, reduction as shown by curve A in Fig. 5. However, the peak intensity of the wtistite phase in the X-ray diffraction pattern increased slowly after the lattice constant attained a maximum value (patterns (b) and (c) in Fig. 1). Thus, it is considered that the expansion of the cell volume of the spine1 structure first takes place due to cation (Fe*‘) movement into interstitial B sites in the spine1 structure, before an appreciable crystal structure of wustite ranging around 2.0- 10 nm is generated. These considerations suggest that in the H, reduction of (1 11)-magnetite, Fe ‘+ ions formed from Fe3+ ions by H, reduction move into interstitial B sites, and some excess Fe” ions form an appreciable amount of NaCl structure which can be detected by X-ray diffractometry, and eventually the wustite phase appears. The chemical analysis for the sample of pattern (b) in Fig. 1 showed that the Fe *+/Fe,,,,, mole ratio was 0.376. In pattern (b) in Fig. 1, we can observe a small peak of wtistite, and therefore this chemically determined value does not accurately indicate the value corresponding to the single phase of the spine1 structure (cation-excess). However, comparing the peak intensity of the wiistite phase with that in pattern (c), the peak intensity of wiistite in pattern (b) is fairly small. Therefore it is a reasonable approximation to say that the sample for pattern (b) is just in a transient state from a single phase cation-excess spine1 structure to a two-phase mixture consisting of wiistite and cation-excess spinel. Based on this approximation, the chemical composition of cationexcess spine1 (cation-excess magnetite) was estimated to be Fe~~,,Fe~‘,,O,_, with 6 = 0.1 (a,, = 0.8407 nm). In the second stage (patterns (c),(d) in Fig. l), wiistite is considered to change into a-Fe, both peak intensities for Fe0 and Fe,O, decreased during the
T. Togawa et al. I Solid State tonics 89 (1996) 279-286
283
cube
b)
octahedron
Fig. 3. TEM micrographs size of 390-440 nm.
of the powders
of (lOO)-magnetite
with a particle size of 430 nm (a) and of
increase in the a-Fe peak intensity (from pattern (b) to (c) in Fig. 1). As can be seen from curve A in Fig. 5, the lattice constant of the spine1 structure lowered with further H, reduction toward an overall reduction time of 6 h, where the wiistite phase is transformed into the a-Fe phase. When wtistite is transformed into cy-Fe (patterns (c) and (d) in Fig. l), the higher value of the lattice constant of the spine1 structure (cation-excess) lowered from 0.841 nm to 0.840 nm. This lowering of the lattice constant suggests a lowering in the concentration of cations (Fe2+) situated in interstitial B sites of the cationexcess spine1 structure. Fig. 2 shows the time variation of the X-ray diffraction pattern of (lOO)-magnetite with H, reduction at 300°C. As can be seen from the pattern after 3 h reduction, no wtistite peaks appeared but a-Fe peaks along with the strong peaks of the spine1 structure were present. This (Y-Fe peak intensity became slightly stronger after further H, reduction
(1I1 )-magnetite (b) with a particle
(Fig. 2). Since no peaks of wtistite were observed throughout H, reduction, it is suggested that the reduction of Fe2+ or Fe3+ ions in the spine1 structure results in the direct formation of U-Fe during the H, reduction of (lOO)-magnetite. As can be seen from curve B in Fig. 5, the lattice constant of the spine1 structure of (lOO)-magnetite was increased during the H, reduction at 300°C. However, the lattice constant did not become as large as that of stoichiometric magnetite (in Fig. 5 the lattice constant for stoichiometric magnetite is indicated by an arrow). Moreover, the lattice constant was nearly constant at 0.8400 nm during the H, reduction. These results indicate that the amount of Fe’+ ions formed from Fe3+ ions, which can move into interstitial B sites, is limited in the H, reduction of (lOO)-magnetite at 300°C. Keeping the lattice constant at 0.8400 nm, the peak intensity for a-Fe increased in the X-ray diffraction pattern during the H, reduction (pattern (c) in Fig. 2). This suggests that excess Fez+ ions
T. Togawa et al. I Solid State Ionics 89 (1996) 279-286
284
lattice constant of H, reduced (ill)-magnetite became significantly larger than that of stoichiometric magnetite, as can be seen from curves A and B in Fig. 5. This shows that in the H, reduction of (111) magnetite, the amount of Fe*+ which can move into interstitial B sites is larger than that of (lOO)-magnetite.
3.3. H2 reduction of Zn(II) ferrite
Fig. 4. Single crystal electron diffraction patterns of spots on photographic films of (lOO)-magnetite (a) and (1 1 1).magnetite (b).
0
2
4
6
Reduction time (h) Fig. 5. Variations in the lattice constants of (11 I)-magnetite (A) and (lOO)-magnetite (B) as a function of the H, reduction time at 300°C.
above a limited amount, which could have moved into interstitial B sites, must be directly transformed into a-Fe. Thus, in H, reduced (lOO)-magnetite, the amount of the Fe*’ ions which can move into interstitial B sites seems to be small. In contrast, the
Fig. 6 shows the time variation of the X-ray diffraction pattern of Zn(I1) ferrite upon H, reduction at 350°C. As can be seen here, the reduction of Zn(I1) ferrite results in the formation of wiistite as in the case of (ill)-magnetite. Zn(I1) ferrite is a normal spinel; the A sites are occupied by Zn*+ ions, and the B sites by Fe 3+ ions [14]. Therefore in H, reduced Zn(I1) ferrite, Fe3+ ions in B sites are reduced. The results shown in Fig. 6 suggest that the reduction of Fe3+ ions in B sites preferentially forms wiistite. This finding suggests that Fe*+ ions, which are formed from Fe3+ ions in B sites by H, reduction, preferentially move into interstitial B sites, the resulting anion array corresponding to the lattice points of the NaCl structure of wiistite. Fig. 7 shows the time variation in the lattice constant of Zn(I1) ferrite with H, reduction time. As can be seen here, the lattice constant of Zn(I1) ferrite rapidly increased with H, reduction. This increase in the lattice constant shows that Fe*+ ions formed from Fe3+ ions in B sites move into interstitial B sites of the spine1 structure of Zn(I1) ferrite, leading to an expansion of the cell volume of the spine1 structure. Since this increase in the lattice constant was observed before the appearance of the distinct peaks of wiktite in the X-ray diffraction pattern (pattern in Fig. 6), it is concluded that the movement of cations (Fe*‘) into interstitial B sites of the spine1 structure occurred before an appreciable crystal structure ranging around 2.0-10 nm was generated. The further reduction of Zn(I1) ferrite resulted in the decomposition of Zn(I1) ferrite into ZnO and Fe0 (wiistite), see pattern (d) in Fig. 6. Moreover, the wiktite peak intensity became very strong after further reduction. Thus, in the case of Zn(I1) ferrite, wiistite, which is formed by Fe*+ ions moved into interstitial B sites during H, reduction, is not so readily transformed into a-Fe; wiistite can exist as a
T. Togawa et al. I Solid State Ionics 89 (1996) 279-286
sites takes place. This would preferentially facilitate the transformation of cation-excess spine1 into NaCl structure material. If this is the reason for the prevention of the transformation of wtistite into LYFe, the easy transformation of wtistite in to (Y-Fe in the case of (11 I)-magnetite would be due to the reduction of Fe3+ ions on A sites. As described above, the transformation of wtistite into a-Fe during H, reduction of (11 I)-magnetite causes a lowering in the lattice constant of the cation-excess spinel. This suggests that excess cations in the spine1 move into the wiistite and/or the cu-Fe phase during the transformation process of wiistite into a-Fe. Since the reduction of Fe3+ ions on A sites in (11 l)-magnetite takes place along with the reduction of iron ions on B sites, some excess cations in the spine1 structure of (11 1)-magnetite would be generated from Fe3+ ions on A sites. The ease in the formation of wtistite into a-Fe in the H, reduction of (11 I)-magnetite would come from the additional generation of excess cations in the spine1 due to the reduction of Fe3+ ions on A sites.
(311) n 8 ZnFQ,
a)
WO) II
b)
.
-,-no
. .
zno u-Fe 0 WO)
(511)
I
60
285
3.4. Cation movement
80
mechanism
2e(Fe&) Fig. 6. Change in the powder XRD patterns during H, reduction of Zn(II) ferrite for 0 h (a), 2 h (b), 6 h (c) and 10 h (d), respectively. H, flow rate = 0.1 dm2 min-'
rather stable phase, compared with the case observed for (ill)-magnetite. Since the A sites are occupied with Zn2+ ions, only the reduction of Fe3+ ions on B
0 CJ
0.843 -
0
Redu&~~ time (h,8
Fig. 7. Variations in the lattice constant of Zn(II)-ferrite function of the H, reduction time at 300°C.
as a
Fig. 8 shows the schematic illustration of the cation movement mechanism in the H, reduction of the (11 l)- and ( lOO)-magnetite. On the surface of the (1 11)-magnetite, a layer of Fe3+ and Fe’+ ions on B sites is covered with an oxygen layer, as shown by Fig. 8(a). If the oxygen ions of the surface layer in Fig. 8(a) are removed by reaction with H, and Fe3+ and Fe2+ ions are reduced, the reduced iron ions are supported by the three oxygen ions existing beneath the layer consisting of iron ions on B sites. Even though these iron ions are surrounded by oxygen ions, this coordination state is very asymmetric. Therefore they will move into a more stable coordination, i.e. to octahedrally or tetrahedrally coordinated sites, i.e. into interstices. As shown in Fig. 8, there exists an interstitial B site beneath the 2nd layer of the oxygen ions for (ill)-magnetite. It is well known that electron hopping occurs among the iron ions on B sites. Therefore, during increasing the lattice constant of the (ill)-magnetite and Zn(I1) ferrite during H, reduction, the electrons of iron ions on B sites are considered to be delocalized among the iron ions on these sites by electron hopping. This
T. Togawa et al. I Solid State lonics 89 (1996) 279-286
286
b)
Fig. 8. Illustration
of reduction
reactions
occurring
on the crystals
means that the Fe2+ ions formed from Fe3+ ions on B sites are not so readily reduced to cr-Fe, since the electrons will be readily transmitted to the iron ions on B sites. This eventually results in the formation of the NaCl structure. Thus, the reduced iron ions on B sites are relatively resistant toward a further reduction to Fe’. These rather stable iron ions can have an amount of the transition time for the movement into interstitial B sites. Thus, in the case where the wiistite phase appears in the H, reduction, the Fe*+ ions formed from Fe3+ ions are considered to move preferentially into interstitial B sites of the spine1 structure. These considerations suggest that in the H, reduction of (1 11)-magnetite and Zn(I1) ferrite, the reduction of the Fe3+ ions on B sites results in the movement of reduced iron ions into interstitial B sites.
Acknowledgments The present work was partially supported by Grant-in-Aid for Science Research No. 03203216 from Ministry of Education, Science and Culture.
of
(11I)-magnetite
(a) and (lOO)-magnetite
(b).
References [II .I. Smith and J.I.J. Wijn, Ferrite (Philips Technical Library, Tokyo, 1965) p. 136. PI H. Flood and D.G. Hill, Ber. Bunsenges. Phys. Chem. 61 (1957) 18. [31 J. Brynestad and H. Flood, Ber. Bunsenges. Phys. Chem. 62 (1958) 953. [41 R. Dieckmann, Ber. Bunsenges. Phys. Chem. 86 (1982) 112. PI Y. Tamaura and M. Tabata, Nature 346 (1990) 255. [61 R. Dieckmann and H. Schmalzried, Ber. Bunsenges. Phys. Chem. 81 (1977) 414. r71 Y. Tamaura, Proc. Int. Symp. Chemical Fixation Carbon Dioxide, Nagoya, 1991, ed. K. Ito (Chemical Society of Japan; Research Group on Fixation of Carbon Dioxide, Nagoya, 1991) p. 167. WI Y. Tamaura, The 6th International Conference on Ferrites (Tokyo, Japan, 1992) p. 195. 191 L.S. Darken and R.W. Gurry, J. Am. Chem. Sot. 68 (1946) 798. [101 T. Kodama, M. Tabata, K. Tominaga and T. Yoshida, J. Mater. Sci. 28 (1993) 547. [Ill I. Iwasaki, T. Katsura, T. Ozawa, M. Yoshida, M. Mashima, H. Haramura and B. Iwasaki, Bull. Volcanol. Sot. Jpn. Ser. II 5 (1960) 9. 1121 JCPDS card 19-629 (Joint Committee on Powder Diffraction Standard, Swarthmore, 1989). [I31 E.R. Jette and F. Foote, J. Chem. Phys. 1 (1933) 29. [I41 E.J.W. Verwey and E.L. Heilmann, J. Chem. Phys. 15 (1947) 174.
a
SOLID STATE
ELSEYIER
IONICS
Solid State Ionics 89 (1996) 279-286
The effect of the crystal orientation on the rate of formation of cation-excess magnetite T. Togawa, T. Sano, Y. Wada, T. Yamamoto, Department of Chemistry, Research Center
M. Tsuji, Y. Tamaura”
for Carbon Recycling and Utilization, Tokyo Institute of Technology, 2-12. I, Ookayatnu, Meguro-ku, Tokyo 152, Japan
Received
17 July 1995; accepted
10 February
1996
Abstract Cation-excess magnetite (Fe,O,_s) has been found to form by passing H, gas through magnetite powder at 300°C with its spine1 structure retained. Cation-excess magnetite iS a metastable phase in the transformation of magnetite into a-Fe. The lattice constant of the cation-excess magnetite was enlarged to a maximum value of 0.8407 nm, which is substantially larger than that of stoichiometric magnetite (a, = 0.8396 nm). The formation mechanism of cation-excess magnetite in the spine1 structure was studied using two specimens of magnetite crystals with (I 11) and ( 100) planes developed. They were referred to as (11 I)- and (lOO)-magnetite. The formation of cu-Fe was decreased over the former, where wiistite and cation-excess magnetite were formed while keeping the fee arrangement of the oxide ions in the solid. On the other hand, cr-Fe was more easily formed over the latter. It is considered that the cation excess state can appear when lattice oxygen is removed faster than the formation of Fe*+ ions coming from Fe3+ ions through the H, reduction and that the cation excess state is stabilized due to electron hopping between Fe’+ and Fe’+ Ions in the B site of the spine1 structure. Keywords: Cation-excess
magnetite;
Spine1 structure;
Electron
hopping;
1. Introduction At room temperature magnetite has an inverse spine1 structure, Fe3+(Fe2+Fe3+)04; one third of the cations occupy tetrahedral interstices in a c.c.p. assembly of O*- ions and equal numbers of Fe*+ and Fe 3+ ions occupy octahedral interstices. It is believed that Fe2+ and Fe3+ ions are randomly distributed over the occupied tetrahedral and octahedral sites at high temperature. A slightly oxidized magnetite has a metal deficit relative to the stoichiometric composition and the formula may accordingly *Corresponding
author. Fax:
0167.2738/96/$15.00 Copyright PI1 SOI67-2738(96)00358-X
+ 81-3-5734-3337 01996
Crystal orientation
be written Fe3_c0, [l-3]. The 5 value increases with decreasing temperature at constant partial pressure of oxygen. It is commonly assumed that the metal deficiency constitutes the presence of unoccupied cation sites. Although very few reports for cation-excess spine1 compounds, Fe,O,_,, are available, there are a few studies on the reaction of gaseous oxide with cation-excess metal oxides [4-61. Dieckmann has studied cation-excess magnetite in the temperature range 900- 14OO”C, where it exists in equilibrium with wiistite [4]. Recently, Tamaura and Tabata have reported that there exists a cation-excess magnetite at around 300°C in the course of the H, reduction of magnetite into a-Fe [5]. They confirmed
Elsevier Science B.V. All rights reserved
280
T. Tqawa
et al.
I Solid
the cation-excess state by means of chemical analysis, X-ray diffractometry and the MSssbauer effect [7,8]. Since this cation-excess state can be obtained by removing lattice oxygen from the spine1 structure, cation-excess magnetite is in an oxygen deficient state compared with the stoichiometric composition of magnetite (Fe,O,). So it may be also called H, reduced magnetite in a cation-excess state. Based on the analysis on the Mijssbauer parameters, they concluded that H, reduced magnetite has a deformed spine1 structure with more A sites occupied than stoichiometric magnetite. The cation-excess magnetite with the deformed spine1 structure is unstable and is strongly inclined to restore the stoichiometric compound. This instability gives rise to a high reduction potential; CO, gas is easily decomposed to carbon and oxygen at 300°C [7,8]. In the phase diagram published by Darken and Gurry [9], coexisting stable phases at around 300°C in the Fe-O system are stoichiometric magnetite (Fe,O,) and (Y-Fe. The cation-excess magnetite can be obtained as a metastable intermediate in the transformation reaction of magnetite into a-Fe by means of the H, reduction at around 300°C; this metastable phase of cation-excess magnetite can exist under a nonequilibrium condition [7,8]. Therefore, the formation of the cation-excess state depends on the reducing conditions such as temperature and H, gas flow rate etc.; the cation-excess state will appear when lattice oxygen is removed with a faster rate than that of reconstruction of cations into the metallic iron crystal structure. The removal rate of lattice oxygen can be expected to be accelerated by increasing the H, flow rate at a constant temperature. Reconstruction of the cations into crystalline metallic iron requires cation movement to make a network of Fe0 such as a cluster. Of course these rates will be associated with the electron-transfer mechanism from cations to H, gas. In previous reports [7,8], we have suggested that the high reduction potential of cation-excess magnetite is due to the existence of electron hopping between Fe’+ and Fe3+ in the octahedral sites of the spine1 structure. The objectives of the present work are to clarify the effect of the crystal orientation of magnetite crystallites on the formation rate of the cation-excess state with the spine1 structure retained and to discuss the formation mechanism of the cation-excess state.
State
lonics
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2. Experimental 2. I. Materials All chemicals used for sample preparation were of analytical grade, and distilled water was used for the preparation of solutions, FeSO, * 7H,O, ZnSO, . 7H,O and NaOH were supplied by Wako Chemical Industries, Ltd. H,, N, and air gases used were research grade. 2.2. Preparation
of Zn(lI) ferrite,
and (lOO)- and
(I I I)-magnetite
Zn(II) ferrite was synthesized by air oxidation of aqueous suspensions of Fe(U) hydroxides and Zn(I1) ions according to the wet method reported previously [lo]. After passing nitrogen gas through distilled water (4.0 dm3) for 30 min, FeSO, .7H,O (300 g) and ZnSO, .7H,O (15 1 g) were added. The pH of the solution was adjusted to 9 by adding a 3.0 mol dmm3 NaOH solution. Air was passed through the alkaline suspension at 80°C for 6 h. The reaction pH was kept constant at 9 during the air oxidation. The product was collected by decantation. After washing with distilled water and acetone successively, the product was dried in a nitrogen gas stream at 300°C. The product was identified by X-ray diffractometry with FeKa radiation (Rigaku, Model RAD2A diffractometer). The X-ray diffraction pattern of the Zn(II) ferrite showed a single phase of the spine1 type compound without any other peaks assigned to a-Fe,O, and iron hydroxides such as cu-FeOOH. The chemical composition of the Zn(I1) ferrite was determined by atomic absorption spectroscopy for Zn(I1) and Fetola, contents, and by calorimetry with 2,2’-bipyridine [l l] for Fe(U) and Fetota, content. Magnetite powders with surface crystal planes of ( 111) and ( 100) were supplied by the Toda Kogyo Company. They were synthesized by air oxidation of Fe(U) ions in aqueous suspension. Hereafter these magnetites will be referred to as (1 1 1)-magnetite and (lOO)-magnetite, respectively. 2.3. H2 reduction The powder samples (l-4 g) were placed in a quartz tube of the reaction cell (diameter: 10 mm,
T. Togawa et al. I Solid State lonics 89 (1996) 279-286
length: 350 mm). The reaction cell was evacuated, and then heated in an electric furnace at 300°C. After passing H, gas through the samples for l- 12 h at 3OO”C, the samples were quenched by quickly placing the reaction cell in a refrigerant of ice/NaCl while passing nitrogen gas through the reaction cell. The samples were taken out under nitrogen atmosphere, and subjected to X-ray diffractometry while preventing oxidation.
b)
281
I
3. Results and discussion 3.1. Crystal orientation magnetite &ace
of the (lOO)- and (111)
X-ray diffractometry showed that the (11 l)- and (lOO)-magnetite powder samples gave only the peaks assigned to those of spine1 type compound, as shown by patterns (a) in Figs. 1 and 2. Fig. 3(a) and (b) are electron-micrographs for ( 11 l)- and ( lOO)-magnetite powders. The particle shapes estimated from the various images projected on the TEM photograph papers are cubes (( lOO)-magnetite) and octahedrons (( 111 )-magnetite). The average particle size estimated from the electron micrographs is 430 nm for (lOO)-magnetite, and 390 nm (a) and 440 nm (b) for (Ill)-magnetite [(a) and (b) are the edges of the octahedron shown in Fig. 31. Fig. 4 shows the electron diffraction patterns for (lOO)-magnetite (a) and (1 1 1)-magnetite (b). These electron diffraction patterns were taken by allowing the electron beam to strike the surface of the single crystal of the (lOO)or (ill)-magnetite perpendicular to its surface. As can be seen from Fig. 4(a), a typical reciprocal lattice for a face-centred lattice, for which the conditions for reflection are taken that h, k and 1 should be either all odd or all even for a particular reflection. Since 020, 200 and 220 appears in the zero layer of the reciprocal lattice (Fig. 4(a)) for a face-centred lattice, the lattice constant a, was evaluated as to be 0.839 nm. This value is nearly equal to the lattice constant of the stoichiometric magnetite (a, = 0.83967 nm) [12]. Thus, the surface of the (lOO)magnetite is oriented toward the (100) direction. Fig. 4(b) shows a typical hexagonal reciprocal lattice. This reciprocal lattice pattern can be projected by taking the diffraction pattern allowing the electron
I
I
I
I
40
50
60
j
28 (Cl.&) Fig. 1. Change in the powder XRD patterns during H, reduction of (11 l)-magnetite for 0 h (a) (non-reduced), 1 h (b), 3 h (c) and 6 h (d) respectively. H, flow rate = 0.1 dm3 min-‘.
beam to strike the single crystal having a facecentred lattice in the (111) direction. The (1 ll)magnetite is the spine1 type compound with a facecentred lattice. Thus, the reciprocal lattice pattern of Fig. 4(b) shows that the surface of the single crystal of (11 I)-magnetite is oriented toward (111) direction, for the electron beam was allowed to strike the surface of the (ill)-magnetite perpendicular to the surface. 3.2. H, reduction of (1 II)- and (100)~magnetite Fig. 1 shows the time variation of the X-ray diffraction pattern of (ill)-magnetite in the H, reduction at 300°C. Small wtistite peaks appeared along with the strong peaks of the spine1 structure at 1 h of H, reduction time (pattern (b) in Fig. 1). During the reduction time from 1 h to 3 h, a small peak for a-Fe appeared along with the small peaks
T. Togawa et al. I Solid State Ionics 89 (1996)
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I
I
I
I
40
50
60
J
20 (Cu&) Fig. 2. Change in the powder XRD patterns during H, reduction of (lOO)-magnetite for 0 h (a), 0.5 h (b) and 3 h (c) respectively. H, flow rate = 0.1 dm3 min-‘.
of wi.istite and the strong peaks of the spine1 structure. After further reduction for 6 h, the peak intensity for a-Fe became stronger, but that for wtistite rather smaller (Fig. l(d)). Thus, in the early stage of the H, reduction of (Ill)-magnetite at 300°C (patterns (a)-(d)), some part of (11 l)-magnetite is transformed into wiistite; the removal of lattice oxygen from the surface of the ( 111) plane of the spine1 structure results in the formation of wtistite with the NaCl structure [13]. Since the wtistite phase was observed by X-ray diffractometry, as shown by Fig. l(b) and (c), it has at least 2.0 nm to 10 nm range of crystal structure. The spine1 structure and the NaCl structure have the same cubic close packing of oxide ions. Therefore, the wtistite crystal structure can be formed only by moving Fe*+ ions, which are formed from Fe3+ ions by the H, reduction, into interstitial B sites of the spine1 structure while maintaining the cubic close packing of the oxide ions (excess oxide ions should be removed by reaction with H, as H,O). On the other hand, as can be seen from curve A in
279-286
Fig. 5 and the X-ray diffraction patterns in Fig. 1, the lattice constant of the spine1 structure increased with an increase in the peak intensity of the wiistite in the X-ray diffraction pattern. Since the expansion of the cell volume of the spine1 structure is accompanied by the wustite phase formation (Fig. l(b)), it is considered that the expansion of the cell volume comes from the movement of Fe*+ ions, which are formed from Fe3+ ions by H, reduction, into interstitial B sites of the spine1 structure. The lattice constant of the spine1 structure rapidly increased with H, reduction as shown by curve A in Fig. 5. However, the peak intensity of the wtistite phase in the X-ray diffraction pattern increased slowly after the lattice constant attained a maximum value (patterns (b) and (c) in Fig. 1). Thus, it is considered that the expansion of the cell volume of the spine1 structure first takes place due to cation (Fe*‘) movement into interstitial B sites in the spine1 structure, before an appreciable crystal structure of wustite ranging around 2.0- 10 nm is generated. These considerations suggest that in the H, reduction of (1 11)-magnetite, Fe ‘+ ions formed from Fe3+ ions by H, reduction move into interstitial B sites, and some excess Fe” ions form an appreciable amount of NaCl structure which can be detected by X-ray diffractometry, and eventually the wustite phase appears. The chemical analysis for the sample of pattern (b) in Fig. 1 showed that the Fe *+/Fe,,,,, mole ratio was 0.376. In pattern (b) in Fig. 1, we can observe a small peak of wtistite, and therefore this chemically determined value does not accurately indicate the value corresponding to the single phase of the spine1 structure (cation-excess). However, comparing the peak intensity of the wiistite phase with that in pattern (c), the peak intensity of wiistite in pattern (b) is fairly small. Therefore it is a reasonable approximation to say that the sample for pattern (b) is just in a transient state from a single phase cation-excess spine1 structure to a two-phase mixture consisting of wiistite and cation-excess spinel. Based on this approximation, the chemical composition of cationexcess spine1 (cation-excess magnetite) was estimated to be Fe~~,,Fe~‘,,O,_, with 6 = 0.1 (a,, = 0.8407 nm). In the second stage (patterns (c),(d) in Fig. l), wiistite is considered to change into a-Fe, both peak intensities for Fe0 and Fe,O, decreased during the
T. Togawa et al. I Solid State tonics 89 (1996) 279-286
283
cube
b)
octahedron
Fig. 3. TEM micrographs size of 390-440 nm.
of the powders
of (lOO)-magnetite
with a particle size of 430 nm (a) and of
increase in the a-Fe peak intensity (from pattern (b) to (c) in Fig. 1). As can be seen from curve A in Fig. 5, the lattice constant of the spine1 structure lowered with further H, reduction toward an overall reduction time of 6 h, where the wiistite phase is transformed into the a-Fe phase. When wtistite is transformed into cy-Fe (patterns (c) and (d) in Fig. l), the higher value of the lattice constant of the spine1 structure (cation-excess) lowered from 0.841 nm to 0.840 nm. This lowering of the lattice constant suggests a lowering in the concentration of cations (Fe2+) situated in interstitial B sites of the cationexcess spine1 structure. Fig. 2 shows the time variation of the X-ray diffraction pattern of (lOO)-magnetite with H, reduction at 300°C. As can be seen from the pattern after 3 h reduction, no wtistite peaks appeared but a-Fe peaks along with the strong peaks of the spine1 structure were present. This (Y-Fe peak intensity became slightly stronger after further H, reduction
(1I1 )-magnetite (b) with a particle
(Fig. 2). Since no peaks of wtistite were observed throughout H, reduction, it is suggested that the reduction of Fe2+ or Fe3+ ions in the spine1 structure results in the direct formation of U-Fe during the H, reduction of (lOO)-magnetite. As can be seen from curve B in Fig. 5, the lattice constant of the spine1 structure of (lOO)-magnetite was increased during the H, reduction at 300°C. However, the lattice constant did not become as large as that of stoichiometric magnetite (in Fig. 5 the lattice constant for stoichiometric magnetite is indicated by an arrow). Moreover, the lattice constant was nearly constant at 0.8400 nm during the H, reduction. These results indicate that the amount of Fe’+ ions formed from Fe3+ ions, which can move into interstitial B sites, is limited in the H, reduction of (lOO)-magnetite at 300°C. Keeping the lattice constant at 0.8400 nm, the peak intensity for a-Fe increased in the X-ray diffraction pattern during the H, reduction (pattern (c) in Fig. 2). This suggests that excess Fez+ ions
T. Togawa et al. I Solid State Ionics 89 (1996) 279-286
284
lattice constant of H, reduced (ill)-magnetite became significantly larger than that of stoichiometric magnetite, as can be seen from curves A and B in Fig. 5. This shows that in the H, reduction of (111) magnetite, the amount of Fe*+ which can move into interstitial B sites is larger than that of (lOO)-magnetite.
3.3. H2 reduction of Zn(II) ferrite
Fig. 4. Single crystal electron diffraction patterns of spots on photographic films of (lOO)-magnetite (a) and (1 1 1).magnetite (b).
0
2
4
6
Reduction time (h) Fig. 5. Variations in the lattice constants of (11 I)-magnetite (A) and (lOO)-magnetite (B) as a function of the H, reduction time at 300°C.
above a limited amount, which could have moved into interstitial B sites, must be directly transformed into a-Fe. Thus, in H, reduced (lOO)-magnetite, the amount of the Fe*’ ions which can move into interstitial B sites seems to be small. In contrast, the
Fig. 6 shows the time variation of the X-ray diffraction pattern of Zn(I1) ferrite upon H, reduction at 350°C. As can be seen here, the reduction of Zn(I1) ferrite results in the formation of wiistite as in the case of (ill)-magnetite. Zn(I1) ferrite is a normal spinel; the A sites are occupied by Zn*+ ions, and the B sites by Fe 3+ ions [14]. Therefore in H, reduced Zn(I1) ferrite, Fe3+ ions in B sites are reduced. The results shown in Fig. 6 suggest that the reduction of Fe3+ ions in B sites preferentially forms wiistite. This finding suggests that Fe*+ ions, which are formed from Fe3+ ions in B sites by H, reduction, preferentially move into interstitial B sites, the resulting anion array corresponding to the lattice points of the NaCl structure of wiistite. Fig. 7 shows the time variation in the lattice constant of Zn(I1) ferrite with H, reduction time. As can be seen here, the lattice constant of Zn(I1) ferrite rapidly increased with H, reduction. This increase in the lattice constant shows that Fe*+ ions formed from Fe3+ ions in B sites move into interstitial B sites of the spine1 structure of Zn(I1) ferrite, leading to an expansion of the cell volume of the spine1 structure. Since this increase in the lattice constant was observed before the appearance of the distinct peaks of wiktite in the X-ray diffraction pattern (pattern in Fig. 6), it is concluded that the movement of cations (Fe*‘) into interstitial B sites of the spine1 structure occurred before an appreciable crystal structure ranging around 2.0-10 nm was generated. The further reduction of Zn(I1) ferrite resulted in the decomposition of Zn(I1) ferrite into ZnO and Fe0 (wiistite), see pattern (d) in Fig. 6. Moreover, the wiktite peak intensity became very strong after further reduction. Thus, in the case of Zn(I1) ferrite, wiistite, which is formed by Fe*+ ions moved into interstitial B sites during H, reduction, is not so readily transformed into a-Fe; wiistite can exist as a
T. Togawa et al. I Solid State Ionics 89 (1996) 279-286
sites takes place. This would preferentially facilitate the transformation of cation-excess spine1 into NaCl structure material. If this is the reason for the prevention of the transformation of wtistite into LYFe, the easy transformation of wtistite in to (Y-Fe in the case of (11 I)-magnetite would be due to the reduction of Fe3+ ions on A sites. As described above, the transformation of wtistite into a-Fe during H, reduction of (11 I)-magnetite causes a lowering in the lattice constant of the cation-excess spinel. This suggests that excess cations in the spine1 move into the wiistite and/or the cu-Fe phase during the transformation process of wiistite into a-Fe. Since the reduction of Fe3+ ions on A sites in (11 l)-magnetite takes place along with the reduction of iron ions on B sites, some excess cations in the spine1 structure of (11 1)-magnetite would be generated from Fe3+ ions on A sites. The ease in the formation of wtistite into a-Fe in the H, reduction of (11 I)-magnetite would come from the additional generation of excess cations in the spine1 due to the reduction of Fe3+ ions on A sites.
(311) n 8 ZnFQ,
a)
WO) II
b)
.
-,-no
. .
zno u-Fe 0 WO)
(511)
I
60
285
3.4. Cation movement
80
mechanism
2e(Fe&) Fig. 6. Change in the powder XRD patterns during H, reduction of Zn(II) ferrite for 0 h (a), 2 h (b), 6 h (c) and 10 h (d), respectively. H, flow rate = 0.1 dm2 min-'
rather stable phase, compared with the case observed for (ill)-magnetite. Since the A sites are occupied with Zn2+ ions, only the reduction of Fe3+ ions on B
0 CJ
0.843 -
0
Redu&~~ time (h,8
Fig. 7. Variations in the lattice constant of Zn(II)-ferrite function of the H, reduction time at 300°C.
as a
Fig. 8 shows the schematic illustration of the cation movement mechanism in the H, reduction of the (11 l)- and ( lOO)-magnetite. On the surface of the (1 11)-magnetite, a layer of Fe3+ and Fe’+ ions on B sites is covered with an oxygen layer, as shown by Fig. 8(a). If the oxygen ions of the surface layer in Fig. 8(a) are removed by reaction with H, and Fe3+ and Fe2+ ions are reduced, the reduced iron ions are supported by the three oxygen ions existing beneath the layer consisting of iron ions on B sites. Even though these iron ions are surrounded by oxygen ions, this coordination state is very asymmetric. Therefore they will move into a more stable coordination, i.e. to octahedrally or tetrahedrally coordinated sites, i.e. into interstices. As shown in Fig. 8, there exists an interstitial B site beneath the 2nd layer of the oxygen ions for (ill)-magnetite. It is well known that electron hopping occurs among the iron ions on B sites. Therefore, during increasing the lattice constant of the (ill)-magnetite and Zn(I1) ferrite during H, reduction, the electrons of iron ions on B sites are considered to be delocalized among the iron ions on these sites by electron hopping. This
T. Togawa et al. I Solid State lonics 89 (1996) 279-286
286
b)
Fig. 8. Illustration
of reduction
reactions
occurring
on the crystals
means that the Fe2+ ions formed from Fe3+ ions on B sites are not so readily reduced to cr-Fe, since the electrons will be readily transmitted to the iron ions on B sites. This eventually results in the formation of the NaCl structure. Thus, the reduced iron ions on B sites are relatively resistant toward a further reduction to Fe’. These rather stable iron ions can have an amount of the transition time for the movement into interstitial B sites. Thus, in the case where the wiistite phase appears in the H, reduction, the Fe*+ ions formed from Fe3+ ions are considered to move preferentially into interstitial B sites of the spine1 structure. These considerations suggest that in the H, reduction of (1 11)-magnetite and Zn(I1) ferrite, the reduction of the Fe3+ ions on B sites results in the movement of reduced iron ions into interstitial B sites.
Acknowledgments The present work was partially supported by Grant-in-Aid for Science Research No. 03203216 from Ministry of Education, Science and Culture.
of
(11I)-magnetite
(a) and (lOO)-magnetite
(b).
References [II .I. Smith and J.I.J. Wijn, Ferrite (Philips Technical Library, Tokyo, 1965) p. 136. PI H. Flood and D.G. Hill, Ber. Bunsenges. Phys. Chem. 61 (1957) 18. [31 J. Brynestad and H. Flood, Ber. Bunsenges. Phys. Chem. 62 (1958) 953. [41 R. Dieckmann, Ber. Bunsenges. Phys. Chem. 86 (1982) 112. PI Y. Tamaura and M. Tabata, Nature 346 (1990) 255. [61 R. Dieckmann and H. Schmalzried, Ber. Bunsenges. Phys. Chem. 81 (1977) 414. r71 Y. Tamaura, Proc. Int. Symp. Chemical Fixation Carbon Dioxide, Nagoya, 1991, ed. K. Ito (Chemical Society of Japan; Research Group on Fixation of Carbon Dioxide, Nagoya, 1991) p. 167. WI Y. Tamaura, The 6th International Conference on Ferrites (Tokyo, Japan, 1992) p. 195. 191 L.S. Darken and R.W. Gurry, J. Am. Chem. Sot. 68 (1946) 798. [101 T. Kodama, M. Tabata, K. Tominaga and T. Yoshida, J. Mater. Sci. 28 (1993) 547. [Ill I. Iwasaki, T. Katsura, T. Ozawa, M. Yoshida, M. Mashima, H. Haramura and B. Iwasaki, Bull. Volcanol. Sot. Jpn. Ser. II 5 (1960) 9. 1121 JCPDS card 19-629 (Joint Committee on Powder Diffraction Standard, Swarthmore, 1989). [I31 E.R. Jette and F. Foote, J. Chem. Phys. 1 (1933) 29. [I41 E.J.W. Verwey and E.L. Heilmann, J. Chem. Phys. 15 (1947) 174.