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phys. stat. sol. (b) 163, 17 (1991)
Subject classification: 61.50; 61.60; 64.70; S1.61; S1.62; S10
Instituteof Chemistry, Academy ofsciences of'the USSR, Urn1 Scientific Centre, Soerdlovsk')
Disorder and Long-Range Order in Non-Stoichiometric Inter- stitial Compounds
Transition Metal Carbides, Nitrides, and Oxides
BY A. I. GUSEV
Contents
I . Introduction
2. Disordered non-stoichiornetric interstitial compounds
2.1 Disorder in non-stoichiometric compounds 2.2 Crystal structure and homogeneity regions 2.3 Special features of the electron structure and equilibrium structural states
3. Ordering in non-stoichiometric interstitial compounds
3.1 Superstructures of non-stoichiometric carbides 3.2 Superstructures of non-stoichiometric nitrides and oxides 3.3 Ordering in solid solutions of non-stoichiometric compounds
4. Conclusion
Rejerences
1. Introduction
All compounds that exhibit a high atomic defect concentration even in the absence of impurity atoms are normally said to be non-stoichiometric. For example, the relative deficiency of non-metallic atoms in a non-stoichiometric MeX,-. compound (with Me and X being metal and nonmetal atoms, respectively) may come from the presence of unoccupied sites in the non-metallic sublattice or, conversely, from a surplus of metal atoms occupying the interstitial sites of the metallic sublattice. Similarly, the relative surplus of non-metallic atoms in the MeX,+, crystal may be due to the location of excessive non-metal atoms in the interstitial crystal lattice sites or due to the presence of unoccupied metallic sublattice sites.
In the general case the equilibrium defect (vacancy or interstitial atom) concentration is an exponential function of temperature [l, 21; this implies absolutely defect-free crystals do
I ) Pervomaiskaya 91, SU-620219 Sverdlovsk, USSR.
2 physica (b) 163:l
18 A. I. G u s ~ v
not exist, whatever the temperature may be. In this context, one may assume that per se the presence of vacancies is no indication of non-stoichiometry at all. An essential sign of non-stoichiometry is a concentration of vacancies that ensures interaction among these. The vacancy or interstitial atom concentration of most of the binary compounds is small and does not exceed 0.1 at%. With this small concentration, the defects are remote from one another and do not interact; therefore they are viewed as non-interacting point particles. Clearly, the concept of non-stoichiometry is inapplicable to these compounds.
Meanwhile, compounds exist in which the defect concentration under normal conditions is well above 0.1 at%. The most known of these compounds is apparently wustite, FeO. This always contains a surplus of oxygen (at 1300 K wustite has the composition Fe,,8sO) due to the presence of vacancies in the irorl sublattice and does not exist in the stoichiometric state. Large departures from stoichiometry with vacancies forming in the metal sublattice are observed in iron and copper sulphides Fe,.,,S and Cu,,,,S. Appreciable departures from stoichiometry with oxygen deficit are typical of some higher transition-metal oxides: TiO,, V,O,, Nb,O,, MOO,, WO,. In these compounds, the oxygen deficit leads to a partial crystal lattice reconstruction that is due to the formation of shear structures; put another way, the observed departures from stoichiometry correspond actually to a homologous series of compounds, e.g., Tino,,- ,.
A special place among the non-stoichiometric compounds is occupied by refractory compounds of group IV and V transition metals with carbon, nitrogen, and oxygen. These compounds possess a simple uniform structure and extended regions of homogeneity. In the literature, compounds of this group are often called interstitial phases, interstitial compounds, interstitial alloys, and variable-composition compounds [3 to 61. The most accurate term to describe these is non-stoichiometric interstitial compounds.
The present paper aims at discussing features peculiar to the crystalline structure of the afore-mentioned compounds in the disordered and odered states. Owing to the high structural vacancy concentration, the non-stoichiometric interstitial compounds are unique materials for exploring atomic ordering. These compounds display a far greater variety of superstructures than any alloy. The properties of the non-stoichiometric compounds depend largely on their structural state (disordered or ordered).
2. Disordered Non-Stoichiometric Interstitial Compounds
The term “interstitial phase” dates back to Hagg [3], who used it in discussing the structure of transition-metal carbides, nitrides, hybrides, and borides. The term was applied in a restricted sense, with reference to only those substances in which the atoms of H, B, C, or N arrange themselves inside the simple metallic lattice. Indeed, a feature peculiar to the structure of the substances considered by Hagg is the presence of a face-centred cubic (f.c.c.) or hexagonal close packed (h.c.p.) metal lattice, while the non-metallic atoms are located at the centres of the octahedral interstitial sites or trigonal prismatic voids of the metallic lattice. At the same time, the metal lattice symmetry of carbides and nitrides differs from the crystal lattice symmetry of the corresponding transition metals; that is, as carbides and nitrides form, the crystal structure of the metal alters. Group IV transition metals (titanium, zirconium, hafnium) with h.c.p. structure form carbides and nitrides with an f.c.c. metallic sublattice. Transition metals with a body-centred cubic (b.c.c.) structure (vanadium, niobium, tantalum, chromium, molybdenum, tungsten) form carbides and nitrides that possess an
Disorder and Long-Range Order in Non-Stoichiomctric lntcrstitial Compounds 19
f.c.c. or h.c.p. metallic sublattice. The alteration of the crystal structure of the metal as a carbide or nitride is formed indicates the presence of sufficiently strong interactions among metallic and non-metallic atoms; therefore, applying the term “interstitial phase” to the above substances is not quite justified. The point is that only restricted interstitial solid solutions are truly interstitial phases.
In the past decades, the notion “interstitial phase” (“interstitial alloy”) has become widespread enough. Interstitial phases are currently understood as a wide range of phases with related structures that arise when atoms of hydrogen, nitrogen, oxygen, carbon, boron, and silicon intrude into the interstices of the crystal lattice formed by transition-metal atoms. The afore-mentioned group of compounds is also referred to as variable-composition compounds, thus emphasizing the fact that they possess wide homogeneity regions within which the composition of compounds may deviate appreciably from stoichiometry. As departure from stoichiometry, i.e. non-stoichiometry, is an intrinsic property of the above compounds, it is more correct to call them “non-stoichiometric interstitial compounds”. This name accentuates that the compounds under discussion (carbides, nitrides, etc.) are precisely compounds rather than interstitial solid solutions; in addition, such a name points out the most important structural features of the compounds, namely the fact that the non-metal atoms occupy the interstitial sites of the metallic sublattice and the composition may largely deviate from stoichiometry.
In the full sense of the word, only carbides, nitrides, and lower oxides of transition metals may be classified as non-stoichiometric interstitial compounds. The point is that stability and limiting hydrogen content of hybrides depend strongly on pressure and temperature, while the type of bond, with the hydrogen content being the same, may change as a function of external conditions. This implies that the distinction between transition-metal hydrides and solid solutions of hydrogen in transition metals becomes vague. As far as borides and especially silicides are concerned, these possess virtually no regions of homogeneity; besides, direct B-B and Si--Si bonds play a rather important role in the afore-mentioned compounds (direct interactions among non-metal atoms are negligibly small in carbides, nitrides, and oxides). Higher oxides are also practically devoid of homogeneity regions. Thus the present paper will treat transition-metal carbides, nitrides, and lower oxides as nm-stoichiometric interstitial compounds.
The structure and properties of non-stoichiometric interstial compounds have been treated in hundreds of original papers, as well as in numerous reviews and monographs, including [l to 91. This abundance of published work enables us to restrict our attention to crystal structure peculiarities most typical for the compounds concerned.
A number of empirical rules for constructing the crystal structures of the compounds under review have been formulated by Hagg [3]. Currently, exceptions from these rules are known; however, the major underlying principles remain valid. According to these rules, non-stoichiometric carbides, nitrides, and oxides form provided the ratio of the atomic radii of the non-metal to those of the metal satisfies the condition 0.41 < R,/R,, < 0.59. Subject to this condition, the non-metal atoms reside in the largest metal lattice interstices; significantly, the interstice should be somewhat smaller than the non-metallic atom that intrudes into it. In consequence, the metallic lattice expands somewhat and its symmetry changes when a carbide, nitride, or oxide is formed. These factors ensure the stability of the arising structure. If the ratio Rx/R,, exceeds 0.59, compounds form which have a more complicated structure and normally possess no homogeneity region. Table 1 summarizes values of the quantity R,IR,, for carbon, nitrogen, oxygen, and transition metals of groups
A. I. Gustv
Table 1 Atomic radii of transition metals, carbon, nitrogen, and oxygen (for coordination number 12) and ratios of atomic radii R,/R,, (with Me metal, X = C, N, 0)
Ti Zr Hf V Nb Ta Cr Mo C N 0
0.1467 0.1597 0.1585 0.1338 0.1456 0.1457 0.1267 0.1394 0.0772 0.0740 0.0660
0.526 0.483 0.486 0.576 0.530 0.529 0.609 0.553
0.504 0.463 0.467 0.553 0.508 0.508 0.584 0.531
0.450 0.4 14 0.417 0.493 0.453 0.453 0.521 0.473
IV to VI. For most of the transition metals the ratio Rx/R,, is less than 0.59, so they form compounds with wide homogeneity regions. For chromium RJR,, = 0.609 and the chromium compounds Cr,,C,, Cr,C,, and Cr,C, have complicated structures and possess no homogeneity regions.
A distinctive feature of the non-stoichiometric interstitial compounds is the quasi- independent existence of the pure metal sublattice that serves as a matrix for all kinds of atoms intruding into its interstitial voids and forming the non-metallic sublattice. Unfilled interstices are viewed not merely as crystal lattice “holes”, but as “holes” that are equivalent to interstitial atoms. With the non-metallic sublattice being filled in part, vacant interstitial voids (structural vacancies) can diffuse through the lattice and behave as real interstitial atoms. Therefore, structural vacancies in non-stoichiometric interstitial compounds may, as a first approximation, be viewed as some analogues of interstitial atoms [l, 2, 4, 9, 101. Indeed, the departure from stoichiometry may well be considered in terms of the theory of substitutional solutions whose constituents are interstitial atoms and structural vacancies.
The defect content of the crystal structure, i.e., the presence of a high structural vacancy concentration, is one of the most important properties of non-stoichiometric interstitial compounds. Structural imperfection has a very considerable effect on the properties of non-stoichiometric compounds. Just for this reason most of the investigations carried out in the seventies and eighties are devoted to determining the dependence of specific properties of non-stoichiometric compounds on the relative metal, non-metal, and structural vacancy content of such systems. At the same time, recent studies show that in discussing the structure and properties of non-stoichiometric compounds, one needs to allow for not only the quantitative ratio between atoms and vacancies, but also the character of their distribution in the crystal lattice. This problem is addressed in [9], which for the first time generalizes experimental and theoretical data about the effect of atomic and vacancy distributions on the structure and properties of non-stoichiometric interstitial compounds.
2.1 Disorder in non-stoichiometric compounds
Crystalline solids exhibit the highest degree of spatial order. In physical terms, an ideal crystal is an ensemble of an infinitely large number of atoms (or molecules) that fill some volume and are uniformly distributed in regular chains and planes. In the general case, the
Disorder and Long-Range Order in Non-Stoichiometric Interstitial Compounds 21
presence of disorder always breaks the regularity of the crystal. However, in some cases the crystal lattice symmetry is preserved even in the presence of disorder. This applies, for example, to substitutional solid solutions. For such solutions one cannot specify the sort of the atom that is located in a given site v ; however, it is possible to specify the probability of a given site being filled with an atom of a particular species.
The same is true of non-stoichiometric interstitial compounds in which the substitutional elements are non-metallic atoms and structural vacancies.
In the non-stoichiometric compounds under discussion, a situation is realized where the number of positions in the lattice is greater than the number of atoms occupying these positions; under certain conditions the distribution of interstitial atoms in the positions available will be random, at least in part. For this reason, disordered non-stoichiometric compounds do not possess the translational symmetry of a crystal of stoichiometric composition. However, the random distribution of interstitial atoms in non-metallic sublattice sites signifies that the probabilities of these sites being filled are equal; as a consequence, all non-metallic sublattice sites are crystallographically equivalent. In other words, disordered non-stoichiometric compounds have the translational symmetry of some lattice of site occupation probabilities.
Most of the non-stoichiometric compounds possessing wide homogeneity regions have a high-symmetry B1 (NaC1)-type structure that is preserved when interstitial atoms and vacancies are randomly distributed. The random distribution results in the non-preservation of the symmetry of the local environment of each individual atom. In turn, the distortion of the local symmetry gives rise to static displacements of both metallic and interstitial atoms.
The cubic symmetry of disordered non-stoichiometric compounds with B1 (NaC1) structure allows only a spherically symmetric distribution of static atomic displacements, for otherwise the crystallographic equivalence of different crystal lattice sites would be perturbed. In disordered non-stoichiometric compounds a symmetric distribution of static displacements is ensured by the random arrangement of vacancies. Currently, static displacements have been determined for most of the transition-metal carbides and some transition-metal nitrides [11 to 171; the results of these studies are generalized in [9,18,19].
For non-stoichiometric compounds containing a considerable number of structural vacancies, the state of thermodynamic equilibrium at and below 300 K is the ordered state; attaining this state normally requires long-term annealing or other special types of heat treatment. Conversely, the disordered state of non-stoichiometric compounds is realized readily enough by quenching, which takes place when compounds are subjected to normal cooling subsequent to high-temperature synthesis. For this reason, an overwhelming majority of experimental papers deals with disordered non-stoichiometric interstitial com- pounds.
The ease with which one can produce disordered non-stoichiometric compounds has provided a basis for the widespread erroneous idea that a disordered (statistical) distribution of interstitial atoms and vacancies is probably the only possible structural state of these compounds, while an ordered distribution of atoms and vacancies is something that is rare enough, exotic. In reality, however, the reverse is true: For non-stoichiometric compounds the disordered state is a state of thermodynamic equilibrium only at temperatures above 1000 to 1500 K, whereas at lower temperatures (below the disorder-order transition temperature) the state of thermodynamic equilibrium is the ordered state; the disordered state is metastable in this temperature range. In this context, the question arises as to how much reliance can be placed upon the numerous theoretical electron energy spectrum
22 A. I . Gust L
calculations that have been made without allowance for thermal excitation (i.e., for temperature 0 K) although only the ordered state of non-stoichiometric compounds is in equilibrium at 0 K.
2.2 Crystal structure and homogeneity regions
Data on the crystal structure and homogeneity regions of disordered non-stoichiometric carbides, nitrides, and oxides are summarized in Table 2.
Most of the non-stoichiometric carbides and nitrides contain vacancies only in the non-metallic sublattice. The presence of structural vacancies also in the metallic sublattice is typical for cubic monoxides of titanium and vanadium, as well as for oxycarbides and oxynitrides of these metals. A defective metallic sublattice is characteristic of nitrides that form in thin films and ultradisperse powders.
Group IV and V transition metals form highest-melting carbides with f.c.c. or h.c.p. metal sublattice. Carbides of group VI transition metals are less stable; molybdenum and tungsten carbides still preserve simple structures, while chromium carbides already have complicated structures with large unit cells. Since chromium carbides display no appreciable departure from stoichiometry, we do not enlarge on these.
Group 1V metals (titanium, zirconium, hafnium) form only TIC,, ZrC,, and HfC, monocarbides with B1 (NaC1) structure; the composition of these monocarbides may deviate largely from the stoichiometric composition MeC,,,,.
In addition to the cubic carbides VC,, NbC,, and TaC,, group V transition metals at high temperatures form disordered lower carbides V,C, Nb,C, and Ta,C with h.c.p. structure of the W,C (L3) type. The homogeneity regions of the monocarbides of group V metals are less extended than the counterparts for the monocarbides of group IV metals, but they are still sufficiently wide. By contrast with other carbides of group IV and V transition metals, the upper bound of the homogeneity region for cubic vanadium carbides VC, under normal conditions is a carbide with a high structural vacancy concentration in the carbon sublattice, namely VC,
For monocarbides of group IV and V transition metals the position of the lower homogeneity region boundaries is individual. Using percolation theory, the author of [20] has made for the first time a theoretical calculation of the lower structural stability boundary of non-stoichiometric carbides.
The carbides of group VI metals, except MoC, and W,C, are practically devoid of homogeneity regions. Data on the structure of molybdenum carbides are highly con- tradictory. For the carbide MoC,, e.g., five structures are known currently, four of these being hexagonal. The specific modifications that arise depend largely on the way the carbide is produced and on the mode of heat treatment. The unstable character of the structure of MoC, is confirmed by the ease with which its structure can be altered by introducing certain alloying additions. For example, small additions of NbC to MoC, stabilize the cubic NaCl structure. while additions of WC stabilize the simple hexagonal WC structure. The high-temperature modification P-Mo,C has a disordered W,C (L’3)-type structure.
Transition-metal nitrides are in mcii iy ways similar to carbides. They are close to carbides in structure and properties. Transition-metal nitrides and carbides have common formation regularities. Titanium, zirconium, and hafnium nitrides (MeN,), just as the corresponding carbides, have a B1 (NaC1)-type structure. Group V transition metals form S-phases -
Disorder and Long-Range Order in Non-Stoichiometric Interstitial Compounds 23
T a b l e 2 Crystal structure and homogeneity regions of disordered non-stoichiometric interstitial compounds
stoichio- homogeneity metric region compo- sition
crystal structure ~
type of compo- lattice period structure sition
a (nm) c (nm)
1 2 3 4 5 6 7
TIC
ZrC
HfC
vc
v2c
NbC
Nb,C TaC
Ta,C
Y-MoC") wc W2Cb) Ti N
ZrN
HfN
VN
V2N 6-NbN')
NbzN TaN ') Ta,N CrN
Cr, N
TiC0.48 -Tic l ,OO
zrc0.60-zrc0.9~
HfC0.56 - HfCl .OO
vc0.65-vc0 87
vc0.42 -Vc0.50
NbC0.70-NbCl.00
NbC0,35 -NbC0,50
TaC0.7, - T G 00
-TaC0.50
M°C0.64 -MoC0.98
WCO.98 - wc1.02
TiN0.38 -TiN1.00 wc0.36- wc0,52
ZrN0.5 5 - ZrN 1 .OO
Hm0.74-HfN1.00
VN0.7, - V N l . O O
VN0.48-vN0.50
NbNo,,8-"bN0.98
NbNo.4,-NbN0.50 TaN0.9, -TaN,.oo ~ ~ N o . , o - ~ ~ N o . 5 o narrow; the boun- daries are unknown
CrN,.,, -CrNo.50
Fm3m
Fm3m
Fm3m
Fm3m
P6,/mmc Fm3m
P6,/mmc Fm3m
P6,/mmc
~ 6 m 2 P6m2 P6,/mmc Fm3m
Fm3m
Fm3m
Fm3m
Fm3m
P6,/mmc P6/mmm P6,/mmc Fm3m
NaCl
NaCl
NaCl
NaCl
w2c NaCl
w2c
NaCl
w2c
wc wc w2c
NaCl
NaCl
NaCl
NaCl
h.c.p. NaCl
w2c
w2c CoSn
NaCl
h.c.p.
TiC,,,, 0.4296 TiC,,,, 0.4328 ZrC,,,, 0.4694 ZrC,.,, 0.4698 HfC,,,, 0.4632 HfC,,,, 0.4639 VC,,,, 0.4126 VC,,,, 0.4153 VC,,,, 0.2902 NbC,,,, 0.4429 NbC,,,, 0.4469 NbC,,,, 0.3127 Taco,, , 0.4412 Taco,,, 0.4456 TaC,,,, 0.3101 Taco,,, 0.3106 MoC,,,, 0.2898 WC,,,, 0.2906 WC,,,, 0.2992 TiN,,,, 0.4210 TiN,,,, 0.4244
ZrN,,,, 0.4565 ZrN,,,, 0.4566 HfN,,,, 0.4518 HfN, ,,, 0.4524 VN,,,, 0.4060 VN,,,, 0.4134 VN,,,, 0.2839 NbN,,,, 0.4383 NbN,,,, 0.4391 NbN,,,, 0.3056 TaN,,,, 0.5191 TaN,,,, 0.3048 CrN 0.4149
CrN,,,, 0.4759
0.4577
0.4972
0.4933 0.4945 0.2809 0.2837 0.4722
0.4560
0.4957 0.29 10 0.49 1 5
0.4438
24 A. I. GUSEV
T a b l e 2 (continued)
stoichio- homogeneity metric region compo- sition
crystal structure
space type of compo- lattice period group structure sition
a (nm) c (nm)
1 2 3 4 5 6 7
MoN narrow; the boun- hexa- MoN 0.5725 0.5608 daries are unknown gonal
Mo2N MoNo.,z-MoNo.,o Fm3m NaCl MoN,,,, 0.4165 WN narrow; the boun- P6m2 wc WN 0.2893 0.2826
W2N narrow; the boun- Fm3m NaCl WN,,,, 0.4126 daries are unknown
daries are unknown T i 0
vo v00.8S-v01.23
NbO narrow
Fm3m NaCl TiO,,,, 0.4192 TiO,,,, 0.4169
Fm3m NaCl VO,,,, 0.4024 VO,,,, 0.4130
Fm3m NaCl NbO 0.440 TaO narrow Fm3m NaCl TaO 0.439
") Molybdenum carbide is assumed to have the composition Mo,C,. There exist several modifications of MoC, with different sequence of packing of metal atom layers.
') A cubic modification of the carbide W,C exists at a temperature above 2800 K. ') Several related modifications of NbN exist, including hexagonal ones. ') Series of polytypes with composition-dependent structure.
mononitrides - with f.c.c. metallic sublattice (except for TaN, which under normal conditions has a hexagonal structure of the CoSn type), as well as &-phases - stoichiometric MeN,.,, nitrides - with h.c.p. structure and lower Me,N nitrides with h.c.p. structure. Note that the cubic phase TaN with B1 structure can be observed in thin films [8, 91, but this phase is unstable in bulk specimens.
The nitrides of group VI transition metals are rather unstable and decompose at temperatures that are not high. The nitrides CrN, Mo,N, and W2N have B1 structure, and MoN, WN, and Cr,N a hexagonal structure. The nitrides of group VI metals have very narrow homogeneity regions, the exact boundaries of which are not known.
A typical representative of the non-stoichiometric monoxides is TiO, which has B1 (NaC1) structure. Large (up to 15 at%) departures from stoichiometry to form structural vacancies in both the oxygen and metal sublattices are possible in this compound. The question as to whether zirconium and hafnium monoxides can exist has not yet been resolved finally. The vanadium monoxide VO, just as TiO, has a B1 structure and contains a large number of vacancies in both sublattices. The cubic monoxides NbO and TaO have narrow homogeneity regions and form when thin metallic films oxidize. The presence of these monoxides in bulk specimens is found only in the presence of higher oxides.
Thus, disordered non-stoichiometric carbides, nitrides, and oxides have mostly cubic or hexagonal structure. These structures may be visualized as successively alternating atomic layers. For example, in the non-stoichiometric B1 (NaC1) structure compounds under review,
Disorder and Long-Range Order in Non-Stoichiometric Interstitial Compounds 25
9 Fig. 1. B1 (NaC1)-type structure and the nearest environment of a metal atom (0) (the non-metallic sublattice sites (0 , are indicated by numerals): 1 to 6 first coordination sphere; 7 to 14 third coordination sphere; the second coordination sphere is formed by twelve metal atoms
planes alternate perpendicular to the [l 1 l],, direction (or perpendicular to the equivalent [lil], , , [ill],,, [Till,, directions) which are formed by only metallic atoms and only non-metallic ones.
In the B1 (NaC1) structure (Fig. l), the non-metal atoms statistically fill the octahedral interstitial sites of the metallic sublattice, each metal atom being octahedrally surrounded by six non-metallic sublattice sites; the second coordination sphere is formed by twelve metal atoms, the third by eight non-metallic sublattice sites. In turn, every non-metallic sublattice site is surrounded by six metallic atoms that form the first coordination sphere of the metal; the second coordination sphere is formed by twelve non-metallic sublattice sites, etc. For the B1 structure, Table 3 presents the distribution of the sites of both sublattices in the coordination spheres at the centre of which a nonmetallic sublattice site is located. If we need to consider a sequence of coordination spheres constructed around a metallic sublattice site, it suffices to replace in Table 3 Me by X and to replace X by Me (this is possible because the B1 (NaCl) structure is invariant under the replacement of one species of atoms by the other).
Fig. 2 Fig. 3
Fig. 2. Octahedral representation of the B 1 (NaC1)-type structure
Fig. 3. W,C (L’3)-type structure with an h.c.p. metallic sublattice: o metal atoms; o statistically half-filled interstitial positions
26 A. I. GUSEV
T a b l e 3 Rl(NaC1) structure: Distribution in the coordination spheres of metallic (Me) and non- metallic (X) sublattices sites
coordination coordinates ”) of a site sublattice number ofthe coordination relative sphere”) that is part of the whose sites coordination number radius of
coordination sphere form the j-th sphereformed (number of the j-th coordination by the sub- sites in the coordination sphere lattice sites j-th coordina- sphere‘)
tion sphere)
.I -
0 1 2
3 4
5
6 I
8
9 10 11 12 13 14
(hk0
000 100 110 111 200
210 21 1 220
~
{ 3 10 31 1
222 320
321 400
0 0 0 X
112 0 0 Me 112 112 0 X 112 112 112 Me 1 0 0 X
1 1/2 0 Me
1 112 112 X 1 1 0 X
312 112 0 X 312 112 112 Me 1 1 1 X
312 1 0 Me 312 1 112 X
2 0 0 X
0 - - 1 1
- 2 2 - - 3
3 -
4 -
- 4
5 - - 5 6 - - 6 7 -
8 -
-
1 6
12
8 6
24
24 12
30
24 24
8
24 48
6
”) The centre of the coordination spheres is a non-metallic sublattice site. b, The coordinates of the rest of the coordination sphere sites can be obtained with the aid of symmetry operations of point group m3m (Oh), ’) The relative radius of the j-th coordination sphere is Rj/a = where a is the unit cell period.
Non-stoichiometric carbides, nitrides, and oxides with B1 structure pertain to a class known as octahedral structures, i.e., structures in which the transition-metal atom is octahedrally surrounded by non-metal atoms or by non-metal atoms and vacancies. Such structures can be pictured as different versions of the articulation of MeX, octahedra; for example, edges join the octahedra in the B1 (NaCl) structure (Fig. 2).
Typically, lower carbides and nitrides possess a hexagonal close-packed structure of the W,C (L’3) type (Fig. 3). In this structure, the non-metallic atoms statistically fill half of all the octahedral interstices of the metallic sublattice. In the WC structure metal atoms form a simple hexagonal lattice in whose trigonal prismatic voids non-metallic atoms reside.
Disorder and Long-Range Order in Non-Stoichiometric Interstitial Compounds 27
2.3 Special features of the electron structure and equilibrium structural states
A distinctive feature of high-melting non-stoichiometric interstitial compounds is the combination of the major properties inherent in metals (simple structure, high thermal and electrical conductivity decreasing with temperature) and covalent compounds (high hard- ness, low plasticity). Another important feature of non-stoichiometric carbides, nitrides, and oxides is the presence of wide homogeneity regions within which these compounds preserve the type of crystal structure. Departure from stoichiometry leads to alteration of all the properties of the compounds under review [2, 4 to 91.
Currently, concepts of the electron structure and chemical bond in non-stoichiometric interstitial compounds have, on the whole, taken shape. According to these concepts [6], the valence band of non-stoichiometric MeX, compounds with B 1 structure includes a low-energy 2s(X) band that contains small contributions of the s-, p-, and d-states of the metal, a funda- mental valence bonding band formed by the strong mixing of 2p(X)-d(Me) functions, a partly filled high-energy conduction band formed preferentially by nd(Me) functions with an admix- ture of 2p(X), (n + 1) p(Me), and ( n + 1) s(Me) functions. In going from carbides to nitrides and oxides, the low-energy 2s(X), fundamental valence bonding 2p(X)-d(Me), and high- energy d, s(Me) bands narrow and shift towards smaller energies. As non-stoichiometric compounds form, a redistribution of individual atomic states comes about which leads to partial charge transport between metallic and non-metallic atoms and thus is responsible for the ionic component of the chemical bond. Results of most of the cluster and band calcula- tions as well as X-ray emission and photoelectron spectroscopy data indicate that the charge is transported from the metal to the non-metal. In the carbide-nitride-oxide series, the charge increases somewhat in magnitude (e.g., for TIC, TIN, and T i 0 the charge on the titanium atom, in units of the electron charge, is + 0.50, + 0.64, and + 0.78), thereby enhancing the ionic component of the chemical bond. Thus a combined covalent-metallic-ionic type of chemical bond is realized in non-stoichiometric interstitial compounds. Vacancy formation in the non-metallic sublattice of the compounds considered gives rise to an extra density-of-electron-states “vacancy” peak in the conduction band below the Fermi level.
Despite the considerable progress that has been made recently in the study of the electronic structure in disordered non-stoichiometric interstitial compounds, this problem has not yet been fully resolved. In this connection, the following must be noted. All theoretical calculations of the electronic structure of non-stoichiometric compounds have been carried out for the ground state (i.e., for the state at OK). However, the disordered state of non-stoichiometric compounds is stable only at high temperatures (above 800 to 1300 K), whereas at low temperatures ordered non-stoichiometric compounds are in thermodynamic equilibrium. Nevertheless, theoretical energy band calculations that allow for ordering in non-stoichiometric compounds are sufficient so far. Of these, we may mention electron structure calculations for ordered defective metallic-sublattice zirconium nitride Zr,N, (Zro.,5N) [21] and ordered titanium carbide Ti,C3 and titanium nitride Ti,N, (TiN0,,J [22, 231. Note that such superstructures do not actually exist. Apart from this, an attempt has been made [24, 251 to calculate the ordering energy of carbides and nitrides proceeding from the electron structure of these compounds in the disordered state; the results obtained for the energies of pair interactions in the non-metallic sublattice have been used to predict the type of ordering liable to arise.
As follows from the foregoing, in calculating the ground state of the electronic subsystem as applied to non-stoichiometric compounds, one needs to consider the equilibrium, i.e.,
A. 1. GUSEV 28
ordered, state of these compounds. Insofar as disordered non-stoichiometric compounds exist, their electronic structure should be calculated with allowance for thermal excitation, i.e., for T > 0 K.
The ultimate goal of quantum chemical energy band calculations of solids (including non-stoichiometric compounds) is to provide a theoretical explanation of the various macroscopic properties on the basis of atomic and electronic concepts. In the general case the solution of this problem is obtained in two stages. The first stage is to determine the electron energy spectrum in terms of the adiabatic approximation, where the nuclei (ionic cores) are assumed to be fixed in the compound considered. In the second stage, in treating equilibrium properties, one has to determine the partition function for all possible positions of the nuclei and to find the thermodynamic potential of the crystal as a function of the corresponding thermodynamic variables, The various techniques of the contemporary quantum chemistry of non-stoichiometric carbides, nitrides, and oxides solve only the first stage of the problem, a fact that makes the findings restricted and incomplete.
It is worth while to amplify on the applicability of the adiabatic approximation to non-stoichiometric interstitial compounds. The major adiabaticity criterion is the absence in the electron spectrum of thermal excitations that have an energy of the order of the nuclear vibration energy hw (w = (vnUcl)/AR is the nuclear vibration frequency, (vnuc,) the mean nuclear vibration velocity, and AR the displacement of nuclei)
hw = h(vnuCl)/AR < AE,.
This criterion fails with substances with metallic conductivity (to which non- stoichiometric carbides and nitrides belong), because near the Fermi surface elec- tronic transitions with an arbitrarily small excitation energy are possible and the energy spectrum has no gap, i.e., AEe = 0 (with AEe being the excitation energy or the energy gap between the energy of the outer-shell (valence) electrons in the ground state and the energy of the first excited level). This means that in describing non-stoichiometric compounds it is necessary, for obtaining the electron energy spectrum of a static lattice, to take into account the electron-phonon interaction that leads to renormalization of the Fermi surface electrons. Existing theoretical, quantum chemical, calculations of the electron structure of non-stoichiometric interstitial compounds disregard this cir- cumstance.
Thus, in describing non-stoichiometric compounds using methods of quantum chemistry and solving the problem posed, the Hamiltonian of the system should involve not only the kinetic energy of electrons and the potential energies of interelectron and electron-nucleus (core) interactions, but also the kinetic energy of nuclei (cores) and the potential energy of nuclear interactions. In the description of disordered non-stoichiometric compounds, special allowance must be made also for the thermal excitation of the system.
The considerations with regard to the features to be taken into account in the description of disordered non-stoichiometric compounds apply in full measure to all systems with substitutional disorder, i.e., to alloys and solid solutions with a disordered distribution of the constituents on crystal lattice sites. Such states normally are thermodynamically metastable (although capable of having large lifetimes), and the problems of describing these have not yet been worked out in sufficient detail.
Disorder and Long-Range Order in Non-Stoichiometric Interstitial Compounds 29
3. Ordering in Non-Stoichiometric Interstitial Compounds
Atomic ordering is a synonym to a structural disorder-order phase transition (in the case of magnetic phase transitions one speaks of magnetic ordering). Ordering as a phase transition is the result of a redistribution of atoms in the sites of the crystal lattice of a substitutional solid solution. In the disordered state, the reciprocally substituting consti- tuents of the solid solution are randomly distributed in the sites of some crystal lattice; the probability of any lattice site to be filled with an atom of a given sort coincides with the concentration of atoms of that sort in the solution. As the temperature is lowered a disorder-order phase transition comes about, as a result of which the sites of the crystal lattice of the disordered solid solution break down into several non-equivalent sublattices. The sublattices of an ordered solid solution differ from one another in the probabilities of their sites being filled with atoms of a given species.
Ordering is possible not only in substitutional solid solutions. This phenomenon may occur also in interstitial solid solutions, if the number of interstitial positions exceeds the number of atoms occupying these sites. When interstitial solutions suffer ordering, unfilled positions and interstitial atoms form a substitutional solution.
As is clear from the foregoing, the presence of structural vacancies in non- stoichiometric interstitial compounds under certain conditions can give rise to ordering. When describing the ordering in non-stoichiometric compounds, interstitial atoms and structural vacancies are viewed as the components of a binary sub- stitutional solution that is formed in the non-metallic sublattice. In the simplest case, ordering in the non-metallic sublattice reduces to its breaking down into two new sublattices. All sites of the first superstructure sublattice will be filled with interstitial atoms, while the sites of the second sublattice will be vacant. If ordering takes place in substitutional solid solutions formed by non-stoichiometric compounds (e.g., MeC,N, - y or MeC,O, - y , etc.), the sites of one of the superstructure sublattices will be filled preferentially with non-metallic atoms of the first sort, while the sites of the other will be filled preferentially with atoms of the second sort.
Ordering in the non-metallic sublattice of non-stoichiometric compounds is accompanied by a lowering of the crystal symmetry, because part of the symmetry transformations of the disordered non-metallic sublattice that bring the filled and unfilled sites into coincidence will not be involved in the symmetry elements of the ordered crystal, as these sites become crystallographically non-equivalent. Therefore, the ordered state is an equilibrium state for non-stoichiometric compounds at low temperatures.
For a long time predominantly different X-ray structure analysis methods were used to determine the position of interstitial atoms in the lattice of non-stoichiometric compounds. However, small amplitudes of X-ray scattering by interstitial atoms as compared to those of metallic atoms often do not permit to determine the position of non-metallic atoms from an analysis of the diffraction reflection intensity. A consequence of this was the widespread opinion that the interstitial atoms in the non-metallic sublattice of non-stoichiometric compounds are always (whatever the conditions may be) distributed statistically, in a random fashion.
Novel experimental techniques, especially the use of neutron diffraction to investigate the crystal structure of non-stoichiometric compounds, have revealed that under certain conditions an ordered distribution of interstitial atoms occurs in these compounds. Structural analysis by neutron diffraction has become one of the most informative tools in probing
A. I. GUSEV 30
non-stoichiometric interstitial compounds, because the intensity of neutron scattering on the nuclei of light elements is comparable with the intensity of scattering by the nuclei of metallic atoms. It has been possible to record the ordering of interstitial atoms in these compounds by invoking other diffraction techniques (electron microdiffraction, scanning radiography) in studying phase diagrams by differential thermal analysis and other physicochemical methods.
3. I Superstructures of non-stoichiometvic carbides
The ordering of interstitial atoms in non-stoichiometric compounds is widespread and has currently been found to occur in most of the compounds being discussed. At the same time, available data on the structure of the ordered phases of non-stoichiometric compounds are far from being complete and are at variance in a number of cases.
On the basis of crystallographic considerations it has been shown [26] that ordered carbide structures are liable to form, which correspond to the compositions Me,C, Me,C,, Me,C5, Me,C,, Me&,, Me ,C4, Me6C,, and Me&,.
Data on the crystal structure of the ordered phases that form in the Me-C (Me = Ti, Zr, Hf, V, Nb, Ta) systems are summarized in Table 4.
According to [27,28], in titanium carbide TiC~~,50-TiCo,71 cooled slowly from temperature 1400 K an ordered cubic phase arises with doubled (compared to the disordered carbide) lattice period. Ordering goes via the transition channel comprising the wave vector ki l ) = {i, 4, $1, which satisfies the Landau criterion for second-order phase transitions. However, a study of the kinetics of the ordering of TIC,,, carbide yielding a Ti,C superstructure [29] has shown this ordering to be a first-order transition; the disorder-order transition temperature is equal to = 1030 K and the order-disorder transition temperature is 1080 K. The authors of [30] conclude that the ordered phase of titanium carbide has trigonal symmetry and corresponds to the composition Ti,C.
According to neutron diffraction investigations [27, 281, the ordered phase of non- stoichiometric zirconium carbide has a cubic unit cell with doubled (compared to the disordered carbide) lattice period. However, [31] states that allowing for the distortions present, the lattice of an ordered zirconium carbide has trigonal symmetry. In studying ZrC,.,, carbide, the authors of [32] have detected no formation of an ordered phase; according to the same data, an ordered Zr,C, (ZrC,,, ,) phase with doubled lattice period forms in a ZrC,,,, carbide.
According to [31], all carbides of group IV transition metals typically exhibit the trigonal symmetry of ordered Me,C,-type phases. The conclusion of [31] on the formation of ordered Me,C,-type phases is questionable.
Studies of the ordering in a defective vanadium monocarbide, VC,, have revealed the existence of a trigonal high-temperature v6c5 modification [33] and a monoclinic low- temperature [34, 351 V,C,-type modification, as well as a cubic V,V, modification with doubled (compared to the disordered carbide) lattice period [31, 361.
Electron microdiffraction [34, 391 and neutron diffraction [15, 40 to 451 studies of the ordering in a non-stoichiometric niobium monocarbide, NbC,, have shown that annealing at temperatures below 1300 K leads to an ordered Nb,C, phase forming over a wide range of compositions near NbC,.,,. To describe the superstructure reflections observed, a trigonal structure similar to v&5 [33] has been proposed [34, 39, 441. The ordering in NbC, has received a very detailed treatment in [15, 40 to 431.
Tab
le 4
C
hara
cter
istic
s of
ord
ered
pha
ses
in M
e-C
sy
stem
s (M
e=T
i, Z
r, H
E, V,
Nb,
Ta,
W)
orde
red
phas
e di
sord
ered
re
f. re
mar
ks
basi
s ph
ase
__
__
~ ~
-
com
po-
non-
met
al c
onte
nt
crys
tal
stru
ctur
e co
mpo
- la
ttice
si
tion
type
si
tion
(at%
) or
regi
on
-
of e
xist
ence
of
the
latti
ce
spac
e un
it ce
ll pa
ram
eter
s*)
orde
red
phas
e ty
pe
grou
p (n
m)
P, I
Q
r
op F o=: 0 0 $
Tiz
C
cubi
c Fd
3m
a =
2a,
Ti
C
b.c.
c.. B
I [2
7 to
291
tr
ansi
tion
tem
- pe
ratu
re e
qual
to
950t
o 10
30 K
. T
hc c
arbo
n su
b-
latti
ce is
ord
ered
in
par
t st
able
bel
ow
1400
K
stab
le b
elow
I4
00 K
Tr
ans rr
1170
K
Tran
s rr 13
40 K
. Is
omor
phic
to
an
orde
red
ThC
,,,,,
phas
e an
neal
ing
tem
pe-
ratu
re 9
00 to
11
00 K
quen
chin
g tem
pe-
ratu
re 1
800
K
P3,2
1 a
= 0
.306
P
1/Za
p/2,
c
= 1
.491
= 21
/3a,
u
= 0
.663
0 z $
an,
~i =
2a,
a
= 2
a,
c =
1.6
26 =
2 fi
ao
Tiz
C
37.0
- 43
.0
trig
onal
T
ic
b.c.
c., B
1
39.0
-43.
0 tr
igon
al
ZrC
b.
c.c.
. B
I
cubi
c cu
bic?
Z
rC
ZrC
b.
c.c.
, B1
b.c.
c.. B
1
rhom
bic,
tY
Pe
c-Fe
2N
hexa
gona
l
Pbcn
a
= 0
.456
7 =
c,,,
h =
0.5
744
= 2
a,,
c =
0.5
026 z V
3an
u =
0.4
997 z p
a,,,
r
c =
0.4
568
= c
,,
u =
0.5
09, c
= 1
.44
P3,,
P3,
C2
a =
0.5
09, b
= 1
.018
, c
= 0
.882
, 7 =
109
’47’
a
= 0
.588
5 z f
iu0
, c
= 1.
443
5 2
l/Z
ao
h.c.
p., L
‘3
~3
3.0
(V
C,,,
) V
2C
h.c.
p., L
’3
z45.
5 (v
co.x
3)
245.
5 (v
C,,,
3)
trig
onal
m
onoc
linic
vc
, vc
, b.
c.c.
, BI
b.c.
c., B
1 7;
,,,,
5 14
20 K
&V
C,
245.
5 (V
C,,
83
) tr
igon
al
b.c.
c., B
1 st
able
bel
ou
I300
K
w I
W
Tab
le 4
(con
tinue
d)
h,
orde
red
phas
e di
sord
ered
re
f. re
mar
ks
basi
s ph
ase
__
co
mpo
- no
nmet
al c
onte
nt
crys
tal
stru
ctur
e co
mpo
- la
ttice
si
tion
(at%
) or
regi
on
-
sitio
n ty
pe
of e
xist
ence
of
the
latti
ce
spac
e un
it ce
ll pa
ram
eter
s*)
orde
red
phas
e ty
pe
grou
p (n
m)
6,-V
C,
45.0
-47.
0 m
onoc
linic
a
= fi
~,/2
, h
= V
%a,,
V
C,
c =
8 l/
Za,,y
=
109
"28'
VSC
, -4
6.6
(VCO
8,) cu
bic
P4,3
2 a
= 0
.833
2 zz
2a,
vc,
Nb,
C
233.
3 (N
bC,,,
) or
tho-
Pn
ma
a =
1.2
36, b
= 1
.089
, N
b2C
rh
ombi
c,
c =
0.4
96
type
<-F
e2N
type
&-F
e2N
N
b2C
~
33
.3
(NbC
,,,)
hexa
gona
l, a
= p
a,,
c =
c, N
b2C
Nb,
C,
243.
0 (N
bC,,,
,) cu
bic
n =
0.4
445
NbC
Nb,
C,
NbC
,,,,
-NbC
,,,,
trig
onal
Nb,
C,
NbC
o,81
-NbC
,,8,
mon
oclin
ic
C2/
m
a =
c =
0.5
4605
=
fiU,/2
>
&/2
>
b =
0.9
4579
=
'J =
109
.47'
NbC
NbC
b.c.
c., B
1 [3
71
stab
le b
elow
b.c.
c., B
1 [3
1, 3
61
stab
le b
elow
h.c.
p., L
'3
1381
lo
w-te
mpe
ratu
re
1500
K
1500
K
orde
red
phas
e
h.c.
p., L
'3
~381
hi
gh-te
mpe
ra-
ture
ord
ered
ph
ase
b.c.
c., B
1 14
61
othe
r in
vest
igat
i- on
s do
not
con
- fir
m th
e ex
iste
nce
of a
n or
dere
d ph
ase
of th
is k
ind.
Th
e res
ults
of [
46]
are
ques
tiona
ble
b.c.c
., B1
[3
4, 3
9, 4
41
sim
ilar
to V
,C,
stru
ctur
e. 7
;,,,,
= 1
313K
b.
c.c.
, B1
[15,
40
first
-ord
er p
hase
to
43
tran
sitio
n: T
rans
?
=
130
4 K,
3
0
AH
,,,,, z 2
.2
kJ/m
ol
2 <
Ta,
C
-33.
3 (T
aco,
,)
hexa
gona
l P5
m t
-2 i
(trig
onal
), 0
ty
pe C
6 3 - U u. w
- - cu
bic
:
Ta,C
, ~
43
.0
(Tac
o,,,)
Ta,C
, Ta
C,,,
, -T
aC,,,
, m
onoc
linic
?
w,c
-3
3.3
(WC
,.,)
hexa
gona
l P3
ml
(trig
onal
), ty
pe C
6
a =
0.4
424
Ta,
C
h.c.
p., L
'3 [4
71
TaC
b.
c.c.
, €31
~4
61
TaC
b.
c.c.
, B1
[49,
501
W,C
h.
c.p.
, L'3
[&
511
carb
on a
tom
s fil
l pi
in a
n or
derc
d fa
- E
shio
n ha
lf th
e oc
- g.
tahe
dral
inte
rsti-
z a
tial
site
s ot
her
inve
stig
ati-
ons
have
not
re-
ve
aled
this
type
of
orde
ring
in
com
men
sura
te
6 or
dere
d ph
ase,
5 ;
who
se ty
pe is
9
clos
e to
Me,
C,
9
For
Tac
o.,,
5 su
pers
truc
ture
s.
3'
7;,,,,
% 11
00 K
. 7
Ord
erin
g is
firs
t- or
der p
hase
tran
- 8.
sitio
n 5
laye
rs f
orm
ed b
y 3
carb
on a
tom
s 2
and
laye
rs fo
rmed
2'
by v
acan
cies
al-
5 te
rnat
e in
the
di-
2 re
ctio
n pe
rpen
di-
2.
cula
r to
the
c-ax
is
E
0
*) a
,, h,
, c,
stan
d fo
r th
e di
sord
ered
bas
is-p
hase
lat
tice
perio
ds.
W
34 A. I. GUSEV
Niobium carbide specimens of different compositions within the region of homogeneity of niobium carbide were obtained [15] by solid-phase sintering of a mixture of powered metallic niobium and powdered acetylene black. The mixture was sintered at a temperature of 2300 K in a 0.001 Pa vacuum for 20 h, the product being intermediately ground every five hours of sintering. To obtain disordered and ordered niobium carbide preparations, the synthesized specimens were subjected to quenching and annealing, respectively.
The variation of the niobium sublattice period in the ordered and disordered phases as a function of NbC, composition is shown in Fig. 4. For the ordered phase the parameter of the f.c.c. sublattice of niobium turned out to be larger than that for the disordered carbide (Fig. 4), an indication that the volume of the crystal varies discontinuously during ordering. This fact, as well as the availability of specimens containing an ordered phase and a disordered phase simultaneously, provides evidence that the ordering in niobium carbides proceeds by the mechanism of a first-order phase transition [15, 41 to 431.
Neutron and X-ray diffraction techniques have been employed [ 15,40 to 431 to determine the structure of the ordered phase of niobium carbide. Neutron diffraction patterns were obtained of both quenched and annealed NbC, specimens (with y = 0.72, 0.75, 0.77, 0.81, 0.83, 0.845, 0.88, 0.93, 0.97, 1.00). Extended annealing of NbC, specimens with y 5 0.77 and y 2 0.93 did not give rise to superstructure reflections in the neutron diffraction patterns whereas the neutron diffraction patterns of annealed NbC, specimens with 0.81 5 y 2 0.88 displayed weak superstructure peaks along with intense structural lines.
044
t - E s 0
04-4
044
044
044
0144
Fig. 4. Period of the basic crystal lattice vs. the composition of NbC, and TaC, carbi- des: 0 disordered state, o ordered state
'0 Y-
Disorder and Long-Range Order in Non-Stoichiometric Interstitial Compounds 35
The neutron diffraction patterns of NbC,,,,, NbC,,,,, NbC,,,,,, and NbC0,88 specimens contained the same system of superstructural lines. An analysis of the intensity and position of those lines showed that only one ordered Nb,C,-type phase was present in the range of compositions NbC0,8, to NbC,,,,; other ordered Nb,C,-type phases similar to V,C, [36] were not revealed outside that region. Nor was Nb,C, detected, the existence of which had been reported earlier [46]. The most elaborate studies have been made of niobium carbide NbC,,,, specimens corresponding to the stoichiometric composition of an ordered Nb,C, phase. The intensities of the superstructure lines in the neutron diffraction pictures were highest for those specimens.
Calculations [15, 41, 431 have shown that the unit cell of an ordered Nb,CS carbide is monoclinic and belongs to the space group C2/m (Fig. 5). The translation vectors of this cell relative to the basal cubic structure are a = (4, -i, - l},,, b = {+, 2, O),,, and c = f L 12,
- 4, l}B,. From the calculations of theoretical neutron diffraction spectra for the monoclinic superstructure Nb,C, established in [15, 41 to 431 and from calculations of earlier models for an ordered Me&,-type phase [33 to 351, it has been found that the best fit to the experimental spectrum is attained for the monoclinic structure with space group C2/m.
The structure of niobium carbide Nb,C, as determined by the authors of [15, 40 to 431 is in many ways similar to the V,C, structures revealed in [33 to 351. The relation between these structures is shown in Fig. 6. If one considers only the non-metallic sublattice (the symmetry of the metallic sublattice does not virtually change during ordering), then complete planes containing only carbon atoms and partially defective ordered planes containing carbon atoms and vacancies (in defective planes, the vacancies occupy one third of all sites and form regular hexagons in the plane) will alternate successively in the [lil]Nac, direction. In Fig. 6 we show the order in which the defective planes are arranged relative to each other allowing to obtain three differing Me&,-type superstructures.
‘NaCI / k -i
[looha, Fig. 5. Position of the monoclinic unit cell of an ordered Nb,C, phase (space group C2jm) in a B1 (NaC1)-type lattice: o carbon atoms, vacancies, 0 niobium atoms [9, 15, 40 to 431
3*
36 A . I. GUSEV
Fig. 6. Models of related Me&,-type superstructures: a) monoclinic (space group C2/m) structure Nb,C, [40 to 431, b) trigonal (space group P3,) structure V,C, [33], c) monoclinic (space group C2) structure V,C, [34, 351, d) position of a Me,C,n quasimolecule in a B1 (NaC1)-type lattice; 0 carbon atoms, 0 vacancies, 0 metal atoms
With the atoms and vacancies of the adjacent defective planes being successively displaced three times relative to each other in the same direction along the vector {i, -4, 1) (in the cubic coordinate system of a B1 (NaC1)-type structure), the model of a monoclinic (space group C2jm) structure arises (Fig. 6a). A successive helical displacement along the vectors {+, -4, l}, (4, - 1, i}, and (1, -4, t} gives rise to a trigonal (space group P3,) V,C,-type structure [33] (Fig. 6b). In the case where the atoms and vacancies of the defective planes are successively displaced along the vectors (i, -+, l} and { 4, - 1, i} (Fig. 6c), the monoclinic (space group C2) superstructure arises which has been proposed for the low-temperature ordered V,C, phase [34, 351. As can be seen from Fig. 6, all the Me&, superstructures considered may be represented by a set of Me,C,o quasimolecules (Fig. 6d) arranged in a certain order and differ from one another only in the relative position of the partially
Disorder and Long-Range Order in Non-Stoichiometric lnterstitial Compounds 37
defective planes. These superstructures are completely identical in the character of short-range order [52].
The studies of the ordering in NbC, [15, 33, 39 to 451 have not revealed a superstructure similar to V,C, [36]; nor has the cubic superstructure Nb,C, been found.
One of the systems in which ordering is most difficult to probe is cubic tantalum carbide TaC,, because the relative intensity of possible superstructure reflections is very low in an X-ray experiment, in view of the large difference in the scattering amplitudes of tantalum and carbon atoms. In a neutron diffraction experiment, strong absorption of neutrons by the massive tantalum nuclei leads to a considerable lowering of the total intensity of the diffraction spectrum; as a consequence, superstructure reflections are difficult to detect.
An electron diffraction study of carbide TaC,,,, [48] has revealed the presence of a diffuse band whose geometry corresponds to a Me,C,-type ordering with a very small degree of order. The authors of [49, 501 have investigated the ordering in tantalum carbide with the use of neutron diffraction and magnetic susceptibility methods. The neutron diffraction patterns of annealed TaC, (0.79 5 y 5 0.90) specimens exhibited weak superstructure reflections. Judging from neutron diffraction data, an incommensurate ordered phase with a composition close to Ta,C, arises in tantalum carbide [50].
The structure of the ordered phase of tantalum carbide has been studied most fully and actually for the first time by the authors of [50]; therefore we wish to enlarge on the results of that paper.
On the neutron diffraction pattern (I . = 0.1694nm) of tantalum carbide, there is an interval of angles, 20 = 19" to 21", corresponding to wave vectors that restrict the first Brillouin zone of the f.c.c. lattice. The presence of superstructure reflections in this range of angles indicates that static concentration waves with wave vectors terminating near the boundary of the first Brillouin zone arise in the crystal. The ordering in TaC, is accompanied by some increase in the period of the metallic f.c.c. sublattice (Fig. 4); the same was observed earlier [41] for the ordering in niobium carbide.
Magnetic susceptibility measurements [49] have shown that the disorder-order transition temperature varies between 1070 and 1120 K, depending on the composition of TaC,. The temperature-induced susceptibility hysteresis observed in the transition region indicates that the ordering in TaC, is a first-order phase transition.
A detailed analysis of the position of superstructure reflections in the neutron diffraction patterns of TaC, [50] has shown that although displaying similar features, the detected tantalum carbide superstructure differs from the known Me&,-type structures of ordered vanadium and niobium carbides.
Non-metallic complete and defective atomic planes alternate in Me&, structures because the arm ki3) = b,/2 of the star {k , } is present in the disorder-order transition channels [9]. The star { k , ) ensures the commensurability of such ordered structures since the interplanar distance corresponding to it coincides with one of the interplanar distances of the disordered basis structure B1. In other words, the positions of the concentration wave kh3) maxima and minima coincide with the (l i l)Bl planes. (The wave vector stars and their arms for the reciprocal space of the f.c.c. lattice are numbered in keeping with [9]; b, = { 1, - 1, 1) is the basis vector of the reciprocal lattice of an f.c.c. crystal).
A comparison of neutron diffraction patterns of tantalum carbide with diffraction spectra of Me,C,-type ordered structures of vanadium and niobium carbides has revealed that the spectrum of tantalum carbide does not contain reflections that correspond to the star { k , ) of the f.c.c. non-metallic sublattice of the basis structure B1. These reflections in the spectrum
38 A. I. GUSEV
of tantalum carbide split up into satellites. An analysis of the satellite position in the neutron diffraction pattern indicates that the satellites may belong to the star ( k , ) . The arms of the star { k 5 } are collinear to those of the star ( k , ) and have running indices such that their position may vary continuously from the zero of reciprocal space to the boundary of the first Brillouin zone, i.e., to the reciprocal space points to which the arms of star { k , } correspond. The presence in the phase transition channel of the arm ki6’ = yb, which has the running index y z 0.47 actually denotes incommensurability of the superstructure in tantalum carbide. Incommensurability results in that the concentration wave corresponding to the star ( k 5 ) has maxima and minima not coinciding with the (lil)Bl planes of the non-metallic sublattice of tantalum carbide. This implies a dramatic decrease in the number of complete planes and an increase in the carbon concentration of the defective planes. According to [50], complete non-metallic planes in an ordered tantalum carbide are encountered approximately after every 16th or 17th non-metallic (lil)Bl plane rather than every other ( l i l )Bl plane, as is the case with commensurate, ordered, Me,C,-type carbides. Thus the translation period in the [li1lB1 direction increases by a factor of 16 to 17, from 0.5112 nm for a commensurate Me,C,-type phase to 8.4 to 8.5 nm in an incommensurate ordered tantalum carbide. The amount of translation depends largely on tantalum carbide composition and heat treatment conditions.
The non-coincidence of the concentration wave maxima and minima with the non-metallic sublattice (lil)Bl planes signifies also that the filling probabilities for the carbon and vacancy positions in an ordered carbide differ appreciably from 1 and 0, respectively. As a consequence, the degrees of long-range and short-range order in an ordered tantalum carbide are far from maximum possible values.
In view of the foregoing, the authors of [50] suppose that the superstructure in tantalum carbide may be represented as some transformation of known Me,C,-type superstructures with space groups C2/m, P3,, and C2. While complete and partially defective non-metallic (lil)Bl planes alternate in the [lil], , direction in commensurate ordered Me,C, structures, the sequence of alternation of the afore-mentioned planes in the same direction is perturbed in the incommensurate superstructure of tantalum carbide. Actually, the superstructure in tantalum carbide is intermediate between a disordered crystalline structure and completely ordered Me,C,-type structures with different space groups.
The incommensurability of the structure means also the absence of the exact stoichiometric composition of the ordered phase (commensurate superstructures have an exact stoichiome- tric composition).
The results of [50] provide altogether novel informative data on the ordering in tantalum carbide TaC, and permit the problem of ordering in non-stoichiometric compounds to be viewed in a different light. Incommensurate ordered phases may be expected to arise also in other non-stoichiometric compounds.
The above results suggest that ordering in cubic non-stoichiometric monocarbides of group V transition metals gives rise predominantly to Me&, phases that have a related monoclinic or trigonal structure or an incommensurate structure similar to these. A common feature of the commensurate superstructures of non-stoichiometric monocarbides of group IV and V transition metals is that complete (defect-free) and partially filled (defective) carbon planes alternate in the [I 11]B1 direction or in directions equivalent to this direction.
A feature peculiar to the ordering in lower Me,C carbides of group V and VI transition metals is that ordered carbon planes with different degrees of filling with carbon atoms alternate in the direction of the c-axis (Fig. 7).
Disorder and Long-Range Order in Non-Stoichiomctric Interstitial Compounds 39
a 1 'W C
Fig. 7. Ordering of carbon or nitrogen atoms in Me,X structures (the interstitial atom layers are arranged perpendicular to the c-axis of the basal disordered W2C structure): a) type W,C (L'3) with statistical distribution of interstitial atoms, b) type C6 (Ta,C, W,C), c) type E-Fc,N (Nb,C, V,N), d) type (-Fe,N (V,C, Nb,C); statistically half-filled interstitial atom positions, 0 interstitial atoms, o vacancies
In Fig. 8 a to d we show fragments of phase diagrams for Ti-C, V-C, Nb-C, and Ta-C systems that exhibit ordering. The phase diagrams of the Ti-C and V-C systems (Fig. 8 a and b) have been constructed with allowance for experimental data for ordering, whereas the phase diagrams for Nb-C and Ta-C (Fig. 8 c and d) have been calculated theoretically [53]. A calculation [53] performed with the use of the order parameter functional method [54 to 561 has shown that the principal ordered phase of non-stoichiometric niobium and tantalum carbides is a Me&,-type superstructure (Fig. 8 c and d). This is in full accordance with available findings. According to the calculation of [53], a low-temperature Me,C,-type superstructure (Fig. 8c) is liable to arise in a non-stoichiometric NbC, carbide over a narrow composition range at a temperature below 900 K.
3.2 Supevstvuctuves of non-stoichiometvic nitvides and oxides
Data on the structures of ordered phases that form in metal-nitrogen systems are presented in Table 5.
The ordered phases of non-stoichiometric cubic titanium, vanadium, and niobium mononitrides possess tetragonal structures and differ noticeably from the corresponding ordered carbides. In the unit cells of ordered titanium and niobium nitrides, the vacancies reside in the sites of the body-centred tetragonal (b.c.t.) lattice. Ordered phases form also on the basis of the lower vanadium nitride V,N (the character of ordering is the same as that in lower carbides, see Fig. 7c) and on the basis of the a-solid solution of nitrogen in vanadium (the ordered phases have tetragonal or weakly-rhombically distorted lattices). The formation of an ordered Nb,N3 phase has been established in [61]. No ordering has been detected in tantalum nitride TaN,, with y 2 0.4. Data on the ordering in zirconium- nitrogen and hafnium-nitrogen systems are not available (except for the assumptions that ordering may occur in Me,,,,N nitrides, which contain vacancies in the metallic sublattice).
40
t c 0 L - 1 I
3000
5 -
C
2000- p-Nb2C+NbC,
06 08
3001
ZOO(
7001
I I
b -
5 I 7
I
Y- Y-
Fig. 8. Portions of the phase diagrams of a) the Ti-C system with a region of existence of an ordered Ti,C phase (according to the data of [27, 281); b) the V-C system with regions of existence of ordered V2C, V,C,, and V,C, phases (according to the data of [31, 33 to 36, 381); c) the Nb-C system with regions of existence of ordered Nb,C, and Nb,C, phases (theoretical calculation [53]); d) the Ta-C system with the region of existence of the ordered Ta,C, phase (theoretical calculation [53])
A common feature of ordered phases based on lower Me,X (X = C, N) carbides and nitrides is that their interstitial-element content does not exceed 33.3 at% X. If the
Disorder and Long-Range Order in Non-Stoichiometric Interstitial Compounds 41
a, o e , z a $ 5
6 a a E .O 0 .t: urn
/I II c i k
> > >
P
13
Tab
le 5
(con
tinue
d)
~~
~ ~~
orde
red
phas
e di
sord
ered
ba
sis
phas
e re
f. re
mar
ks
com
po-
non-
met
al c
onte
nt
crys
tal
stru
ctur
e si
tion
(at%
) or
regi
on
-
-~
of ex
iste
nce
of
latti
ce
spac
e or
dere
d ph
ase
type
gr
oup
unit
cell
para
met
ers*
) (n
m)
com
po-
latti
ce
sitio
n ty
pe
&V
N,
44.0
-47.
0 te
trag
onal
V,N
~
33
.3
(VN
,,,)
hexa
gona
l,
Nb,
N,
~4
3.0
(NbN
,,,,)
tetr
agon
al
I4/m
mm
Ta,
N
~1
1.1
te
trag
onal
type
&-F
c,N
Ta,,N
, ~
11
.6
tetr
agon
al
u =
c =
0.8
150
%
2% =0
.438
2 %
u,,
c =
0.8
632
4
2a0
a =
b =
4a,
, c
= 4
.02
4~
~
a =
b =
6a,
, c
= 2
.08u
0
stab
le b
elow
80
0 K
V
N
b.c.
c., B
l [3
11
V,N
h.
c.p.
[3
4, 3
91
NbN
b.
c.c.
, B1
[31,
611
Ta
f.c.c
. [6
21
Ta
f.c.c
.
__
*)
a,,
h,, C,
stan
d fo
r th
e di
sord
ered
bas
is-p
hase
latti
ce p
erio
ds.
stab
le b
elow
17
00 K
su
pers
truc
ture
w
ith a
tet
rago
nal
dist
ortio
n of
the
ta
ntal
um s
ubla
t-
tice
supe
rstr
uctu
re
with
a te
trag
onal
di
stor
tion
of t
he
tant
alum
sub
lat-
tic
e
Disorder and Long-Range Order in Non-Stoichiometric Interstitial Compounds
E c 0
I- * m - * m
2
c o g m 5; w" 2 .w 2
43
c
P
Tab
le 6
(co
ntin
ued)
orde
red
phas
e di
sord
ered
re
f. re
mar
ks
basi
s ph
ase
__
__
__
_
~~
com
po-
non-
met
al c
onte
nt
crys
tal
stru
ctur
e co
mpo
- la
ttice
si
tion
(at%
) or
regi
on
sitio
n ty
pe
of e
xist
ence
of
the
latti
ce
spac
e un
it ce
ll pa
ram
eter
s*)
orde
red
phas
e ty
pe
Erou
P (n
m)
1 2
3 4
5 6
7 8
9
Zr,O
= 2
5.0
hexa
gona
l, P6
,22
u =
0.5
630
b6a0
, a-
Zr
h.c.
p.
[65,
691
7;,,,,
=
700
to
type
Ni,N
c
= 0
.519
8 90
0 K
Z
rO,
25.9
- 29
.2
hexa
gona
l P3
12
a =
k'%o,
c =
co
a-Z
r h.
c.p.
[7
01
7;,,,,
=
1270
K.
(tri
gona
l)
[71]
exp
ects
tha
t w
hen
ZrO
, fo
rms
an in
ter-
m
edia
te p
hase
ar
ises
whi
ch i
s st
able
up
to
710
K
Hfo
, 6
Y 1
4.0
Y 1
1.1
hexa
gona
l
tricl
inic
a-H
f h.
c.p.
vana
dium
sub
latti
ce
V
f.c.c
u
= b
= 0
.311
0,
c =
0.2
994,
2
= fl
= 9
0.3'
, y =
90'
laye
rs fo
rmed
by
oxyg
en a
tom
s an
d va
canc
ies
and
laye
rs f
or-
med
by
vaca
ncie
s al
one a
ltern
ate
in
the
dire
ctio
n pe
r-
pend
icul
ar t
o th
e c-
axis
long
-per
iodi
c st
ruct
ure
?
Disorder and Long-Range Order in Non-Stoichiometric Interstitial Compounds 45
Y
2 0 0
V
5 k
- m r- Y
c: c! D
0, ? U
F c c - v,
- m
d d c! 0 D cc:
D z 0 >
II / I
d m-
c? '9
II II
0 u
00 CCI
0
hl W - 3 0
0 0
d Q P 3
P
c? 4
9 0, D z
P
Tab
le 6
(co
ntin
ued)
m
orde
red
phas
e di
sord
ered
re
f. re
mar
ks
basi
s ph
ase
-
com
po-
nonm
etal
con
tent
cr
ysta
l st
ruct
ure
com
po-
latti
ce
sitio
n (a
t%) o
r re
gion
si
tion
type
of
exi
sten
ce o
f th
e la
ttice
sp
ace
unit
cell
para
met
ers*
) or
dere
d ph
ase
type
gr
oup
(nm
)
1 2
3 4
5 6
7 8
9
NbO
, z
66.7
Ta,O
= 1
4.3
Ta,O
zz
20.0
Ta,
O
N 33
.3
tetr
agon
al
sim
ple
cubi
c
= 0
.664
5, c
= 0
.480
5 N
b
a =
0.4
21
NbO
tetr
agon
al
14Ja
a
= 1
.371
x 2
l/Za,
, N
bO,
tetr
agon
al
a =
0.3
363.
c =
0.3
254
Ta
c =
0.5
985
v 2
c,
orth
o-
rhom
bic
a =
0.3
61, b
= 0
.327
, Ta
c
= 0
.320
tetr
agon
al,
a =
0.6
675,
c =
0.4
721
Ta
type
Cu,
O
*) a
,, b,
, c,
stan
d fo
r th
e di
sord
ered
bas
is-p
hase
lat
tice
perio
ds.
f.c.c
. [7
7]
met
asta
ble,
fo
rms
at 7
00 to
80
0 K
b.
c.c.
, B1
[78]
in
the
ord
ered
st
ruct
ure,
one
ni-
obiu
m a
tom
is
abse
nt in
the
ver-
te
x of
the
cube
, an
d on
e ox
ygen
at
om is
abs
ent i
n th
e ce
ntre
of t
he
cell;
[78
] vie
ws
NbO
as
an in
de-
pend
ent
phas
e [7
91
tetr
agon
al,
rutil
e-t y
pe
f.c.c
. [8
0]
met
asta
ble,
fo
rms
belo
w
800
K
form
s be
low
80
0 K
form
s at
700
to
-
f.c.c.
[S
O]
met
asta
ble,
?
800
K
c1 2
f.c.c.
[8
1]
met
asta
ble,
/
Disorder and Long-Range Order in Non-Stoichiometric Interstitial Compounds 41
Fig. 9. Structures of a) a-Ti and b) an ordered Ti,O phase 1641: 0 empty octahedral interstitial sites of the h.c.p. sublattice of a-titanium, o octahedral interstitial sites occupied by oxygen atoms, 0 titanium atoms
/-
b
interstitial-element (X) content is higher than 33.3 at%, only the disordered h.c.p. structure of lower carbides and nitrides is stable.
A large number of ordered phases has been detected among the transition-metal monoxides and suboxides (Table 6).
In [63, 641, the existence of ordered Ti,O and Ti,O phases has been established by a radiographic method. A neutron diffraction study [65] has confirmed the ordered distribution of oxygen atoms in the octahedral interstitial sites of the h.c.p. lattice of titanium. Supporting evidence for the presence of ordered Ti,O and Ti,O phases in the Ti-0 system has been provided also by a microstructural investigation and an analysis of composition-property diagrams; the studies were made on specimens annealed for hundreds and thousands of hours [66]. The structure of titanium oxide Ti,O is depicted in Fig. 9. The distribution pattern of oxygen atoms in ordered Ti,O, Ti,O, and Ti,O structures is shown in Fig. 10.
As a function of composition and annealing temperature, several ordered phases are liable to form in cubic titanium monoxide TiO, [66 to 681.
The structures of ordered Zr,O, ZrO,, and HfO,,, phases are similar to those of the corresponding titanium suboxides. A feature peculiar to the structure of ZrO, [70] is the presence of three oxygen atom position types differing in occupation probabilities.
Typically, lower vanadium oxides have ordered phases with a monoclinic primitive or centred structure [73 to 751; if the oxygen content of the oxide is less than that of VOo,25, ordered phases form which have a triclinic [60] or tetragonal [72, 731 structure. A common
Fig. 10. Distribution pattern of oxygen atoms in a) ordered Ti,O, b) Ti,O, and c) Ti,O phases [65]: 0 empty octahedral interstitial sites of the h.c.p. sublattice of a-titanium, o octahedral interstitial sites occupied by oxygen atoms
A. I. GUSEV 48
feature of all these phases is an ordered distribution of oxygen atoms in the octahedral interstitial sites of the metallic sublattice.
Niobium and tantalum suboxides are viewed as ordered solutions of oxygen atoms in the b.c.c. lattice of niobium and tantalum; all of these solutions are metastable [77, 79, 801.
Depending on oxygen content and heat treatment conditions, diverse ordered structures with cubic, tetragonal, rhombic, and monoclinic lattices may arise as cubic titanium monoxides suffer ordering. The large variety of ordered phases of cubic titanium and vanadium monoxides is due apparently to the presence of structural vacancies in both the oxygen sublattice and the metallic sublattice of these compounds. A detailed description of the ordered structures that form in cubic titanium monoxide has been provided in [82].
3.3 Ordering in solid solutions of non-stoichiometric compounds
Few reports are available on the ordering in substitutional solid solutions based on transition-metal carbides, nitrides, and oxides (Table 7).
A neutron diffraction investigation of titanium and zirconium carbonitrides, MeC,N,, [83,84] has shown that ordering is absent in low-vacancy carbonitrides (with x + y 2 0.75). With a large vacancy content (x + y < 0.74), the observed ordering is due to different probabilities of the 16(c) and 16(d) positions being filled with carbon and nitrogen atoms; however, no separation into a carbon sublattice and a nitrogen sublattice occurs. According to [84], the substitution of nitrogen for carbon in TiC,N, leads to a lowering of the disorder-order transition temperature. When ordering takes place in titanium oxycarbide, the hypothetic ordered phase TiC,,,N,,, is characterized by the presence of a carbon sublattice and an oxygen sublattice [88].
In Ti,V,-xCO,s solid solutions with 0.2 5 x 5 0.3, the ordered anti-CaC1,-type phase has been detected [85, 901. The authors of [91] have found that the intrusion of hydrogen in a defective titanium carbide TIC, (0.55 5 y 5 0.71) does not alter the type of the cubic superstructure that forms.
4. Conclusion
An analysis of the experimental data obtained in the past twenty years, as well as of review papers [l, 6, 9, 11, 31, 92, 931, shows that atomic ordering is a widespread phenomenon among the non-stoichiometric interstitial compounds. The change in the structural state of a non-stoichiometric compound comes from the non-metallic atoms being redistributed in the interstitial sites of the metallic sublattice. The disordered state of non-stoichiometric compounds is stable and in thermodynamic equilibrium at a temperature above 1300 to 1600 K. At a temperature below the disorder-order transition temperature T,,,,, the disordered state is metastable and can be preserved by quenching. The ordered state of non-stoichiometric compounds is in thermodynamic equilibrium at T < Trans; this state can be produced as a result of an annealing whose duration and temperature are suacient for the diffusive redistribution of lattice atoms.
A moderately deep insight has been gained into the crystal structure of ordered phases in non-stoichiometric carbides of group V transition metals and in non-stoichiometric oxides. The structures of ordered non-stoichiometric nitrides have far less been studied.
Tab
le 7
C
hara
cter
istic
s of
ord
ered
pha
ses
that
for
m i
n so
lid s
olut
ions
of
pseu
dobi
nary
car
bide
-nitr
ide,
ca
rbid
e-ox
ide,
ca
rbid
e-ca
rbid
e,
and
carb
ide-
hydr
ide
syst
ems
- .. 8 or
dere
d ph
ase
diso
rder
ed
ref.
rem
arks
-
basi
s ph
ase
-
corn
po-
nonm
etal
con
tent
cr
ysta
l str
uctu
re
com
po-
latti
ce
sitio
n (a
t%) o
r re
gion
si
tion
type
of
exi
sten
ce o
f th
e la
ttice
sp
ace
unit
cell
para
met
ers*
) or
dere
d ph
ase
type
gr
oup
(nm
)
TiC
,N,
x +
y s 0
.72
cubi
c Fd
3m
a =
2a0
Ti
C,N
, b.
c.c.
, B1
[83,
841
se
cond
-ord
er
phas
e tr
ansi
tion.
the
tran
sitio
n te
mpe
ratu
re is
10
43 K
For
TiC
q.:o
No.
,z
ZrC
,N,
x +
y 5
0.74
cu
bic
Fd3r
n a
= 2
a0
ZrC
,N,
b.c.
c., B
1 [8
3]
VC,
N,
0.46
s x
+ y
s 0.6
4 he
xago
nal
P?m
l a
= a,
6,
c =
co
V,C
h.
c.p.
, L’3
[8
0]
0.11
2 x 5
0.4
6 0.
04 5
y 5
0.3
5
VC,
N,
cubi
c,
P4,3
2 a
= 2
a0
V&
, ty
pe
VC,
N,
b.c.
c., B
1 [8
6]
cool
ing
from
te
mpe
ratu
re
1720
K. A
nnea
ling
betw
een
770
and
1070
K g
ives
ris
e to
a r
hom
bic
<-Fe
,N-t
ype
supe
rstr
uctu
re
anne
alin
g te
m-
pera
ture
900
to
1300
K. T
he s
truc
- tu
re o
f va
na-
dium
car
boni
trid
e
has
been
in
vest
igat
ed
VCO
.66N
O.2
5
P a
ul
0
Tab
le 7
(con
tinue
d)
orde
red
phas
e di
sord
ered
re
f. re
mar
ks
basi
s ph
ase
__
co
mpo
- no
n-m
etal
con
tent
cr
ysta
l str
uctu
re
com
po-
latti
ce
sitio
n (a
t%) o
r re
gion
si
tion
type
of
exi
sten
ce of
the
la
ttice
sp
ace
unit
cell
para
met
ers*
) or
dere
d ph
ase
type
gr
oup
(nm
)
vco.5
00.5
cubi
c Fd
3m
a =
2a0
Ti
C,O
, b.
c.c.
, B1
[87]
th
e pr
epar
a-
tions
stu
died
w
ere
hete
ro-
phas
e
cubi
c
tetr
agon
al
a =
0.4
280
TiC
,O,
b.c.
c., B
1 [8
8]
the
stru
ctur
e of
tit
aniu
m o
xyca
r-
bide
TiC
0,44
O
,,,,
has
been
in
vest
igat
ed
a =
0.5
85 R
Z a,
fi,
VC
,O,
b.c.c
., B1
[8
9]
c =
0.4
14 z a
, th
e st
ruct
ure
of
vana
dium
oxy
- ca
rbid
e V
C,,,
, O
o, 59
has
bee
n in
vest
igat
ed; a
n or
dere
d va
canc
y di
stri
butio
n in
va-
na
dium
sub
latti
ce
is e
xpec
ted.
The
pr
esen
ce o
f va
- ca
ncie
s in
the
no
n-m
etal
lic s
ub-
latti
ce h
as n
ot
been
tak
en i
nto
?
acco
unt.
The
re-
r
sults
of
[89]
are
0
chal
leng
eabl
e 5 2
Disorder and Long-Range Order in Non-Stoichiometric Interstitial Compounds 51
E 2 W
z1 4 A
0 z + V l l 2
x * VII
a x VII 2 0 0 0
VII c? 10 w
VII
VII
0
VIIVII a
vl
A*
52 A. I. G u s ~ v
Still less is known of the ordered solid solutions of non-stoichiometric compounds. Therefore, the non-stoichiometric interstitial compounds afford ample scope for further research in atom-vacancy ordering.
The objective of the present review was to discuss structural aspects of disorder and long-range order in non-stoichiometric interstitial compounds and to bring out general and distinctive features of the superstructures that form in these compounds. The paper provides crystallographic information about ordered and disordered non-stoichiometric compounds; these data are prerequisites for studying the effect of the structural state on the properties of the afore-mentioned compounds and for working out techniques to be used in the theoretical description of structural disorder-order phase transitions. These vast topics are of independent significance and will be treated elsewhere.
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(Received April 23, 1990)
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