Short-Range Order in Nonstoichiometrie Transition Metal Carbides, Nitrides, and Oxides

Review Article phys. stat. sol. (b) 156, 11 (1989)

Subject classification: 61.60 and 75.30; 76.60; S1.61; 51.62; S10

Iqastitute of Chemistry, Academy of Sciences of the USSR, Ural Scientific Centre, Sverdlovskl)

Short-Range Order in Nonstoichiometric Transition Metal Carbides, Nitrides, and Oxides

BY A. r. GUSEV

Contents 1. Introdzrctioic

2. Short-range order in nonstoichiometric compounds

2.1 NMR studies 2.2 Diffuse neutron and electron scattering investigations of short-range order 2.3 Magnetic susceptibility and short-range order

3. Short-range order in the metallic sublattice of carbide solid solzitions

4. Local atomic displacements

5. Conclusiont

References

1. Introduction

Recently there has been a growing interest in studying the structure of compounds such as transition metal carbides and nitrides. This is associated primarily with the application of highly sensitive experimental techniques which have provided evidence that the distribution of nonmetallic atoms in these compounds is by no means a statistical one, as was otherwise believed earlier. The increased interest in nonstoichiometric carbides and nitrides is due also to properties inherent in these. The afore-mentioned compounds belong to a group of highest-melting materials (most of them have a melting point that exceeds 3500 K and the melting point of tantalum carbide is 4270 K, the highest melting temperatures known) and are simultaneously superconductors with a superconducting transition temperature of down to 17 K. In addition, these materials exhibit high hardness, one which approximates that of diamond. A further distinctive feature of nonstoichiometric compounds is their capa- bility to form unrestricted mutual solid solutions whose physical and chemical charac- teristics may be altered appreciably by substitution of metallic and nonmetallic atoms. The properties of nonstoichiometric compounds are directly related to their structural state, i.e., to the presence or absence of correlations, short-range or long- range order in the relative arrangement of atoms.

The present paper attempts a t systematizing the comparatively scanty but important results obtained in the past twenty years of research into the local distribution

l) Pervomaiskaya 91, SU-620219 Sverdlovsk, USSR.

12 A. I. GUSEV

and displacements of atoms in nonstoichiometric compounds and their solid solutions. However, before proceeding to discuss these results, it would be worthwhile to charac- terize in general the terms and consider the major features of the crystal structure and chemical bond in nonstoichiometric transition metal compounds, for this information will be needed in the following.

Nonstoichiometric compounds are frequently referred to as interstitial phases (alloys), since this term implies that a feature peculiar to their structure is the presence of a face-centred cubic (f.c.c.) or hexagonal close-packed (h.c.p.) metallic lattice in which the interstitial sites are occupied by nonmetallic interstitial atoms. A t the same time, the symmetry of the metallic lattice in carbides and nitrides differs from the crystal lattice symmetry of the corresponding transition metals, i.e., as the carbides and nitrides form, the crystal structure of the metal undergoes changes (an exceptmion is cobalt). The group I V transition metals (titanium, zirconium, and hafnium) having h.c.p. structure form carbides and nitrides with an f.c.c. metallic sublattice. The transition metals with body-centred cubic (b.c.c.) structure (vanadium, niobium, tantalum, chromium, molybdenum, and tungsten) form carbides and nitrides with an f.c.c. or h.c.p. metallic sublattice. The change in the crystal structure of the metal as a carbide or a nitride is formed indicates the presence of sufficiently strong interactions between metallic and nonmetallic atoms, so applying the term “interstitial phase” to these substances appears to be not quite justified. The point is that in the narrow sense of the word only restricted interstitial solid solutions are true interstitial phases.

I n the past decades the concept “interstitial phase” (“interstitial alloy”) has become widespread enough. It implies a broad range of phases with related structixres that arise as a result of the intrusion of hydrogen, nitrogen, carbon, oxygen, boron, and silicon atoms in the interstices of the crystal lattice constituted by transition metal atoms. Classed among the interstitial phases are both interstitial solid solutions and a vast family of such compounds as transition metal hydrides, carbides, nitrides, oxides, and borides, and also the mutual substitutional solid solutions formed by these compounds. The above group of compounds is also called variable-composition compounds, emphasizing thereby that they possess wide regions of homogeneity within which the composition of the compound may deviate appreciably from stoichiometry while the type of crystal structure is preserved. As departuie from stoichiometry, i.e., nonstoichiometry, is a distinctive property of these compounds, it would be more correct to call them interstitial-phase-type nonstoichiometric compounds. Such a name emphasizes that the substances under discussion (carbides, nitrides, etc.) are compounds rather than interstitial solid solutions and points out simultaneously their salient structural peculiarities - location of nonmetallic atoms a t the interstitial sites of the metallic sublattice and feasibility of a large departure from stoichiometry.

Detailed information on the nonstoichiometric compounds is available in mono- graphs [l to 51.

In the full sense of the word, only transition metal carbides, nitrides, and lower oxides may be ranked within the nonstoichiometric compounds. The point is that the stability and limiting hydrogen content of hydrides may vary as a function of external conditions. Therefore, the distinction between transition metal hydrides and solid solutions of hydrogen in transition metals becomes vague, As far as borides and particularly silicides are concerned, these have virtually no homogeneity regions ; besides, direct B-B and Si-Si bonds play a rather important role in these compounds (direct interactions among nonmetallic atoms in carbides, nitrides, and oxides are negligibly small). Nor do higher oxides possess homogeneity regions. Thus the present paper

Short-Range Order in Nonstoichiometric Carbides, Pu'itrides, and Oxides 13

treats transition metal carbides, nitrides, and lower oxides as interstitial-phase-type nonstoichiometric compounds.

Nonstoichiometric transition metal monocarbides, mononitrides, and monoxides, MeX, have a B1-type (NaC1) structure. I n a B1-type structure, nonmetallic atoms statistically fill the octahedral interstitial sites of the metallic sublattice. This results in each metallic atom being octahedrally surrounded by six nonmetallic sublattice sites in the first coordination sphere ; the second coordination sphere is constituted by twelve metallic atoms, the third by eight nonmetallic sublattice sites. Lower carbides and nitrides, Me,X, exhibit a W2C(L'3)-type hexagonal close-packed structure in which nonmetallic atoms statistically fill half the octahedral interstices.

An analysis of existing theoretical concepts and of the numerous calculations in terms of band and cluster methods has enabled the authors of [3, 61 to formulate the major structural features of the valence band of nonstoichiometric compounds with Bl(NaC1) structure. According to [3, 61, the valence band incorporates a low-energy 2s(X) band that contains small contributions by the s-, p-, and d-states of the metal, a fundamental valence-bonding band formed by strong mixing of 2p(X)-d(Me) functions, a partially filled high-energy conduction band formed preferentially by nd(Me) functions with an admixture of 2p(X), (n + 1) p(Me), and (n + 1) s(Me) functions. The redistribution of individual atomic states, as nonstoichiometric compounds form, leads to partial charge transport between metallic and nonmetallic atoms and is responsible for the ionic component of the chemical bond.

Thus, a combined covalent-metallic-ionic type of chemical bond is realized in nonstoichiometric compounds. As the vacancy content is increased, the populations of single Me-X bonds enhance, but these bonds decrease in number as the charge localized on the atoms diminishes; this leads to a decrease in the ionic component of the chemical bond. I n addition, with departure from stoichiometry there is an increase in the width and population of the d-band of the metal and a narrowing of the 2p(X) band occurs; this may be viewed as an enhancement of the Me-Me interactions and as a weakening of the covalent component of the Me-X bonds.

2. Short-Rango Order in Nonstoichiometrie Compounds

The presence of structural vacancies and the possibility of their being redistributed in the lattice in conjunction with the combined type of chemical bond are responsible for the sufficiently wide occurrence of both correlations and order in the relative arrangement of atoms in nonstoichiometric compounds.

I n describing crystalline solids one often talks of long-range order. What is implied in this case is a one-to-one correspondence between the value of some physical quantity a t an arbitrary point and the value of the same quantity a t any other (including an infinitely remote) point of the crystal. The presence of such a correspondence is due to the periodicity of the crystal lattice. Clearly, complete long-range order signifies an absolutely regular distribution of atoms in crystal lattice sites.

Any kind of disorder disturbs crystal symmetry and leads to a lowering of the degree of long-range order. The simplest type of disorder occurs in ideal binary substitutional solid solutions. I n an ideal solid solution the substitution of a species B atom for a species A atom occurs without distortion of the crystal lattice and the sites in which such substitution has come about are distributed in space in a random fashion, statistically. In this case the probability of detecting an atom of a given species in any crystal lattice site coincides with the concentration of atoms of that species. Thus, the filling of a given site with an atom of a particular species does not depend on the mode of filling of other sites, both adjacent and remote. In a real crystal atoms of

14 A. I. GUSEV

different species differ in mass and dimensions and therefore the interaction energies between like and unlike atoms differ appreciably, too. It is for this reason that statistical disorder in a crystalline solid may be realized only a t sufficiently high temperatures, when the forces of interatomic interaction are small compared with the energy of thermal vibrations. This state may be ‘‘frozen” by quenching, i.e., rapid cooling to low temperatures a t which the diffusion mobility of atoms is negligibly small.

I n nonstoichiometric compounds a statistical distribution of interstitial atoms and structural vacancies is also obtained by means of rapid cooling from high temperatures. Over a long period of time this disordered state was believed to be unique for nonstoichiometric compounds and only in the past decades has it become clear that this is not so. It has turned out that with nonstoichiometric compounds subjected to comparatively slow cooling, the statistical distribution gives rise to correlations in the arrangement of atoms and vacancies at neighbouring lattice sites and, as a consequence, atoms of a given species may be surrounded preferentially by atoms of the same species or by atoms of a different species (vacancies), i.e., short-range order may be observed. P u t another way, short-range order characterizes the local distribution of atoms in a crystal.

Thus, the atomic arrangement in the crystal lattice of nonstoichioinetric compounds may be characterized with the help of long-range and short-range order. The present paper focuses on recent investigations dealing with short-range order and the allied local atomic displacements in refractory nonstoichiometric compounds and their solid solutions.

Major experimental techniques for investigating short-range order in interstitial phase-type nonstoichiometric compounds are nuclear magnetic resonance (NMR), diffuse neutron or electron scattering (in studying the distribution of interstitial atoms), and diffuse X-ray scattering (when the distribution of metallic atoms is studied). None of these methods is versatile in the sense that it may be applied to every nonstoichiometric compound to derive information about how the atoms of all species are distributed. The most restricted range of applicability to nonstoichiometric compounds is that of the NMR technique, the restrictions being due to purely physical causes - magnetic resonance signals may be produced only by nuclei with a nonzero spin I , which possess a magnetic moment. For this reason the NMR technique can be instrumental in investigating nonstoichiometric compounds of vanadium, niobium, and tantalum, as the most common isotopes of these metals have nuclei with I + 0. Conversely, NMR cannot be employed to study nonstoichiometric compounds of group IV transition metals (titanium, zirconium, and hafnium), since the most common isotopes of these metals have nuclei that are devoid of a magnetic moment.

As concerns the interstitial atoms (carbon, nitrogen, oxygen), the nuclei of the common I z C and I6O isotopes possess a zero spin and permit no observation of magnetic resonance. The most common nitrogen isotope 14N possesses a magnetic moment ( I = 1) ; however, no NMR investigations of 14N nuclei in nonstoichiometric compounds have been carried out so far. The body of information about the nature and structure of nonstoichiometric carbides could be largely extended by NMR investigations of the carbon isotope I3C nuclei spin 1/2. Such investigations may be regarded as highly promising.

I n those cases where the utilization of the NMR method is fundamentally possible, this method proves a very efficient tool in studying nonstoichiometric compounds, because of its high sensitivity to minor changes in the symmetry of the environment of the atoms on which resonance is observed.

Short-Range Order in Nonstoichiometric Carbides, Nitrides, and Oxides 15

2.1 N M R studies

The NMR technique enables the determination of the relative position of interstitial atoms and structural vacancies in the nonmetallic sublattice of nonstoichiometric compounds, i.e., the investigation of short-range order in the above compounds. On the other hand, it permits revealing some of the features peculiar to the energy band structure of nonstoichiometric compounds.

To facilitate further discussion we introduce some notations. The nearest environment of a metallic atom in a carbide MeC, or nitride MeN, with B1 structure comprises three coordination spheres : the first and third coordination spheres are formed by nonmetallic sublattice sites and may contain vacancies, whereas the second coordination sphere is constituted by twelve metallic atoms and contains no vacancies. With this environment of the metallic atoms, the position of the latter may be conveniently designated by a subscript and a superscript that indicate the number of vacancies in the first and third coordination spheres, respectively. For example, the notation Me: denotes that the first coordination sphere of the metallic atom is complete (devoid of vacancies) and the third coordination sphere contains two vacancies.

Of the nonstoichiometric compounds, vanadium compounds have received the most detailed study. The authors of [7] studied VC, samples with y = 0.66, 0.7, 0.8, 0.84, and 0.875, i.e., over the entire region of homogeneity of vanadium carbide. The NMR spectra of 51V in carbides VCo,875, VC,,,,, and VC,,,, exhibited several resonance lines whose intensity changed with varying VCy composition. A comparison of the total intensity of resonance lines in metallic vanadium and its carbides showed that NMR detects practically all vanadium nuclei in VC, irrespective of the composition of this compound. At the same time, the presence of vacancies in the carbon sublattice of VC, had an appreciable effect on the relative intensity of the iesonance lines observed. The authors of [7] placed the lines and their satellites in the NMR spectra of 51V in VC, in correspondence with the positions of vanadium atoms, which have determined vacancy configurations in the first coordination sphere. Such an identification was possible because the electric field gradient on vanadium nuclei that have no vacancies in the first coordination sphere should be considerably less than that with one or a few vacancies present in the first coordination sphere. On account of this, the observed lines were placed in correspondence with the vanadium atom positions V, (without vacancies in the first coordination sphere), V,, V,, and V, (with one, two, and three, vacancies in the first coordination sphere, respectively).

In [7] an equation was also obtained which established a relationship between the number of vacancies, n, in the nearest environment of a vanadium atom and the jump of the Knight shift,

K , - Kn-I = nK,, where KO = 0.07 to 0.09%, i.e., the jump of the Knight shift is proportional to the number of vacancies in the first coordination sphere of the vanadium atom in VC,.

To ascertain the character of the vacancy distribution in vanadium carbide the authors of [7] placed the experimentally determined distribution of the probabilities P(n) of vanadium atom positions with different numbers of vacancies, n, in correspondence with the random probability distribution calculated by the formula

P(n) = q ( l - y)" y6--" . (2) For all the VC, samples investigated, the experimental distribution turned out to differ from the random distribution. Thus, for example, as the carbon content of the carbide is decreased, the probability of V, positions with a complete (vacancy-free) environment in the first coordination sphere diminishes much more rapidly than i t

16 A. I. GWSEV

does when the distribution is random, and V, positions occur by far more rarely than in the case of a random distribution. The small number of V, positions and the absence of vanadium atom positions with more than three vacancies in the first coordination sphere was attributed to the presence of a repulsion between neighbonring vacancies. Thus the authors of [7] experimentally established for the first time the presence of short-range order in one of the nonstoichiometric compounds, viz., in vanadium carbide.

Also in [7] the NMR method was employed for the first time to investigate the ordered phase of a nonstoichiometric compound. The NMR spectrum of 51V in VC,,,,, differed appreciably from those of vanadium carbides of other compositions. Specif- ically, the ratio of the line intensitites corresponding to the V, and V, positions was equal to 1/3. If the relative intensity of each line is proportional to the relative number of positions corresponding to it, such an intensity ratio may be explained if three fourths of all the vanadium atoms in the carbide VC,,,,, have one vacancy in the first coordination sphere and the rest of the vanadium atoms have a complete environment. Such vacancy distribution implies the presence of a V,C,-type superstructure with space group P%32 or P4132. It should be noted, however, that, according to [7] , such a vacancy distribution model describes a real lattice not quite accurately. Indeed, in such a superstructure with helicoidal vacancy arrangement there are only Vi and V: positions, while a study of the satellites present in the NMR spectra reveals the existence of three V, position types, which the authors of [7] failed to discriminate.

Interesting results, which permitted one more ordered structure to be revealed, were obtained from NMR studies of annealed vanadium carbide VC,,,, single crystals [S, 91. The NMR spectra of 61V in VC,.,, were resolved into lines that corresponded to vanadium atom positions V, and V,. An analysis of the satellites allowed the resolution of the intense line V, into lines corresponding to vanadium atom positions Vy, V:, and Vl. The intensity ratio of the lines V: and Vf in the spectra obtained was 2 : 1, so i t may be assumed that in the carbide investigated the number of vanadium atom positions V: is twice that of positions Vf. The large intensity of the lines corresponding to positions V: and V! enables these lines to be viewed as positions of the ordered structure of vanadiuni carbide. In an electron diffraction investigation of VC,.,,, the authors of [S] proposed the trigonal symmetry of an ordered vanadium carbide; the availability of NMR data on the ratio of the number of V: and Vl positions enabled them to describe the distribution of carbon atoms and vacancies in a trigonal unit cell. The presence of weak lines corresponding to V, and Vy positions was attributed to a departure of the composition of the carbide investigated from the composition of an ideal ordered phase VC0,,,(V6C,). Note that the presence of these lines may be associated also with the deviation of the degree of long-range order in the samples investigated from the maximum degree of long-range order. NMR investigations [7 to 91 permitted to obtain a qualitative picture of the features

peculiar to the distribution of carbon atoms and vacancies in the nonmetallic sublattice of vanadium carbide. A further step was an attempt to derive quantitative information about the short-range order in VC2/ by minimizing the NMR spectra of ‘lV [lo, 111. The experimental NMR spectra obtained in [lo] for VC,.,,, VC,.,,, VC,,85, vco.,6, and VC,.,, are similar t o those described earlier [7] and contain resonance lines that correspond to V,,, V,, and V, positions. Proceeding from the assumption that the complicated lineshape is due to the presence in the crystal of nuclei with different local environment (presence of different metal atom position types) and that the magnitude of the signal (its intensity) is proportional t o the number of positions of


a given type, the calculated intensity value was represented as

where Pr is the probability of a cluster with the i-th configuration of the nearest environment of the resonance nucleus constituted by nonmetallic atoms (for nonstoichiometric compounds with B1 structure, an octahedron of six nonmetallic sublattice sites with a metallic atom a t the centre is considered as a basis cluster figure), Ai is the multiplicity of the i-th configuration of the basis cluster figure, and I i (v ) is the mathematical model of an ideal spectral line corresponding to the i-th configuration of the nearest environment of the resonance nucleus (in [lo] the lines were described by a Gaussian distribution). The probabilities Pi were expressed as a series in pairwise and many-particle correlation parameters. Minimization of the NMR spectra of 51V in VC, combined with successive allowance for the correlation parameters, starting with binary and finishing with sextic correlation parameters, showed that the best description of the spectrum is provided when allowance is made for pairwise and triple correlations. Allowance for higher-order correlation moments did not lead to a substantial improvement of the theoretical spectrum. According to the results obtained, the pairwise correlations E~~ describing the distribution of vacancies on two neighbouring sites, located relative to one another in the first coordination sphere, are negative in the entire region of homogeneity of vanadium carbide. This means that in the entire region of homogeneity of VC, the probability of finding vacancies that are nearest neighbours is less than the probability in the case of a statistical distribution. The short-range order parameter for the first coordination sphere, al, is equal to E ~ ~ / ~ J ( 1 - y) and amounts to about -0.1 ; it decreases slowly with increasing deviation from stoichiometry and exhibits a small minimum near the composition VCo,,B. Thus, the calculation results for the correlations and short-range order parameter in vanadium carbide are, on the whole, consistent with the conclusions [7 t o 91 that VCv has a short-range order which is due to vacancy “repulsion”. At the same time, the authors of [ l l ] have wrongly asserted that the occurrence of ordered vanadium carbide phases is due to the presence of a definite type of short-range order. As shown in [5, 12 to 141, long-range order is liable not t o arise in nonstoichiometric compounds with Bl(NaC1) structure even in the presence of maximum short-range order in the first and second nonmetallic coordination spheres ; therefore, the presence of short- range order in these compounds is only a necessary but not a sufficient condition for long-range order t o form.

The effect of composition on the NMR spectra of 51V in VN, (0.98 2 y 2 0.706) has been studied in [16]. The total NMR signal intensity, normalized to the amount of vanadium, turned out to be constant with all nitrogen concentrations. This signifies that in VN, nitrides (just as in VC, carbides) all vanadium nuclei make a contribution to the resonance absorption line, thereby permitting a quantitative analysis of the spectra obtained.

The resonance peak observed in the spectra was resolved into two lines that correspond to a complete position of vanadium atoms, V,,, and to positions with one and more vacancies in the first coordination sphere. A comparison of the experimental relative intensity of the line corresponding to V, positions in the nitride VN, (with n 2 1 being the number of vacancies in the nearest environment of a vanadium atom) with the theoretical intensity calculated for a random distribution by the formula

2 physics (b) 156/1

18 A. I. GUSEV

where I. is the integral spectral intensity and x the number of vanadium atoms that are within the range of influence of the vacancy, showed that in the region of small vacancy concentrations x = 6, i.e., the observed effects are determined chiefly by the interaction in the first coordination sphere of the vacancy.

At high vacancy concentration the experimental resonance line intensity deviated systematically from the intensity calculated in the approximation of the random distribution of nitrogen atoms and vacancies. This indicates that the distribution of nitrogen atoms and vacancies in the nonmetallic sublattice of vanadium nitride is characterized by some degree of order, but is not statistical. The authors of [15] deem that the character of vacancy distribution (presence of only two position types, V, andV,) that they observed for a VNo,,5 nitride corresponds to an ordered V,N, phase.

A quantitative analysis of short-range order in vanadium nitride was made in [Ill, following the same scheme as that for vanadium carbide. From the minimization of NMR spectra, the parameters of pairwise and ternary correlations were found to vary monotonically as a function of nitride VN, composition. According to the calculation results, the parameter E ~ , is negative over the entire range of homogeneity of VN, and decreases, whereas E~~ is positive and enhances with increasing number of vacancies. This implies that in vanadium nitride there is a short-range order that displays a preferential location of vacancies a t opposite vertices of an octahedron, six sites of which form the first coordination sphere of the vanadium atom.

Cubic vanadium monoxide VO, was investigated by broad-line NMR in the temperature range between 1.4 and 300K [16, 171. The NMR spectra of 51V in VO, exhibit only one line, whose width increases as the oxygen content of the oxide VO, is raised. Measurements have shown also that the amount of the Knight shift is independent of temperature and VO, composition.

The NMR method has been employed to investigate short-range order not only in nonstoichiometric vanadium compounds, but also in solid solutions, primarily vanadium oxycarbides and oxynitrides [ l l , 18, 191. The NMR spectra of 51V in oxycarbides VC2;02/ represent an assembly of four lines, the width and position of which are virtually independent of oxycarbide composition. A comparison of the integral intensities of the NMR spectra of 61V in carbide VC,,,, and oxycarbide VC,O, (x + y = 0.75 to 0.94) has enabled the authors of [11, 181 to conclude that all vanadium nuclei contri- bute to the oxycarbide spectrum. In oxygen-rich vanadium oxycarbides (y 2 0.4) vacancies are present not only in the nonmetallic, but also in the metallic sublattice ; however, it has not been possible to reveal an influence of vanadium vacancies on the integral NMR spectrum intensity.

An analysis of the NMR spectra allowed the lines observed in these to be placed in correspondence with the nonequivalent vanadium atom positions. As noted by the above authors, the line intensity is determined by the oxygen content of the samples investigated. According to [11, 181, all of the lines observed correspond to vanadium atoms being completely surrounded by carbon and oxygen atoms a t once, but the ratio between the number of carbon atoms and the number of oxygen atoms may be different : e.g., [VC,O,], [VC,O,], [VC,O,], [VC303]. The spectral line intensities are determined by the probabilities of the corresponding positions. A comparison of the observed probability values with values calculated on the assumption of a statistical distribution of carbon and oxygen atoms has shown that the distribution of nonmetal atoms is close to random for VCo,6100,14 and VC0,,,O,, 15 and differs appreciably from a random one when the oxygen content is high.

In the NMR spectra of 61V in VN,O,, the authors of [ I l l observed three lines, two of which (by analogy with VCzO,) were placed in correspondence with the positions [VN,OII and [VN,O,I.


An analysis of the NMR spectra of 61V in the solid solutions VC,O,, VNzOl,, and VC,N, [ll] permitted the conclusion that the replacement of one of the atoms forming the nearest environment by an atom of a different species leads to an increase in Knight shift as compared with a complete homogeneous environment. The largest amount of Knight shift is attained when the atoms of different species are equal in number in the first coordination sphere of the vanadium atom. Note that in vanadium carbide, nitride, and oxide the Knight shift increases in magnitude with growing number of vacancies in the nearest environment of the vanadium atom. I n their entirety, these data suggest that introducing any impurity defect into the nearest environment of the vanadium atom should lead to an increase in Knight shift in comparison with a complete homogeneous environment (by impurity defects we imply those particles which are least in number in the first coordination sphere of the metal atom).

Few papers are available which employ the 93Nb nucleus NMR to study nonstoichiometric niobium compounds. Of these papers, [20] may be mentioned which has used the NMR method to carry out a phase analysis of niobium nitride NbN,. According to the findings, each of the niobium nitride phases has intrinsic lines in the NMR spectrum; owing to this feature, spectral analysis permits t o ascertain which of the phases are present in the sample investigated. The authors of [20] note also that the quadrupole interactions in niobium nitride are stronger than in vanadium nitride.

NMR spectra of 93Nb in NbC, have been obtained by the authors of [7]. All of the spectra exhibited only one line, the intensity of which decreased rapidly in going from NbC,.,, t o NbC,,,,: as the carbon content was lowered further the intensity of the observed line decreased weakly. This means actually that as the composition deviates from stoichiometry, a considerable part of the niobium atom positions become un- observable. The latter circumstance comes from the presence of appreciable electric field gradients for some niobium atom positions and from second-order quadrupole broadening. The sole line observed in the NMR spectra of 93Nb in NbC, was attributed by the authors of [7] to Nbt positions with niobium atoms surrounded completely by carbon atoms.

A thorough investigation of short-range order in NbC, by the method of high- sensitivity, pulsed NMR spectroscopy has been carried out in [12, 21 to 231. Non- stoichiometric niobium carbide NbC, samples (0.995 2 y 2 0.75) were synthesized by solid-phase sintering of niobium and carbon powders. To produce niobium carbide in states with different degrees of order, the samples synthesized were subjected to heat treatment under different conditions, which differed in temperature, annealing time, and cooling rate. The heat treatment conditions had an appreciable effect on the phase composition of the preparations: annealing a t temperatures above the order-disorder transition temperature (for NbC,,,,, according to [ 121, !!!'trans = 1304 K) followed by quenching (regime a) led to the production of disordered niobium carbide samples. Annealing a t a temperature of 1300 K followed by slow cooling (regime c) led to NbC, (c) neutron diffraction patterns exhibiting superstructural reflections that testify the presence of an ordered phase. Annealing a t a temperature of 1300 K followed by rapid cooling (regime b) was done to produce samples containing an ordered and a disordered phases simultaneously. Henceforth, to contract notation, the heat treatment regime used will be specified in brackets, after the composition of niobium carbide; for example, NbC,,, (b).

All of the superstructural reflections observed by the authors of [21 to 231 in the neutron diffraction patterns of NbC, samples were interpreted in a monoclinic structure with space group C2/m, determined in [24 t o 261 for an ordered Nb,C, phase. 2'

20 A. I. GUSEV

The absorption spectra of s3Nb nuclei in NbC,, recorded [12, 21 to 231 by a Bruker SXP4-100 pulsed spectrometer a t frequencies of 22.0, 21.5, and 14.01 MHz, are presented in Fig. 1 and 2. Most of the spectra display a line 0. The position of this line coincides completely with that of theline observed in [7] in the NMR spectra of niobium carbide. With increasing defect content of the carbon sublattice in NbC, the intensity of this line decreases to a point where it vanishes; however, the line 0 reappears in spectrum IV (Fig. l), which corresponds to NbC,,,, (b). The slow lowering (compared with the data of [7]) of line 0 intensity in going from NbC0.905 to NbCo.845 and the increase in line 0 width do not allow this line 0 to be referred to niobium atom positions with a complete environment up to the third coordination sphere, as was done in [7]. It is logical to assume that the line 0 in spectra I to IV (Fig. 1) pertains to both complete niobium atom positions Nbg (Fig. 3a) and niobium atom positions that contain vacancies in only the third coordination sphere, for example, Nbk (Fig. 3b), Nb;, etc.

Fig. 2, showing the variation of the spectrum shape as a function of the degree of order, presents NMR spectra recorded a t two frequencies, 21.5 and 14.01 MHz, for s3Nb nuclei in NbC,.,, samples subjected to heat treatment under three regimes. Spectrum I , corresponding to an NbC,.,, (a) sample, contains on either side of the line 0 a broad peak; spectra I1 and 111, which correspond to Nbc,,,, (b) and NbC,,,, (c) samples, exhibit five peaks, in addition to the line 0. Spectra IV and V belong to the

QOZT -

IT A B o o' B'A'

m A B o 0' B'A'

z

P A B 0'0 0' D i d B'A' I

Ho

H- H- Fig. 1 Fig. 2 Fig. 1. The variation of the 93Nb NMR spectrum shape with the composition of the niobium carbide NbC, (v, = 21.5 MHz, H , = 2.069 T): (I) NbC,.,,,, (11) NbC,,,,, (b), (111) NbC,,,,,(b), (IV)

Fig. 2. The effect of ordering on the NMR spectra of 93Nb in NbCo,83 prepared in different heat treatment regimes: (I, IV) regime a, (11, V) regime b, (111) regime c (I to 111: vo = 21.5 MilHz

N~C,.,,W [12,231

Ho = 2.069 T ; IV, V vo = 14.01 MHz, H, = 1.346 T) [12, 231


Fig. 3. The most probable niobium atom positions in niobium carbide NbC,: a) position Nb;, b) position Nbi, c) position Nb!, d) position Nb:, e) position Nby, f ) position Nb! (0 carbon atom, 0 vacancy, 0 resonating niobium atom) [23]

same samples as spectra I and 11, but were recorded a t a lower frequency, the number of peaks in spectrum V amounting to eight (Fig. 2).

The small relative intensity (about 4%) of the line 0 (spectrum 11, Fig. 2) pertaining to niobium atom positions Nb, and the approximately equal amounts of niobium atom positions Nb, and Nb, for NbC,.,, suggest that the peaks observed in spectrum I1 are due primarily to niobium atom positions Nb, with one vacancy in the first coordination sphere (Nb, positions are not observed in keeping with the second-order quadrupole effect).

An analysis [21 to 231 of the peak position in NMR spectra as a function of frequency (Fig. 2, spectra I1 and V) has shown that only peaks 0' may be satellites and the central transition line for these peaks is the line 0. The position of the rest of the peaks varies as a function of fre,quency ; therefore these peaks are central transition lines rather than satellites. NMR spectral line broadening at low frequencies is an indication of the preference of quadrupole interactions compared with magnetic ones. Therefore, according to [21 to 241, the positions of peaks A, B, A', B' (spectrum 11) and A, B, A', B', C', D' (spectrum V) coincide with those of the singular points of split central transition lines (satellites of these lines are not observed in keeping with the first- order quadrupole effect).

Asymmetry and large values of the electric field gradient result in a rather complicated variation of the peak positions in NMR spectra obtained at different frequencies. For this reason, the authors of [12, 20 to 221 have performed a calculation enabling them to separate the peaks in spectra 11, 111, and V (Fig. 2 ) belonging to different central transition lines. Calculation results [ 12, 231 have shown that apart from the line 0, four extra central position lines may be singled out in the NMR spectra of carbide NbC,,,, (Fig. 2). The positions of the singular points of central transition lines and the allied hyperfine interaction parameters (quadrupole coupling frequency vQ, electric field gradient asymmetry parameter qas, Knight shift K ) are given in Table 1.

To relate the lines observed in spectra I1 and V (Fig. 2 ) to particular niobium atom positions; the authors of [12, 231 have carried out a model calculation of the asym-

Tab

le 1

H

yper

fine

inte

ract

ion

para

met

ers

of 9

3Nb

NM

R f

ipec

tra

in n

iobi

um c

arbi

do N

bC,,8

, and

loca

l env

iron

men

t of

niob

ium

ato

ms

[11, 2

1, 2

31

niob

ium

la

bel o

f re

sults

of

mod

el c

alcu

latio

ns [

la]

sing

ular

poi

nts

of c

entr

al

expe

rim

enta

l res

u1t.s

[21,

231

at

om

posi

tion

posi

tion

(Fig

. 3)

l’Q/lP

C - 4

01 9

%~

Pir

d s

pect

ra

’Q

‘Vex

p IP

u-

Pol

K

lines

in 93 N

b N

MR

as

(MH

x)

(MH

z)

( %)

I!,, =

21.

5 I?

,, =

14.

01

(MW

(M

H4

Nb:

a

0 0

0.09

0

0 0

0 -

-

-0.1

1 +_

0.0

5

Nby

C

1.05

0

0.10

0

A, A’

A, A

’ 1.

9 5

0.3

0

1.8

-0.2

5 &

0.0

5 N

b:

b 0.

2 0

0.13

0

0, 0

’ 0,

O‘

0.3

& 0

.07

0 1.

5 -0

.11 f 0.

05

Nb:

d

1.05

0.

2 0.

07

0.66

A

, B’

A, B

‘ 2.

0 f 0.

3 0.

2 &

0.0

5 1.

9 -0

.14

0.05

N

b:

e 1.

05

0.4

0.00

7 0.

33

B, 0

’ B

, C‘

2.1

5 0

.3

0.4 f 0.

05

2.0

0.25

& 0

.05

Nb:

f

1.05

0.

6 0.

001

0 B

, 0’

B, D

‘ 2.

2 0.

3 0.

5 &

0.1

2.

1 0.

5 &

0.1

?

u


metry parametei yas, of the relative value of quadrupole coupling frequency vQ/(qc-qol (lqc - qnl is the absolute value of the charge difference of a carbon atom and a vacancy, expressed in units of the electron charge), and of the probabilities of different niobium atom positions, Ptisord and Pya, in the disordered and ordered states of niobium carbide NbC,.,,, respectively. For the calculated results [12, 233, see Table 1 (0 is a vacancy).

The large difference between eisord and Pya indicates that the NMR spectrum of a disordered phase should differ substantially from that of an ordered phase. For example, the most probable niobium atom positions for a disordered carbide NbC,.,, are the positions Nbt and Nb; (Fig. 3) with a complete first coordination sphere, whereas the most probable position for an ordered phase is Nb: (Fig. 3d) and there should be no niobium atom positions with a complete first coordination sphere at all.

I n juxtaposing vQ and qas data obtained from a model calculation and from experiment, the authors of [23] preferred the equality of the calculated asymmetry parameter $:lc to the experimental asymmetry parameter r&. From such a juxtaposition, they managed to find experimental values of the hyperfine interaction parameters vq, yas, and K (Table 1) for different niobium atom positions.

Allowance for the environment of niobium atoms to within three coordination spheres has enabled the authors of [12, 231 to explain the presence of all the peaks observed in the NMR spectra of "Nb nuclei in carbide NbC0.83. According to their data, only two niobium atom position types, viz., Nb: and Nb; (Fig. 3d and e), are possible in a completely ordered niobium carbide NbC,,,,. An NMR investigation of carbides NbC,,,, and NbC,,,, has shown that the character of short-range order in these compounds is the same as that in an ordered Nb,C,-type phase. A Knight shift calculation has shown that for niobium carbide, just as for other nonstoichiometric compounds, the quantity K augments with increasing number of vacancies in the environment of the metal atom. The difference of Knight shifts for different niobium atom positions indicates different populations of niobium atom d-orbitals as a function of the nearest environment.

In their entirety, the results [12, 21 to 231 provide evidence that the character of short-range order in a nonstoichiometric niobium carbide is determined mainly by the mutual repulsion of vacancies in the carbon sublattice.

2.2 Difjttse neutron and electron scatteriiig investigations of short-range order

Few papers are available which deal with a diffuse scattering study of short-range order in nonstoichiometric compounds. Using experimental results [29] on the intensity of diffuse electron scattering by vanadium carbide VC,,,,, the authors of [27, 281 have calculated the short-range order parameters ol(hkE) for eight nearest coordination spheres of the nonmetallic sublattice; a vacancy was chosen to be the centre of the coordination spheres. The results of the determination of the parameters a as well as of the number of carbon atoms, nc, and vacancies, no, in each coordination sphere are summarized in Table 2 (for comparison, the same quantities have been calculated for an ordered V,C,-type phase with trigonal superstructure).

Examination of the data presented in Table2 shows that the character of the distribution of carbon atoms and vacancies in the nonmetallic sublattice of VCO.,, differs appreciably from a statistical one. Thus, e.g., the number of vacancies in the first two coordination spheres is much less than the number this carbide should have in the case of a random distribution. As the radius of the coordination sphere is increased, the value of ol oscillates, decreasing gradually in magnitude, The authors of [27, 281 have interpreted t,he findings regarding the vacancy and carbon atom distri-

Tab

le 2

Shor

t-ra

nge

orde

r pa

ram

eter

s an

d di

stri

buti

on o

f ca

rbon

ato

ms

and

vaca

ncie

s in

car

bide

VC

,,,,,

idea

l or

dere

d ca

rbid

e V

,C,,

and

car-

bi

des

NbC

,,,,,

Tic,

,,,,

and

NbC

,,,,

coor

dina

tion

coor

dina

- V

C,,,,

[27]

v,c

, ~7

1

NbC

n.73

[31l

T

iC0.

76

r3l1

N

bCn.

sn [

32l

sphe

re*)

ti

on n

umbe

r

j (h

kl)

1070

K

1270

K

~ a,**)

"I

nc

nu

"I

nc

nn

"I

"c nu

"1

nc

nn

110

200

211

220

310

222

32 1

40

0

12 6 34

12

24 8 48 6

- 0.

178

0.17

6 0.

008

0.04

1

- 0.

260

-0.1

71

- 0.

070

0.14

3

10.6

1.

4 -0

.2

5.7

0.3

-0.2

14

.8

9.2

0.2

8.9

3.1

0 17

.3

6.7

0 7.

0 1.

0 -0

.2

38.5

9.

5 -0

.1

3.8

2.2

0.2

12 6 16

10

20 8 44

4

- 0.

095

- 0.

275

0.05

1 0.

072

0.04

4 - 0.

030

- 0.

020

0.03

0

9.6

2.4

5.6

0.4

16.6

7.

4 8.

1 3.

9 16

.7

7.3

6.0

2.0

35.7

12

.3

4.2

1.8

- 0.

005

0.01

3 0.

006

0.02

5

0.00

3

- 0.

080

- 0.

003

- 0.

007

9.2

2.8

4.9

1.1

18.0

6.

0 9.

1 2.

9 18

.3

5.7

5.9

2.1

36.7

11

.3

4.5

1.5

- 0.

089

0.04

4 0.

024

0.22

8

0.25

4

- 0.

158

-0.1

94

- 0.

053

- 0.1

31

- 0.5

07

- 0.0

97

- 0.

347

- 0.

299

- 0.

150

- 0.

102

0.01

8

*) V

acan

cy i

s a

cent

re o

f a

coor

dina

tion

sph

eres

; coo

rdin

atio

n sp

here

radi

us is

R, =

(a/

2) vh

Z +

k2 +

P, w

here

a is

the

par

amet

er o

f th

e f.c

.c.

**)

The

aut

hors

of

[32]

hav

e de

term

ined

the

sho

rt-r

ange

ord

er p

aram

eter

s w

rong

ly.

carb

on s

ubla

ttic

e in

MeC

,.

S hort-Range Order in Nonstoichiometric Carbides, Nitrides, and Oxides 25

butions in the nonmetallic sublattice as a consequence of the mutual repulsion of vacancies and noted that a similar character of short-range order should be observed in niobium and tantalum carbides. A conclusion as to the similarity of short-range order in nonstoichiometric group V transition metal monocarbides has been drawn also in [30]. Note that according to [30] i t is impossible to produce absolutely disordered (without short-range order) samples of vanadium carbide VC, with y > 0.78.

A study of short-range order by the method of diffuse neutron scattering by carbides Tic,,,, and NbC,., has been made in [31]. The distribution of the intensity of diffuse neutron scattering by single crystals of specified composition was measured at room temperature in the (110) plane of the reciprocal lattice in the diffraction vector range 2 nm-l < 2nq < 40 nm-l. Prior t o measurements, Tic,,,, and NbC,.,, single crystals had been annealed in vacuum for several days. The results obtained provide evidence that atomic displacements from the positions of an ideal undistorted lattice take place. Using the Fourier transformation, the authors of 11311 determined the short-range order parameters for eight nearest coordination spheres of the nonmetallic sublattice of carbides Tic,,,, and NbC, ,, (Table 2). From the data given in Table 2, it is seen that the character of the vacancy distribution in titanium carbide TIC,.,, and niobium carbide NbC0,73 is basically the same and is due to the mutual “repulsion” of vacancies, like in VC,. The authors of [31] note also that the correlations in niobium carbide span more than nine coordination spheres. The results of [31] are in agreement with the qualitatively similar results obtained in [30] in a diffuse neutron scattering investigation of NbC, powders.

The same method has been used by the authors of [32] in studying the short-range order in carbide NbC,,,,. The sample investigated was produced in two states. The ordered state of carbide NbC,,,, was produced as a result of annealing a t 1070K followed by quenching (according to [32 ] , the transition temperature to the ordered state, Ttrans, for NbC,.,, is equal to 1250 K which is in good agreement with the data [12, 261). Adisordered NbC,,,, carbide was produced as a result of annealing a t T = = 1270 K > TiranB followed by quenching. The short-range order parameters were determined for twelve nonmetallic sublattice coordination spheres (Table 2 presents values of 01 for eight coordination spheres). I n the sample annealed a t 1070 K, the type of short-range order corresponds to an ordered Nb,C, structure with a small value of the long-range order parameter. Quite clearly, in [32] a formal approximation of experimental data for a disordered carbide NbC,,,, has yielded short-range order parameters that go beyond physically admissible limits. For example, a, = -0.194, a, = 0.228, andn, = 0.254 (with an ordered Nb,C,-type phase forming in the carbide NbC,,,,, only the following short-range order parameters are possible : a5 = 0, -0.16 5 < n, < 0, and 0.16 2 01, > 0). For a disordered carbide NbC,,,, the authors of [32] httve obtained also several physically inadmissible values of 01 (e.g., (x2 = -0.507!). At the same time, the authors of [32] draw a fundamentally wrong conclusion about the presence of direct pairwise interactions of carbon atoms and formally calculated the energies of these interactions. However, this contradicts the numerous investigations according to which direct interactions of carbon atoms, ... C-C ..., are practically absent in the nonmetallic sublattice of carbides [ l to 61 and only indirect carbon atom interactions realized via metal atoms, i.e., ... C-Me-C ..., are possible.

and ThC,,,, samples [30] has permitted the determination of the correlations in the distribution of vacancies in three coordination spheres (with a vacancy a t the centre). According to these data, the number of vacancies present in the first coordination sphere is insignificant (less than in the case of a statistical distribution), vacancies are altogether absent in the second coordination sphere, and the third coordination sphere contains

A diffuse neutron scattering investigation of polycrystalline ZrC,

26 A. I. GCSEV

a considerably larger number of vacancies than in the case of a statistical distribution

The diffuse electron scattering method has been employed in studying short-range order in lower vanadium and niobium carbides V,C and Nb,C [33]. The type of order (long-range or short-range) in these structures is due to the distribution of interstitial atoms into regular C-0-C-0- ... series parallel to the c-axis. Comparing experimental and calculated values of diffuse scattering intensity has enabled the authors of [33] to ascertain possible types of order in Nb,C and V,C.

On the whole, available information on short-range order in the nonmetallic sublattice of nonstoichiometric compounds suggests that the correlations in the distribution of interstitial atoms and vacancies have received a fairly good study only in niobium and vanadium compounds, the most detailed study having been made of niobium and vanadium carbides. The character of short-range order in the nonmetallic sublattice of nonstoichiometric carbides is due chiefly to vacancy repulsion. In the carbides of group V transition metals the sites of the first two coordination spheres (with a vacancy a t the centre) are occupied preferentially by carbon atoms, whereas the vacancies reside preferentially in the third coordination sphere - this character of short-range order corresponds to the distribution of atoms in ordered Me&,-type phases. In group IV transition metal carbides the character of short-range order is associated with vacancies being practically absent in the second coordination sphere and present in large amounts in the third coordination sphere. Short-range order in lower hexagonal Me,X-type carbides and nitrides is due to the formation of X - ~ - X - ~ - series aligned with the c-axis.

( m1 - - - 0 . 0 5 , ~ ~ = -0.20,013 = 0.075).

2.3 Magnetic susceptibility and short-range order

The application of the magnetic susceptibility method for analyzing the atomic distribution in a crystal and determining short-range order dates back to [12, 34, 351 devoted to the study of nonstoichiometric niobium carbide, NbC,.

Measurements of the magnetic susceptibility x in niobium carbide samples of different compositions over a temperature range between 300 and 1300 K revealed x t o depend on the mode of heat treatment of NbC, carbides with compositions 0.81 5 I y 0.88. I n magnetic susceptibility measurements for NbC, samples quenched from temperatures well above the disorder-order transition temperature and display- ing no superstructural reflections in the neutron diffraction patterns, x lowered dis- continuously (Fig. 4) down to the susceptibility of ordered samples of the same com-

T O4 2 2 0 s P -04 R Y

-08

\

-72

Fig. 4. Influence of ordering on the magnetic susceptibility x of NbC, 83 (the arrows show the direction of temperature variation) : (1) quenched disordered state, (2) ordered state [35]

-%6

300 500 700 900 1700 1300 TlKl -


position produced following the procedure described in [12, 241. Subsequent to x ( T ) measurements, the neutron diffraction patterns exhibited superstructural reflections indicating an ordering transition to have taken place during the measurement. Also, measurements showed that the susceptibility value for samples containing simultaneously a disordered and an ordered phases coincides practically with that for NbCU samples of the same cornpositmion that contain only an ordered phase, i.e., the susceptibility is independent of the ordered phase content. Since the formation of long- range order is accompanied by the production of short-range order, the independence of x of the ordered phase content and, a t the same time, the variation of x during heat treatment, indicate that the magnetic susceptibility is determined mainly by the formation of short-range rather than long-range order.

The concept of the determining influence of short-range order on magnetic susceptibility has been corroborated by a calculation [12, 341 of the short-range order parameters from experimental data obtained in [la]. The gist of the computation was as follows. Originally, in keeping with the procedure employed in [34], the temperahre dependences of magnetic susceptibility for disordered and ordered NbC, samples of different compositions were used to estimate the quantities Pi(y), i.e., the probabilities that niobium atom positions exist in NbC, which have the i-th configuration of the carbon atoms and vacancies constituting the nearest environment of the metallic atom. According to [12, 341, to correctly describe the carbide MeC, in the composition range 1.0 2 y 2 0.80, it suffices to consider a total of four positions: a position with the metal atom being completely surrounded by six carbon atoms, this position has the probability Po; a position with one vacancy in the octahedral environment of the metal atom, the probability of this position is Pl; positions with two nonadjacent and two adjacent vacancies, these positions have the probabilities Pz and P3, respectively.

The short-range order parameter 011 in the j-th coordination sphere of the nonmetallic sublattice may, according to [12 to 141, be determined by the formula

where Pg?:j-'o and PF-'& are the probabilities of a C-• pair forming in the compound investigated and in a completely disordered compound, respectively. For carbide MeC, the binominal probability is Ac-uPF!~ = 2y( l - y), and the real probability of a C - 0 pair forming in the j-th coordination sphere can be found from the knowledge of Pt(y) by the formula

( 6 ) Ac-&l-o (j) = 2 n p i P t ( y ) , i

where nij) is the fraction of C - 0 pairs in the j-th coordination sphere of a position with an i-configuration, Ac-• = 2 the multiplicity of a C-• pair, and Ai the multiplicity of the position of the metal atom having i-configuration of the nearest environment composed of carbon atoms and vacancies. I n the case under consideration

and = A,Pl(y)/3 $- 2&pz(Y)/3 $- &P3(y)/2

~C-Opgln = 21Pi(y)/3 $- 2&p&)/3 . As a calculation has shown, the values of a1 and a, for disordered NbC, samples are close to zero, whereas the counterparts for annealed niobium carbide samples are negative and differ appreciably from zero. The negative values of 0 1 ~ and a2 point to

28 A. I. GUSEV

a mutual “repulsion” in 0-0 and C-C pairs; it is this repulsion that determines the character of short-range order.

The magnetic susceptibility method has been used also in [36] t o probe ordering in a nonstoichiometric tantalum carbide, TaCu. The temperature dependences of magnetic susceptibility for tantalum carbide samples in disordered and ordered states were similar to the x ( T ) relations obtained in [12, 34,351 for niobium carbide. The authors of [36] employed magnetic susceptibility data for tantalum carbide to calculate the short-range order parameters. According to the calculation results, the short-range order parameters in the first and second coordination spheres of the nonmetallic sublattice of an ordered tantalum carbide are negative (Table 3).

Table 3

a1 - 0.064 -0.132 - 0.153 - 0.059 - 0.063 - 0.080 - 0.040 -0.126 -0.137 - 0.043 - 0.080 - 0.086

For ordered Me,C,-type (MeC,.,,) phases the short-range order parameters a1 and a2 are equal to -0.20, in good agreement with the values obtained for NbC,.,, and

a certain discrepancy between the values of aI and a2 calculated from the magnetic susceptibility and the maximum possible values of these parameters indicates that no complete ordering was attained in the niobium and tantalum carbide samples investigated in [34 to 361.

On the whole, the results obtained in magnetic susceptibility studies [12, 34 to 361 of short-range order in carbides NbC, and TaC, are in good agreement, and also give a good fit to data of NMR investigations of short-range order in niobium carbide [2l to 231.

3. Short-Range Order in the Mctsllic Sublattice of Carbide Solid Solutions

The previous section dealt with short-range order in the nonmetallic sublattice of nonstoichiometric compounds. Of considerable theoretical and practical interest are the results of studies of the atomic distribution in the metallic sublattice of solid solutions of nonstoichiometric carbides of different transition metals.

The diffuse X-ray scattering method has been repeatedly employed in probing short-range order in the metallic sublattice of solid solutions formed by transition metal carbides close to stoichiometry. According to [37], the most substantial result in the investigation of short-range order in MeIC-MeIIC solid solutions is the establish- ment of the fact itself that the distribution of metal atoms in metallic sublattice sites is not random and the detection of the effect of carbon on the character of correlations in the arrangement of metal atoms.

Pioneer investigations of the local atomic distribution in the metallic sublattice of solid solutions NbC-TaC and ZrC-HfC of equimolar composition have been carried out in [38,39]. Theabsence of vacancies in the carbon sublattice and the closeness of the atomic radii of interchangeable metals (Nb and Ta, Zr and Hf) suggested the absence of static distortions in the lattice of the solid solution. The short-range order parameters for three nearest coordination spheres of the metallic sublattice were determined from the intensity of diffuse scattering. For the first and second coordination spheres the short-range order parameters turned out to be positive, the values of a1 and a2 increasing with decrease in quenching or annealing temperature (Table 4).


Tab le 4 Short-range order parameters in the metallic sublattice of carbide solid solutions and alloys

composition heat treatment short-range order parameters ref. and temperature a1 (K)

"2 "3

annealing, 2000 annealing, 1800 annealing, 1500 annealing, 1800 annealing, 1500 quenching, 1600 annealing, 1400 quenching, 2200 quenching, 1800 quenching, 2200 quenching, 1500 quenching, 1500 quenching, 1500 quenching, 2200 quenching, 1500 quenching quenching quenching, 2100 quenching quenching, 2100 quenching quenching, 2200 quenching, 2400 quenching, 2400 quenching, 2400 quenching, 2400 quenching, 2400 quenching, 2400 quenching, 2400

0.058 0.089 0.120 0.11 0.14 0.13 0.16 0.06 0.09

- 0.06 0.066 0.063 0.091

0.019 0.07 0.03

- 0.01 - 0.23

0.03 -0.10 - 0.06 - 0.09 - 0.07 - 0.06 - 0.05 - 0.06

- 0.06

0.05 0.07

0.025 0 0.049 0 0.071 0 0.10 0.02 0.13 0.04 0.09 0.02 0.08 - 0.04 0 0 0 0

0.103 0.047 0.017

0.03 0 0 0 0.02 0.01

- 0.02 - 0.02 0.04 0 0.16 0.04 0.15 0.03

The positive values of 0 1 ~ and a2 evidence that in the first and second coordination spheres of a metal atom of a given species the content of metal atoms of the same species is higher than that in the case of a statistical distribution, i.e., stratification occurs in the solid solutions investigated. Short-range stratification is observed also in Nb-Ta [40].

The short-range order parameter a3 for the third coordination sphere is negative and close to zero, so it may be assumed that the correlations in the arrangement of atoms in the metallic sublattice apply to only two closest coordination spheres. Addi- tional annealing of solid solution samples, and also doping them with 1 molyo WC, led to an enhancement of the correlations in the relative arrangement of metal atoms [38, 391.

In [all diffuse scattering was studied on equimolar solid solutions Tic-NbC, Tic-TaC, TiC-WC, VC-NbC, and VC-WC and on related equiatomic metallic Ti-% alloys, V-Nb (36 a t yo Nb), and V-W (36 a t % W). The short-range order parametera, turned out t o be positive for Ti-Ta alloys and negative for V-Nb and V-W alloys.

30 A. I. GUSEV

For all the carbide solid solutions investigated, except for VC-WC, the parameter a1 is negative, whereas for VC-WC a,> 0. A comparison of the short-range order parameters a, in carbide solid solutions and allied metallic alloys shows that the parameter a1 in Tic-NbC, VC-WC, and Tic-TaC solid solutions is opposite in sign to that in Ti-Nb [42], V-W, and Ti-Ta alloys (Table 3).

Short-range order in TiC-WC solid solutions has been investigated [43, 441. Diffuse X-ray scattering measurements were performed on samples of different composition subjected to heat treatment under slightly differing regimes (sintering a t different temperat,ures, quenching, furnace cooling, etc.). According to the results obtained, all samples of Tic-WC solid solutions with a complete carbon sublattice exhibit very strong short-range order in the first coordination sphere (a, < 0), the parameters a2 and a3 for the second and third coordination spheres being close to zero (Table 4). This means that correlations in the arrangement of titanium and tungsten atoms are realized a t small distances within one to three coordination spheres. In all the cases additional annealing of samples led to an increase in the degree of short-range order. No state with a metal atom distribution close to statistical was registered.

Increasing the tungsten content of Tic-WC solid solutions entails an enhancement of the degree of short-range order in the first coordination sphere. The character of the distribution of metal atoms is largely affected by the carbon content. With carbon content departing from stoichiometry the short-range order parameters for three nearest coordination spheres become positive (Table 4). A t the same time, the character of diffuse X-ray scattering by noiistoichiometric solid solution suggests the presence in the structure of two short-range order types, one with a1 < 0 and the other with a, > 0. In this context, the authors of [43, 441 assume that along with the formation of segregations of metal atoms of the same species, regions of short-range order with a structure typical of a stoichiometric solid solution are retained in a nonstoichiometric solid solution Ti,WI-,C,.

In 11451 the type of short-range order in the metallic sublattice of unrestricted carbide solid solutions was placed in correspondence with the calculated value of maximum elastic strain energy, En,. It turned out that in solid solutions Me:Me:I,C (0.56 2 2 x 2 0.45), for which the short-range order parameter in the first coordination siheie is negative (a1 < 0), the maximum strain energy Em was larger than 3.7 kJ/mol. A positive short-range order parameter, OL > 0, is typical of carbide solid solutions with small En,. For example, for solid solutions of the systems ZrC-HfC and NbC-TaC

0 (Table 4) and En, is equal to 0.5 and 0.1 kJ/mol, respectively [45]. The conclusions drawn in [45] regarding the relation of the type of short-range order in unrestricted stoichiometric carbide solid solutions to the value of maximum elastic strain energy are tentative in character and call for further support. At the same time, the attempt 1451 to establish such an interrelation is very fruitful.

4. Local Atomic Displacements

I n the disordered state, nonstoichiometric compounds are characterized by a random interstitial atom and vacancy distribution in nonmetallic sublattice sites and therefore do not possess the translational symmetry of a crystal of stoichiometric composition. However, a random distribution of interstitial atoms in the sites of the nonmetallic sublattice means that the probabilities of its sites to be filled are equal; as a consequence, all the nonmetallic sublattice sites are crystallographically equiv- alent. Put another way, disordered nonstoichiometric compounds have the translational symmetry of some site-occupation-probability lattice.


Most of the nonstoichiometric compounds possess wide regions of homogeneity and have a high-symmetry B1-type (NaC1) structure that is preserved when the distribution of interstitial atoms and vacancies is disordered. A disordered distribution results in the nonconservation of the symmetry of the local environment of each atom. In turn, a local-symmetry distortion gives rise to static displacements of both metal atoms and interstitial atoms. Static displacements imply deviations of atoms from positions corresponding to the ideal basis structure of a compound, i.e., imply some quantity un = R, - r,, where Rn and F, are the radius vectors defining the position of an atom n in a real and an ideal lattice, respectively. In contradistinction to dynamic displacements arising from thermal atomic oscillations, static displacements are a consequence of the asymmetric angular distribution of bonds between adjacent atoms and of differences in the energies of these bonds.

The cubic symmetry of disordered nonstoichiometric compounds allows only a spherically symmetric distribution of static atomic displacements, for otherwise the crystallographic equivalence of different crystal lattice sites would be perturbed. I n disordered nonstoichiornetric compounds a symmetric distribution of static displacements is ensured by a random vacancy distribution. Allowing for the radial distribution of static displacements and exploiting the analogy with dynamic displacements, we may proceed to consider mean-square static displacements.

During ordering, a lowering of crystal symmetry occurs which is accompanied by a symmetry change in the local environment of atoms (note that for an ordered phase of stoichiometric composition the symmetry of the local environment coincides with that of the crystal). Since, in the ordered state, the symmetry of the local environment is not cubic, static displacements of atoms arise during ordering. The amount of static displacements for the disordered and ordered states of the crystal is a t variance, as the local environments of atoms in these states do not coincide.

Determining the amount and direction of static displacements for each atom in nonstoichiometric compounds with different structural states permits a specification of equilibrium positions, i.e., enables one to determine the crystal structure accurately and fully. Comparison of the amount of static displacements corresponding to different structural states of the crystal allows one also to ascertain possible mechanisms of an order-disorder phase transition and to analyze the behaviour of interparticle interactions in the crystal during ordering.

Major methods employed to determine atomic displacements are X-ray and thermal neutron diffractions. There is no fundamental difference between these techniques, since elastic neutron scattering on atomic nuclei and X-ray scattering on static crystal imperfections are described essentially by the same formulae and may be treated in terms of a unified theory. At the same time, a difference in the parameters of the theory may lead to very important advantages of the procedure.

As is known, the amplitudes of X-ray scattering by individual atoms are determined by the number of electrons in the atoms and increase monotonically with increasing number of the element in the Periodic Table, whereas the amplitudes of neutron scattering by nuclei of different atoms vary irregularly. As applied to nonstoichiometric compounds, such as transition metal carbides and nitrides, this means that for X-ray scattering a major contribution to the intensity of diffraction peaks will be made by transition metal atoms, so information about the nonmetallic sublattice is practically lost when radiography is used to determine displacements. I n the neutron diffraction method the amplitudes of atomic scattering of neutrons by various con- stituents of nonstoichiometric compounds are comparable in magnitude (e.g., the neutron scattering amplitudes for zirconium, hafnium, niobium, tantalum, carbon, and nitrogen are equal to 0.71 x 10-l2, 0.78 x 0.71 x 0.70 X

32 A. I. GUSEV

0.665 x cm, respectively). Therefore, when thermal neutron diffraction is used, a contribution to the intensity of the reflections observed is made by both metallic and nonmetallic atoms. It might be wellpointed out that the neutron diffraction technique of determining displacements is, generally speaking, more accurate than the radiographic method. This is because the neutron scattering ampli- tude is independent of the scattering angle and no correction for extinction needs to be introduced ; apart from this, the absorption of neutrons by a substance is considerably lower than that of X-rays. Thus, in the investigation of nonstoichiometric transition metal compounds the neutron diffraction technique for determining displacements is to be preferred.

The diffraction line intensity is weakened by both dynamic (thermal) and static displacements of atoms from the sites of an ideal crystal lattice; in the absence of ordering both displacement types cause the intensity to decrease by the same law. Indeed, the presence of dynamic displacements makes the atom positions depend on time; but since the frequency of the radiation scattered is several orders of magnitude higher than the frequency of thermal oscillations of atoms, it may be assumed that scattering occurs on fixed atoms occupying some instantaneous positions. From this, it follows that for a diffraction experiment there are no fundamental differences between dynamic and static displacements. One just needs to take account of the superposition of these displacements ; as a result, the spectrum intensity will depend on the sum of their mean-square values,

and 0.94 x

(7) 2 2

(UZ) = (Ustatie) + (~2dyn) . Isolation of dynamic displacements from the total ones is possible, e.g., when using the temperature dependence of diffraction reflection intensities.

Currently, static displacements have been determined for only several carbides and nitrides, with nearly all of the investigations performed on disordered compounds.

In [46] the X-ray diffraction method was employed to determine displacements of the atomic complex ( u ~ M ~ x , ) in the regions of homogeneity of titanium, zirconium, and hafnium carbides (in studying nonstoichiometric compounds whose atoms have scattering amplitudes close in magnitude, i t is practically impossible to find displacements for atoms of each species ; therefore one determines the mean-square displacement of the atomic complex (&MeXy), i.e., displacements averaged over both sub- lattices). Measurements were carried out a t room temperature, when (according to [46]) the thermal oscillations of atoms are small and dynamic lattice distortions may be neglected. For this reason, the authors of 11461 set (UiMeX,) = ( u;t,tic ~\z~x,). Accord- ing to the findings, as the defect content of the carbon sublattice is increased, the static lattice distortions in the carbides investigated enhance appreciably (Table 5). The largest static displacements observed are those of titanium carbide; in going to zirconium and hafnium carbides the displacements decrease in amount. Analyzing the mode of variation of the crystal lattice periods and the displacements as a function of carbide composition permitted to conclude that a more dramatic change in crystal lattice period is accompanied by a larger value of static displacements.

The neutron diffraction method was employed in [47] to determine static distortions in titanium, zirconium, vanadium, niobium, and tantalum carbides. Effective mean-square displacements were found from experimental diffraction reflection intensity data. For carbides close to stoichiometry the static distortions should be small, as the lattice is practically complete; therefore (&MeC) = ( u & , ~ M ~ c ) for stoichiometric compositions. Support for this assertion comes from the good agreement between the measured values of (ugMec) and the values of the quantity ( U & & ~ ~ C ) ,

calculated in [48] from elastic constants in the Debye approximation (/=) equals


T a b l e 5 Mean-square displacements rv) ( 10-1 nm) in transition metal nonstoichiometric compounds MeX,

composition V<UiMeXy) V<uitatic MeX,) ra ym ref.

0.160 ~461 TiCO.EE 0.184 ~461 Tic,. 60 0.199 ~461 TiCn.ge 0.084 [471 TiCn.m 0.104 0.065 [471 TiCn.81 0.119 0.087 [471 TiCn.6, 0.142 0.118 [471 TiCn.ge 0.103 0.084 0.109 [521 ZrCn.sn 0.066 [461 ZrCn.7, 0.120 ~461 zrCn.eo 0.138 ~461 ZrCn.9, 0.075 [471 ZrCn.9, 0.101 0.067 [a71 Zrcn.87 0.110 0.080 1471 ZrCn.e, 0.142 0.120 [471 ZrCn.6, 0.147 0.126 [a71 HfCo.9, 0.080 c461 HfCo.7, 0.091 ~461 Hfc,.m 0.095 ~461 VCn.a5 0.096 0.045 C471 VCn.814 0.106 0.063 [471 NbC1.0 0.081 [471 TaC1.n 0.091 1471 TiN0.98 0.089 0.078 0.100 [491

ZrNi .n 0.085 0.061 0.112 [491 HfN0.!38 0.081 0.049 0.126 fa91

TiNn.98 0.105 0.081 0.096 c521

TiCO.**NI, 1 0.107 r521

8.1 pm for TiC, 6.9 pm for ZrC, and 7.2 pm for NbC). Starting from the assumption that the Debye temperature depends weakly on carbide composition, the authors of [47] assume that dynamic lattice distortions change insignificantly with decreasing carbon content and that, to a first approximation, in MeC, the composition dependence of these distortions may be neglected. This allows the calculation of the static distortions,

(8 ) 2 2 2 2 2

(UstaticMeC,) = (%MeCv) - (UdynMeC) (UXMeC,) - (UZMeC) . The values of displacements as obtained in [47] for titanium carbide were appreciably smaller than those determined in [46] by the X-ray method; for zirconium carbide the results [46, 471 are in better agreement (Table 5).

In [49] the neutron diffraction method was used to determine the mean-square displacements (u&,a), (&Me), and (u&) in nearly stoichiometric nitrides TIN,.,,, ZrN,,,,, and HfN,,,, (Table 5). The results [49] on (u;) for titanium and zirconium nitrides give a fairly good fit to the value of dynamic displacements calculated [50] for these compounds from elastic constants in the Debye approximation (/(u&) = = 8.3 pm for TiN and is 8.0 pm for ZrN). 3 physica (b) 156/1

34 A, I. GrrsEv

The values and directions of relative static atomic displacements in carbides ZrC,.,, [30], Tic,.,,, and NbC,.,, 1'311 were determined when investigating diffuse neutron scattering. In carbide ZrC,,,, zirconium atoms are displaced from vacancies by 4.0 pm. In titanium and niobium carbides metal atoms are displaced from a vacancy toward a carbon atom by 3.0 and 3.3 pm, respectively, whereas carbon atoms withdraw from each other (in the first coordiiiation sphere the spacing between adjacent carbon atoms increased by 1.1 pm for Tic,.,, and by 0.6 pm for NbC,.,,).

Local displacements of atoms located around a carbon vacancy in niobium carbide NbC, (0.95 2 y 2 0.88) were determined in [51] in an X-ray investigation of single crystals. To separate dynamic and static displacements, X-ray spectra of carbides NbC,,,, and NbC,,,, were obtained a t two different temperatures, 299 and 479K. That enabled the authors to find for the above temperature interval the mean thermal Debye-Waller factor which was equal to 2M = In ( ITJIT,) . The value of dynamic displacements (u&,), according to [all, decreases with increasing number of vacancies in the carbon sublattice of NbC, and may be described by a linear dependence (( z i 2 )

in pmz),

(212yn NbC,) = 27 - 50( 1 - y) . (9)

As X-ray diffraction enables one to extract information only on the metallic sublattice, the authors of [51] assumed that a vacancy is surrounded by six niobium atoms in the first. coordination sphere and by eight niobium atoms in the second, whereas the metal atom displacements in the subsequent coordination spheres do not lead to alterations of the X-ray diffraction patterns. It was also assumed that the vacancies are separated widely enough and therefore the position of niobium atoms is affected by a single vacancy only. Note that this assumption holds good only for niobium carbide compositions close to stoichiometry, but practically does not work for the NbC,,,, sample investigated in [61].

The value of stat,ic displacements of niobium atoms for the first and second eo- ordination spheres was (&,tic)l = 1250 pm2 and (&,tic)2 = 90 pm2, respectively. The directions of displacements were also found: the niobium atoms in the first coordination sphere are displaced from the vacancy, while in the second coordination sphere they approach it. Although the niobium atom displacements in the second coordination sphere are appreciably smaller than those in the first coordination sphere, the authors of [51] suppose that the displacements (&atic)2 compensate for the displacements in the first coordination sphere and lead to a decrease in the lattice period of niobium carbide as the vacancy concentration is increased. Such an inter- pretation of experimental evidence is challengeable, since the coordinates of carbon atoms have not been determined in the paper under discussion and a complete body of information on the position of niobium atoms is absent, i.e., there is little or no information about the crystal lattice period.

A neutron diffraction determination of the displacements ( u ~ M ~ x ~ ) , ( u & ~ ~ ) , and ( U ~ X ) in titanium carbide TIC,,,,, titanium nitride TIN,,,,, and titanium carbonitride TiCo.48No,51 has been carried out in [52 ] . The paper quotes values of the quantity B = 87c2(~%)/3, proceeding from which we calculated the values of the mean-square displacements (z&) (Table 5). The values of displacements for titanium carbide and nitride agree well with the data of [47, 491, which were obtained also from a neutron diffraction experiment. The authors of [52] note that when investigating titanium carbide, they did not reveal any deviation of the diffraction reflection intensities from those calculated in the model of random displacements of atoms from the positions of an ideal lattice.


Detailed neutron diffraction studies of the atomic displacements in the region of homogeneity of disordered niobium carbide NbC, (0.75 5 y 5 1.00) have been made in [5, 12, 53, 541.

The presence of structural vacancies in a nonstoichiometric niobium carbide causes a distortion of the symmetry of the local environment of crystal lattice sites and leads inevitably to the appearance of static displacements of both metallic matrix atoms and carbon atoms. When special heat treatment conditions [12] are used to ensure a random distribution of structural vacancies in the crystal lattice of niobium carbide, the crystal lattice symmetry of this carbide remains the same as that of a defect-free carbide NbC,,,,, i.e., a disordered nonstoichiometric carbide NbC, has some “mean” high-symmetry lattice, the parameters of which vary as a function of composition. In this case t,he static distortions taking place may be represented in terms of mean- square static atomic displacements referred to the sites of such a “mean” lattice.

An X-ray diffraction determination [12] of the concentration dependence of the parameter of a cubic unit cell in disordered carbide NbC, (0.75 5 y 5 1.00) has shown that this dependence exhibits a substantially nonlinear behaviour and has an inflection point in the region of the composition NbC,.,, (Fig. 5, curve 1).

To determine mean-square static and dynamic displacements, the authors of [53,54] employed the neutron diffraction method. The amplitudes of neutron scattering by niobium and carbon atoms are close in magnitude, and in this case the influence of disordered atomic displacements on the diffraction effects may be taken into account by using a Debye-Waller factor averaged over atoms of different species. I n the first approximation the quantity M involved in the Debye-Waller factor may be viewed as a linear function of displacements ( t i : ) ,

M = f $q2(&) , (10)

q = I Q / = 2{sin [€J(hkZ)]}/A being the length of the reciprocal lattice vector. However, measurement of the intensity of diffract,ion peaks in neutron diffraction

patterns of disordered NbC, samples and plotting M N ~ c ~ = f(q2) relations revealed this function to be nonlinear in the region of large q-values [53, 541, the deviations from linearity occurring toward increasing M for all the niobium carbide samples in-

1 0.447

- 0.446

2 s Q

80 0445 - -

- 0.444

0443 0 0.1 0.2 0.3

I-y - Fig. 5. Dependences of the crystal lattice period (l), dynamic (2), and static (3) mean-square atomic displacements on the structural vacancy content in disordered niobium carbide NbC, ; (4) static displacements approximated by (14) for small vacancy content [54]

3’

36 A. I. GUSEV

vestigated. Allowing for the fact that deviations of the dependence MNbC, = f ( q 2 ) occur also for a stoichiometric niobium carbide, the authors of [53, 541 attributed the observed deviations to the anharmonicity of dynamic vibrations of atoms in niobium carbide. This conclusion is consistent with the findings for the heat capacity of a disordered niobium carbide studied in [55], where the presence of appreciable anharmonic effects a t T = 300 K was noted.

To separate the contributions of dynamic and static atomic displacements to <&), the authors of [5,53,54] used the concentration dependence of dynamic displacements. An attempt to determine this dependence was undertaken in [51] ; according to [51], the composition dependence of dynamic displacements is given by (9). However, as has already been stated, (9) found in [51] applies rather to dynamic displacements of metallic sublattice atoms. Incomplete allowance for the displacements of the lighter carbon atoms led to highly underrated values of dynamic displacements.

Earlier [5 , 551, in the investigation of the low-temperature heat capacity of a disordered niobium carbide, a superposition model was used which took into account both acoustic and optical vibrations. I n terms of this model the distribution function of atomic vibration frequencies has the form

where rs is the number of atoms in the primitive cell of the crystal, 6 is the Dirac delta function, and wD and wE are the Debye and Einstein frequencies.

By carrying out integration over all possible frequencies, the authors of [5] obtained a formula that contained the contributions of acoustic and optical vibrations to dynamic displacements of atoms,

with m = ( m ~ b + ymc)/(l + y), mNb, mc the masses of niobium and carbon atoms;

@(z) = (l/z) J’ 1: dz/(exp 1: - I) , 8 D and €iE the characteristic Debye and Einstein

temperatures. The concentration dependence of dynamic displacements (in pm2) of niobium and

carbon atoms in carbide NbC,, calculated in [53, 541 with the use of (12) and the values of OD and OE obtained in [5, 551, reads

X

0

2 (UdynNbC,) = 106 - 36(1 - y) ,

and attests t o larger values of dynamic displacements than reported in [51]. According to (13), the value of dynamic displacements decreases with increasing vacancy concentration in the niobium carbide structure (Fig. 5, curve 2). This may be explained by the fact that the contribution of carbon atom displacements varies as a result of a decrease in carbon content in nonstoichiometric niobium carbide.

The availability of data on the total mean-square atomic displacements ( u i ) in NbCy and the concentration dependence of dynamic displacements (13) permitted the determination of the dependence of mean-square static displacements ( u&,tic) on the composition of niobium carbide (Pig. 5 , curve 3). Inspection of Fig. 5 shows that


the static displacements increase in value as the composition of niobium carbide deviates from stoichiometry, a local minimum being observed in the region of the carbide composition NbC,.,,. This minimum arises apparently from the presence of short-range order in the arrangement of carbon atoms and vacancies; in [53, 541 this order could not be avoided even by quenching of the samples. Note that the composition range where an (&,tic) minimum is observed corresponds to a region in which, according to [12, 24 to 261, an ordered monoclinic Nb,C, phase forms after prolonged annealing.

According to [56], in the region of small defect concentrations and in the absence of correlations in the relative arrangement of atoms, the dependence of the amount of mean-square static displacements on the concentration of defects inducing static displacements, c, has the form

where No = R-l(dV/V dc)2, R-1 is a proportionality coefficient, a the lattice parameter, and V the unit cell volume.

In niobium carbide NbC, structural vacancies are the defects that cause static lattice displacements, so c = (1 - y)/2. By substituting the values of the NbC, cubic lattice parameter into (14) and performing numerical differentiation of the cubic cell volume with respect to the defect concentration, it may be shown that (14) approximates well the observed static displacements only in the composition range 0.97 < < y < 1.00 (Fig. 5, curve 4). The value of R-l determined in [54] for this range of niobium carbide compositions is approximately two orders of magnitude higher than those calculated in [56] for interstitial nontransition metal alloys. This testifies to larger static atomic displacements as compared with alloys and to the predominance of strong localized interactions in niobium carbide.

Examination of Fig. 5 shows also that for large defect concentrations the experimental values of displacements become appreciably less than the theoretical ones. This is probably because the overlap of point-defect-produced perturbations was disregarded in [56] in the derivation of (14) ; such an overlapping occurs in niobium carbide NbC, in the case of a large departure from stoichiometry and should neces- sarily lead to a growth rate of static displacements decreasing with vacancy concentration.

5. Conclusion

As follows from the material presented, the local distribution of atoms in nonstoichiometric compounds has been the subject of intensive research in the past ten or fifteen years. Results of investigations of the structure of nonstoichiometric transition metal compounds and their solid solutions by diffraction, resonance, and other experimental techniques show that short-range order in the relative arrangement of atoms is of sufficiently frequent occurrence in these substances. Particularly successful has been the application of the nuclear magnetic resonance method featuring a high sensitivity to minor symmetry changes in the nearest environment of the atoms on which resonance is observed. Magnetic susceptibility studies of short-range order in nonstoichiometric compounds must also be noted ; earlier the magnetic susceptibility method was not used for this purpose.

An analysis of the experimental data available shows that the character of short- range order in nonstoichiometric compounds is due chiefly to the mutual “repulsion”

38 A. I. GUSEV

of vacancies or like interstitial atoms. An indication of this is the negative sign of the short-range order parameters for the nearest coordination spheres of the nonmetallic sublattice of the compounds discussed. Investigations of short-range order in the metallic sublattice of carbide solid solutions have revealed the presence of correlations in the relative arrangement of metallic atoms and shown also that the carbon content affects not only the value, but also the sign of the short-range order parameter for the first coordination sphere of the metallic sublattice.

Meanwhile it should be noted that so far a considerable share of nonstoichiometric compounds has remained outside the scope of attention of researchers. The local distribution of atoms has been studied in most detail in nonstoichiometric vanadium compounds, viz., carbides, nitrides, oxides, and also in their solid solutions. A practically complete body of information about short-range order, obtained by use of different experimental methods, is available for nonstoichiometric niobium carbide. The atomic distribution in the crystal lattice of ordered Me,C,-type (Me-V, Nb, Ta) carbides has been treated in detail in [57]. According to [57], superstructures such as Me&, are similar in the character of short-range order in the first, second, and third coordination spheres, which are formed by nonmetallic sublattice sites around a metal atom. As concerns other nonstoichiometric compounds, investigations of short-range order in these are represented by a few papers. Carbides and particularly nitrides of group IV transition metals are least studied. One of the reasons for this is apparently the inapplicability of the NMR method for investigating these compounds, as the atomic nuclei in group I V metals possess no magnetic moments. Therefore, in studying nonstoichiometric compounds of group IV transition metals, it is expedient t o apply as widely as possible diffuse X-ray, neutron, and electron scattering methods, and also the NMR of 13C and 14N nuclei.

The nonconservation of the symmetry of the local environment of atoms in nonstoichiometric compounds leads to the occurrence of static displacements of both metallic and interstitial atoms. The amount and direction of static displacements are different for compounds in the disordered and ordered states. Comparison of the displacements that correspond to different structural states of a compound permits a better understanding of the character of interparticle interactions. However, experimental research in this direction has thus far been restricted to disordered nonstoichiometric compounds.

Thus, experimental investigations of the local atomic distribution in nonstoichiometric compounds have been launched in the past decade, but the problem of ex- ploring short-range order in these compounds is far from being resolved. This applies primarily to ordered phases of nonstoichiometric compounds, and also to nonstoichiometric compounds synthesized by simultaneous application of high pressure and temperature, i.e., under thermobaric conditions. Producing ordered phases by a redistribution of atoms in the lattice [4, 51 and also thermobaric synthesis of nonstoichiometric compounds, which is associated with the filling of structural vacancies with atoms [4, 58 to 611, are new promising methods for appropriate synthesis of materials with tailored properties. Investigations of short-range order and of atomic displacements will help to ascertain the mechanisms of transformations that occur in these substances, and also to elucidate the peculiarities of interatomic interactions and the effect of short-range order on the properties of the compounds concerned. Major experimental techniques to be employed as extensively as possible in studying short-range order in nonstoichiometric compounds are high-sensitive NMR spectroscopy, diffuse X-ray, neutron, and electron scattering, and structural neut-ron diffraction analysis.

Short-Range Order in Pu’onstoichiOmetric Carbides, Nitrides, and Oxides 39

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40 A. I. GUSEV: Short-Range Order in Nonstoichiometric Carbides, Nitrides, Oxides

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Documents

Short-Range Order in Nonstoichiometrie Transition Metal Carbides, Nitrides, and Oxides