A. I. GUSEV and A. A. REMPEL: Local Static and Dynamic Atomic Displacements 453

phys. stat. sol. (b) 154, 453 (1989)

Subject classification : 81.50 ; 61.12 ; S1.61

Institute of Chemistry, Academy of Sciences of the USSR, Ural Branch, Sverdlovskl)

Local Static and Dynamic Atomic Displacements in Disordered Niobium Carbide

BY A. I. GUSEV and A. A. REMPEL

A neutron diffraction study is made of disordered niobium carbide NbC, (1.00 2 y 2 0.75) in the region of its homogeneity. The dependences of the mean-square static and dynamic displace- ments of atoms on the composition of niobium carbide are determined. It is suggested that in niobium carbide localized metal-carbon bonds predominate over delocalized metallic interactions.

Es wird eine Neutronenbeugungsuntersuchung an fehlgeordnetem Niobiumkarbid NbC, (1,OO 2 2 y 2 0,75) im Homogenititsbereich durchgefuhrt. Die Abhingigkeiten der Mittelwerte der statischen und dynamischen Verschiebungen der Atome von der Zusammensetzung des Niobium- karbids werden bestimmt. Es wird angenommen, daD in Niobiumkarbid lokalisierte Metall- Kohlenstoffbindungen uber delokalisierte metallische Wechselwirkungen vorherrschen.

1. Introduction

Disordered nonstoichiometric interstitial compounds MeX, (transition metal carbides and nitrides) are characterized by a random distribution of interstitial atoms and vacancies at nonmetallic sublattice sites and therefore do not possess the translational symmetry of a crystal of stoichiometric composition MeX,.,,. However, the random distribution of interstitial atoms a t nonmetallic sublattice sites means that the prob- abilities of these sites to be filled are equal; as a consequence, all sites of the nonmetal- lic sublattice are crystallographically equivalent. Thus disordered nonstoichiometric compounds exhibit the translational symmetry of some lattice of site occupation probabilities.

Most of the nonstoichiometric MeX, compounds possessing wide regions of homo- geneity have a high-symmetry B1-type structure which is preserved when the inter- stitial atoms and vacancies are distributed randomly. However, the formation of structural vacancies and their disordered distribution lead to a distortion of the sym- metry of the local environment of each single atom and give rise to static displace- ments of both metallic and interstitial atoms. Static displacements are deviations of atoms from positions corresponding to the ideal structure of a compound. Note that the cubic symmetry of disordered nonstoichiometric compounds with B1 structure allows only a spherically symmetric distribution of static atomic displacements, otherwise the crystallographic equivalence of different crystal lattice sites would be disturbed. In disordered nonstoichiometric compounds a symmetric distribution of static displacements is assured by a random distribution of vacancies in the lattice. Determining the amount and direction of displacements provides a more accurate description of the crystal structure of compounds.

The purpose of the present paper is to investigate atomic displacements in a dis- ordered nonstoichiometric niobium carbide NbC,.

l ) GSP-145, Pervomaiskaya 91, SU-620219 Sverdlovsk, USSR.

454 A. I. GUSEV and A. A. REMPEL

2. Experiment

Samples of niobium carbide NbC, in the disordered state were synthesized by sintering at a temperature above 1800 K followed by quenching to room temperature a t a cool- ing rate of 100 to 300 K min-l. Such heat treatment assured a random vacancy dis- tribution in the niobium carbide crystal lattice, the symmetry of which remained the same as that of the lattice of defect-free (vacancy-free) niobium carbide NbC,,,. Thus a disordered nonstoichiometric niobium carbide NbC, has some “mean” high-symmetry lattice. In this case the static distortions that occur may be represented in terms of mean-square static atomic displacements referred to the sites of this “mean” lattice.

A radiographic determination of the dependence of the “mean-lattice’’ parameter on the composition of disordered niobium carbide NbC, (1.0 2 y 2 0.75) showed this dependence to be substantially nonlinear and to have a point of inflection in the region of carbide NbC,,,, (Fig. 1, curve 1).

Mean-square static and dynamic displacements of atoms from positions of the “mean” lattice were determined by neutron diffraction (wavelength of the mono- chromatized neutron beam 1 = 0.1643 nm). A neutron diffraction determination of displacements is more accurate than radiography, as the neutron scattering amplitude is independent of the scattering angle and is determined only by the species of the site- filling atom; apart from this, using neutrons permits one to neglect the effect of ex- tinction on the intensity of diffraction reflections.

Diffraction reflection intensity slackening is due to both dynamic (thermal) and static displacements of atoms from the sites of the ideal crystal lattice. In the absence of ordering both displacement types cause the intensity to slacken by the same law. From this follows that in a diffraction experiment there is no fundamental difference between dynamic and static displacements and one need only allow for their super- position. As a result, the spectral intensity will depend on the sum of the mean-square values of these displacements,

(1)

For nonstoichiometric MeX, compounds with B1 structure the intensity of diffrac- tion reflections in the neutron diffraction pattern of a polycrystalline specimen may

<uE> = ( 4 t a t i c ) + ( ~ & n > .

(I-y., - Fig. 1. Dependences of (1) crystal lattice period, ( 2 ) dynamic and (3) static mean-square atomic displacements on the structural vacancy content in disordered niobium carbide NW,; (4) static displacements approximation by (14) for small vacancy content [l]

Local Static and Dynamic Atomic Displacements in Disordered NbC, 455

be represented as

Iexp = KLPF2 (2) with K being the instrumental constant, L = (sin 0 sin 20)-l the Lorentz factor which includes the geometry of the neutron diffraction experiment, P the multiplicity factor equal to the number of reciprocal lattice vectors having the same length as the dif- fraction vector.

The structure factor F2 which is the square of the structure amplitude F , reflects a specific arrangement of atoms in a unit cell of the crystal. The structure amplitude (with allowance for the displacements of atoms from the sites of the ideal lattice) may be written in the form

where xj, yj, and zi are the coordinat,es of the j-th atom in a unit cell, and f i is the ampli- tude (atomic factor) of scattering by the j-th atom. The last cofactor in equation (3) is the Debye-Waller factor. The summation in (3) is carried out over all sites of the unit cell. For nonstoichiometric MeX, compounds with B1 structure (3) transforms to

The + and - signs refer to reflections from planes with even and odd Miller indices, respectively.

Expressions (2) and (4) enable us to find the quantities M for atoms of each species by solving the system (4) for even and odd Miller indices. In practice, such a separation is feasible only provided the amplitudes of neutron scattering by metal and nonmetal atoms differ appreciably, otherwise the structure amplitudes F and accordingly t,he experimental intensities of reflections from atomic planes with odd Miller indices are so small that the accuracy in determining the intensity of such reflections proves insufficient to calculate the quantities M . I n this context, when probing nonstoichio- metric compounds with f M e and f x close in magnitude, one determines the quantity MM~x, averaged over both sublattices. If the scattering amplitudes f m e and fx are close in magnitude and lMbfe - Mx/ < 1, the structure amplitude F may be repre- sented as

F = 4(fMe k Y f x ) exp (-MiVeX,) . (5) The amplitudes of neutron scattering by niobium and carbon atoms are close in

magnitude ( f ~ b = 0.71 x 10-'2 and f c = 0.665 x 10-l2 cm), so the influence of dis- ordered atomic displacements in niobium carbide on diffraction effects may be taken into account by using the Debye-Waller factor averaged over different atoms. Allow- ing for (2) and (5), the intensity of diffraction reflections with even Miller indices for a disordered niobium carbide reads

where 0(hkZ) is the scattering angle and exp (-2MN~cy) the mean Debye-Waller factor for carbon and niobium atoms. Since the quantity M involved in the Debye-Waller factor is proportional to the displacements (ug) , an analysis of experimental diffrac- tion reflection intensity data by used of (6) permitted the determination of the value of total displacements averaged over niobium and carbon atoms, (u:) z f [ (u&b) + + < u ~ c ) l .

456 A. I. GUSEV and A. A. REMPEL

3. Discussion

The integral intensities of diffraction reflections with even Miller indices as determined from experimental neutron diffraction patterns of disordered niobium carbide NbC, are given in Table 1.

To a first approximation, the quantity M involved in the Debye-Waller factor may be viewed as a linear function of displacements (&) [l], i.e.

where q = IqI = (2 sin [O(hkZ)]}/A is the length of the reciprocal-lattice vector. How- ever, measurements of diffraction peak intensities in neutron diffraction patterns of dis- ordered carbide NbC, and plotting M N ~ Q , = f (q2) curves revealed that these depend- ences exhibit nonlinear behaviour in the region of large values of q2, with the depend- ences deviating from linearity toward increasing M’s for all the niobium carbide sam- ples investigated.

The nonlinearity of the function M N ~ J C , = f ( q 2 ) may arise both from the nonadditiv- ity of static (u;ttatic) and dynamic (u&) displacements and from the presence of an- harmonicity in the atomic vibrations. In these cases, according to [l],

As an analysis of (8) shows with nonadditive static and dynamic displacements the term depending on q” of degrees higher than second is negative for all the niobium carbide samples studied. As the observed deviation of M from linearity is positive, it cannot be explained by the nonadditivity of static and dynamic displacements. Allow- ing for the fact that the function M N ~ c , = f ( q 2 ) deviates from linearity also for a stoichiometric niobium carbide NbCl.,,, the deviations observed may be attributed to the anharmonicity of dynamic atomic vibrations in niobium carbide.

To separate the contributions of static and dynamic atomic displacements to (uz) one can use the concentration dependence of dynamic displacements. An attempt to determine this dependence was undertaken in [2], the authors of which investigated the effect of temperature on the integral intensity of diffraction reflections observed in neutron diffraction patterns of niobium carbide NbC,. According to [2], the depend-

Table 1 Relative intensities I(hkZ) of diffraction reflections (hkZ) in neut’ron diffraction patterns of disordered niobium carbides NbC,

composition I(hkZ)

I(200) I(220) 4222) I(420) I(422)

590 645 869 903 898 930

1000 997 846 927 874 916 947 1000 853 894

321 423 436 458 407 406 45 1 417

209 277 274 304 272 274 279 249

527 1000 987 995

1000 974 986 939

1000 892

1000 991 969

1000 918

1000

Local Static and Dynamic Atomic Displacement's in Disordered NbC, 457

ence of dynamic displacements on the composition of niobium carbide NbC, has the form

( d y n > - - -0.005 x 10-2 nm2, AC where AC = 1 - y. This dependence may be represented in a more convenient form for practical use,

(UZyn) = 0.0027 - 0.005(1 - y) x 10-2 nm2 . (9) Since the authors of [2] determined the dynamic displacements by X-ray diffraction, the dependence found applies most probably to dynamic displacements of metallic sublattice atoms, because of the small scattering power of a carbon atom as compared with that of a niobium atom. As a result of an incomplete allowance for the displace- ments of light carbon atoms, the authors of [2] obtained highly understated values of dynamic displacements.

Earlier [3 to 51, in studying the low-temperature heat capacity of disordered nio- bium carbide, a model of the spectrum of natural crystal lattice vibrations was used which took into account both acoustic and optical vibrations. I n terms of that model the distribution function of atomic vibration frequencies is of the form

where 0 is the number of atoms in the primitive cell of the crystal (in the case under consideration (T = 2), 6 is the Dirac delta function (6(w - wE) = 0 if w + wE and 8(co - oE) = 1 if 6) = 6 3 ~ ) ~ (oD and cog are the Debye and Einstein frequencies.

The mean-square deviation of an atom from the equilibrium position, i.e., the dynamic displacement, is determined by

with P ( W , T) = (hto/2) cth (hw/Sk,T) being the mean energy of the linear oscillator and m the effective mass (in the case of niobium carbide, m = (mNb + ymc)/(l + y)), mNb and m, the masses of niobium and carbon atoms. Performing integration over all possible frequencies with allowance for the distribution function g(o) (lo), we obtain formula that takes into account the contributions of the acoustic and optical vibra- t,ions to the dynamic displacements of atoms,

1 9h2 [ (BTD) 60E (:;)I 2k,m 48, 05 2 ( u d y n ) = __ -+- @ - f -c th - ,

where

d4 = - x I S expx x d x - 1 0

and 8D and OE are the characteristic Debye and Einstein temperatures. As calculated with (12) and allowance for the values of OD and OE according to the

data of [3 to 51, the concentration dependence of the dynamic displacements of nio- bium and carbon atoms in carbide NbC, at T = 300 K is

(uZ,,) = 0.0106 - 0.0036(1 - y) x nm2 , (13)

458 A. I. GUSEV and A. A. REMPEL

indicating larger values of dynamic displacements than those reported in [2]. Accord- ing to (13), the dynamic displacements decrease in value as the structural vacancy concentration is increased (Fig. 1, curve 2). This is probably due to the relative contri- bution of carbon atom displacements diminishing with decreasing carbon content in nonstoichiometric niobium carbide.

The aiTailability of data on the total mean-square atomic displacements (&> in NbC, and on the concentration dependence of dynamic displacements (13) permitted to determine the dependence of mean-square static displacements ( utttatic) on the composition of niobium carbide (Pig. 1, curve 3). Inspection of Fig. 1 shows that 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 NbC,,,,. This minimum arises apparently owing to the presence of short-range order in the ar- rangement of carbon atoms and vacancies; the presence of short-range order could not be avoided even by quenching the samples. Note that the composition range in which a (&tatic) minimum occurs corresponds to the region in which, according to [6, 71, an ordered monoclinic Nb,C, phase forms after prolonged annealing.

According to [l], 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 in solid solutions on the concentration c of defects inducing static displacements is

2 c(1 - G) a2M, 2n2 (%static) = ,

where M,, = R-l(dV/V dc)2; R-I is the proportionality coefficient, a the unit cell parameter, and V the unit cell volume.

Equation (14) establishes a relationship between the variation of the dimensions of the “mean” lattice and the static displacements of atoms and the proportionality coefficient R-l = M,(dV/V dc)-2 specifies the response of the crystal to the introduc- tion of defects. An analysis of (14) shows that a large value 0fR-l corresponds to the case where defects lead mainly to local atomic displacements and have an insignificant effect on the crystal lattice parameters. A smaller value of the quantity R-l should be observed with the unit cell dimensions varying substantially during defect formation and is apparently possible when the defects exert a long-range influence on the posi- tion of atoms in the lattice. This allows the quantity R, which is the inverse of the coefficient R-l, to be viewed as some effective range (radius) of the interaction of crystal atoms.

Defects that cause static lattice distortions in niobium carbide NbC, are structural vacancies; therefore c = (1 - y)/2. By substituting into (14) the values of the cubic cell parameter for NbC, and performing numerical differentiation of the cell volume with respect to defect concentration, i t may be shown that (14) approximates well the observed static displacements only in the composition range 0.97 < y < 1.00 (Fig. 1, curve 4). The quantity R-l, determined for this range of niobium carbide composi- tions, is approximately a factor of 2 larger than the counterpart calculated in [l] for interstitial alloys of nontransition metals. This testifies to larger static atomic displacements compared with those observed in alloys and indicates that strong local- ized interactions prevail in niobium carbide. These interactions in NbC, are apparently covalent metal-nonmetal bonds. Insofar as delocalized metallic bonds are present in NbC, they are weaker than those in alloys.

The examination of Fig. 1 shows also that the experimental values of displacements for large defect concentrations are appreciably lower than the theoretical ones. This is apparently so because the overlap of the perturbations produced by point defects

Local Static and Dynamic Atomic Displacements in Disordered NbC, 45 9

was disregarded in the derivation of (14). Such overlapping occurs in a carbide NbC, with large structural vacancy content and should necessarily lead to a decrease in the rate of growth of static displacements with increasing vacancy concentration.

References [l] M. A. KRIVOGLAZ, X-ray and Neutron Diffraction in Non-Ideal Crystals, Nankova Dumka,

[a] T. H. METZOER, J. PEISL, and R. KAUFMAN, J. Phys. F 13, 1103 (1983). [3] A. A. REMPEL and A. I. GUSEV, Zh. fiz. Khim. 62, 2841 (1988). [4] A. I. GUSEV and A. A. REMPEL, Structural Phase Transitions in Nonstoichiometric Compounds

[5] A. A. REMPEL and A. I. GUSEV, Proc. All-Union Conf. Calorimetry and Chemical Thermo-

[6] A. A. REMPEL, A. I. GUSEV, V. G. ZUBKOV, and G. P. SHVEIKIN, Dokl. Akad. Nauk SSSR

[7] A. I. GUSEV and A. A. REMPEL, phys. stat. sol. (a) 93, 71 (1986).

Kiev 1983 (in Russian).

Izd. Nauka, Moskva 1988 (in Russian).

dynamics Vol. 2, Novosibirsk 1986 (p. 143).

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(Received January 17, 1989)