Sequence of phase transformations in the formation of superstructures of the M6C5 type in nonstoichiometric carbides

ISSN 1063�7761, Journal of Experimental and Theoretical Physics, 2009, Vol. 109, No. 3, pp. 417–433. © Pleiades Publishing, Inc., 2009.Original Russian Text © A.I. Gusev, 2009, published in Zhurnal Éksperimental’noі i Teoreticheskoі Fiziki, 2009, Vol. 136, No. 3, pp. 486–504.

417

1. INTRODUCTION

Cubic carbides MCy (0.65 < y ≤ 1.00) of Group Vtransition d metals are strongly nonstoichiometricinterstitial compounds [1]. In MCy carbides with the

basic cubic (space group Fm m) structure B1, carbonatoms are located in octahedral interstitial sites of themetal sublattice with the formation of a face�centeredcubic (fcc) nonmetal sublattice. Depending on the rel�ative carbon content y, the carbon atoms can occupyall or only a part of the interstitial sites. The unoccu�pied interstitial sites are referred to as the structuralvacancies (�). In nonstoichiometric carbides, thestructural vacancies and carbon atoms form a substitu�tional solid solution in the nonmetal sublattice. A highconcentration of structural vacancies, which at thelower boundary of the homogeneity region can reach30 at % or higher, is a prerequisite for the atomic–vacancy ordering in the MCy (MCy�1 – y) carbides.Actually, the distribution of carbon atoms and struc�tural vacancies � over lattice sites can be disordered orordered. The disordered state of carbides is thermody�namically equilibrium at a sufficiently high tempera�ture (T ≥ 1500 K) but can be easily retained by quench�ing to room temperature and exists as a metastable stateat low temperatures. The equilibrium state of nonsto�ichiometric carbides at T < 1300 K is an ordered state,which is attained as a result of special long�termannealing with a slow decrease in temperature.

In nonstoichiometric cubic (space group Fm m)carbides MCy of Group V transition d metals (M = V,

3

3

Nb, Ta) with a relative carbon content in the range0.79 ≤ y ≤ 0.88, the formation of M6C5�type super�structures (Fig. 1) with different symmetries and dis�tributions of carbon atoms and � vacancies over latticesites has been experimentally observed at temperaturesbelow T = 1300–1500 K [2–33].

However, the available experimental data are verycontradictory. For example, it was revealed using elec�tron diffraction, electron microscopy, and NMR spec�troscopy that the axial ordered trigonal phase V6C5 isformed in the vanadium carbide VC0.84 [2–4]. Accord�ing to the symmetry, this superstructure belongs tospace group P31 or enantiomorphic space group P32.More recently, Karimov et al. [5, 6] confirmed the for�mation of the ordered trigonal phase V6C5 with the useof structural neutron diffraction. The influence ofordering on the properties of vanadium carbide wasinvestigated by Lipatnikov et al. [7], who performed anX�ray diffraction analysis of the V6C5 phase and sug�gested that it has a trigonal structure. Khaenko et al.[8, 9] carried out X�ray diffraction studies of annealedVC0.82 crystallites and also revealed that they containthe ordered trigonal (P3112 or P3121) phase V6C5.

Almost simultaneously with the investigations per�formed in [2–4], the structure of nonstoichiometricvanadium carbide was examined by electron diffrac�tion [10–13]. Billingham et al. [10] found thatVC0.77⎯VC0.85 compounds have a base�centered mono�clinic (C2(B2)) superstructure V6C5 rather than thetrigonal superstructure and demonstrated that almostall electron diffraction patterns described in [2] can be

ORDER, DISORDER, AND PHASE TRANSITIONIN CONDENSED SYSTEM

Sequence of Phase Transformations in the Formationof Superstructures of the M6C5 Type in Nonstoichiometric Carbides

A. I. GusevInstitute of Solid State Chemistry, Ural Branch, Russian Academy of Sciences,

ul. Pervomaіskaya 91, Yekaterinburg, 620990 Russiae�mail: [email protected]@ru

Received March 18, 2009

Abstract—A symmetry analysis has been made of monoclinic and trigonal superstructures of the M6C5 type,which are formed in nonstoichiometric cubic carbides MCy with a B1 structure. Channels of disorder–ordertransitions MCy M6C5 have been revealed, and the distribution functions of carbon atoms in the M6C5superstructures have been calculated. The atomic–vacancy ordering in nonstoichiometric cubic carbides ofvanadium VCy and niobium NbCy has been investigated using neutron diffraction and X�ray diffraction anal�yses. It has been shown that, as the temperature decreases, the Group V transition metal carbides MCy canundergo two physically admissible sequences of transformations associated with the formation of M6C5phases.

PACS numbers: 61.50.Ks, 61.66.Fn, 64.70.Kb, 81.30.Hd

DOI: 10.1134/S1063776109090064

418

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 109 No. 3 2009

GUSEV

explained in terms of the monoclinic symmetry of theV6C5 phase rather than the trigonal symmetry. By ana�lyzing the sequence of alternation of layers formed bycarbon and vacancies in the monoclinic (C2 (B2)) andtrigonal (P31) superstructures, Billingham et al. [10]noted that one more monoclinic superstructure withspace group C2/m (B2/m) is possible. The unit cell ofthe monoclinic superstructure V6C5 with space groupC2/m should be two times smaller than that of thesuperstructure with space group C2. According toLewis et al. [12], the domains of the ordered phase ofthe VC0.83 carbide are characterized by a “double”structure, which includes monoclinic and trigonalatomic packings.

The structure of annealed carbides VC0.78, VC0.80,VC0.84, and VC0.86 was investigated using electron dif�fraction by Hiraga [13]. According to [13], the distri�bution of carbon atoms and vacancies in the VC0.84

carbide corresponds, to a greater extent, to the mono�clinic (C2) model of the V6C5 phase [10] as comparedto the trigonal model [2]. Moreover, for the V6C5

superstructure, Hiraga [13] proposed the orthorhom�

bic unit cell with the parameters a = ( /2)aB1 and

b = ( /2)aB1 equal to the corresponding parame�ters of the monoclinic (C2) unit cell and the axis c =

(4 )aB1 parallel to the [111]B1 direction. The long�period superstructure V44C37 was proposed for orderedcarbides VCy (y > 0.84) formed in the range of compo�sitions between the V6C5 and V8C7 phases.

The existence of the ordered monoclinic (C2)phase V6C5 was confirmed by the neutron diffractioninvestigation [14].

Kesri and Hamar�Thibault [15] studied the Fe–V–Csystem in the eutectic region between the austenite andvanadium carbide and established that the solidification

6

18

3

of cast iron containing 3.2 wt % C and 9.0 wt % V isaccompanied by the precipitation of trigonal vana�dium carbide V6C5. Upon solidification of cast ironinvolving less than 2 wt % C and more than 10 wt % V,the vanadium carbide at the center of eutectic grainshas the monoclinic (C2/m) structure V6C5 and theordered cubic (P4332) phase V8C7 surrounded by theFe3C cementite shell is contained in the outer part ofgrains. According to [15], the structure of the mono�clinic phase V6C5 is similar to that of the ordered mon�oclinic niobium carbide Nb6C5 (C2/m (C12/m1))[16–19]. The X�ray diffraction study of the crystalstructure of annealed vanadium carbides VC0.83 andVC0.79 [20, 21] demonstrated that both samples con�tain the ordered V6C5 phase, which can be monoclinic(C2/m) and trigonal (P31). The point is that idealmonoclinic (C2/m) and trigonal (P31) superstructuresof the M6C5 type in a powder diffraction experimentare characterized by a set of superstructure reflectionsidentical in positions and intensities [1, 22]. It shouldbe noted that only the trigonal (P31) model of the V6C5superstructure was considered in [5–7]. It is quite pos�sible that a different inference regarding the symmetryand space group of the ordered V6C5 phase could bemade with due regard for the monoclinic (C2/m)model.

Thus, three different superstructures of the M6C5type for the ordered nonstoichiometric vanadium car�bide VCy with a carbon content in the range 0.79 ≤ y ≤0.83 have been described in the literature (Fig. 1): onetrigonal (P31) and two monoclinic (C2 and C2/m)structures.

As can be judged from the electron microscopydata [2, 3], the ordered trigonal phase V6C5 is stable ina narrow temperature range close to the temperatureof the transition to the disordered state, because thisphase is disordered under the action of an electron

(a) (b) (c)

M

M

M

M

MM

M

M

M M

M

M

M

M

M

M

M

MCC

C

C

CC

C

C

CC

C

C

CC

CC

C

C

C

C

C

CC

CC

M

MM

M

M M M

M

MMM

M MM M

MM

MM

M

MM M

MMM

M

M MM

M MM M

MMM

MMMM

C C C

C C

C

CC

CC

C

C C

CC

C

CC

CC

CCC

CC C

CCC

CC

M C C

C CC

CC

C

CCC

C

CC

C

CC

CCC

CC

Fig. 1. Unit cells of (a) trigonal (P31), (b) monoclinic (C2), and (c) monoclinic (C2/m) superstructures of the M6C5 type.


SEQUENCE OF PHASE TRANSFORMATIONS IN THE FORMATION 419

beam [3]. The disordered monoclinic (C2) phase V6C5[12] is most likely formed at lower temperature ascompared to the trigonal phase V6C5. Indeed, the neu�tron diffraction data [5, 6, 14] show that the mono�clinic phase is stable in the temperature range 1050–1370 K and the trigonal phase exists at temperaturesfrom 1140 to 1520 K.

As follows from the results obtained in [15], theother monoclinic (C2/m) superstructure V6C5 exists inapproximately the same temperature range as the trig�onal superstructure V6C5. According to the averagedexperimental estimates, the lower and upper bound�aries of the homogeneity region of the ordered V6C5phase at a temperature of approximately 1000 K cor�respond to the VC0.75–0.77 and VC0.82–0.86 carbides,respectively. Some uncertainty of the boundaries(especially, the upper boundary) is associated with dif�ferent conditions of ordering annealing of the VCy car�bide in different experimental investigations.Annealed samples of the vanadium carbide VCy withcarbon contents y ≥ 0.83 contain two ordered phases,V6C5 and V8C7.

The amount of experimental data on superstruc�tures of nonstoichiometric niobium carbide NbCy issmaller than that for vanadium carbide. The ordering

in the cubic (Fm m) niobium carbide NbCy was inves�tigated using electron [11, 12, 15, 23] and neutron[16–19, 24–26] diffraction. It was revealed that theordered Nb6C5 phase is formed over a wide range ofcompositions in the vicinity of the NbC0.83 carbide. In[11, 12, 15, 25, 26], the observed superstructure reflec�tions were described within the trigonal structure modelsimilar to that used for the V6C5 superstructure [2]. Inaddition to neutron diffraction, Landesman et al. [25]studied the Nb6C5 phase by X�ray diffraction (CuKβ

radiation) and experimentally revealed the trigonalsplitting of the (440)B1 structure reflection, which indi�cates a trigonal distortion of the cubic metal sublattice.

The neutron diffraction investigation of theannealed single crystal NbC0.83 confirmed the trigonalstructure of the Nb6C5 phase and revealed a noticeabledisplacement of niobium and carbon atoms from allpositions of the ideal structure and the presence of car�bon atoms with a probability of 0.52 at the lattice sitesthat should be completely vacant in the ideal trigonalsuperstructure Nb6C5. Although it was demonstratedin [25, 26] that the observed superstructure reflectionsdo not correspond to the monoclinic superstructureNb6C5 with space group C2, the other monoclinic(C2/m) structural model was not discussed.

In [16–19], the trigonal and two monoclinic struc�tural models of the ordered niobium carbide phasewere analyzed and, with the use of the neutron andX�ray diffraction data, the inference was made that theordered monoclinic (C2/m) phase Nb6C5 is formed innonstoichiometric niobium carbide in the concentra�tion range NbC0.81–NbC0.88 upon annealing at tem�peratures below 1350 K. The X�ray diffraction study of

3

the ordered niobium monocarbide phase [27, 28]showed the presence of the Nb6C5 phase with a hypo�thetical trigonal symmetry. More recently, the sameauthors [29] drew the conclusion that the Nb6C5 phaseis monoclinic and belongs to space group C2/m.

Among the vanadium, niobium, and tantalum car�bides, the cubic tantalum carbide TaCy is the mostcomplex object for the study of the ordering. Experi�mental data on the ordering of the TaCy carbides arealmost absent because the direct investigation of thestructure of their ordered phases by diffraction meth�ods is complicated. In X�ray experiments, the relativeintensity of possible superstructure reflections is verylow due to the large difference between the scatteringamplitudes for Ta and C atoms. In turn, in the case ofneutron diffraction, the neutron absorption by massivetantalum nuclei is very large and, therefore, a consid�erable accumulation of signals is required to revealpossible superstructure reflections.

The electron diffraction study of the TaC0.83 car�bide [23] revealed diffraction fringes with the geome�try corresponding to an M6C5�type ordering with avery low degree of order. According to the neutron dif�fraction data [30–33], the ordering of the TaCy carbideleads to the formation of an incommensurate orderedstructure similar to the M6C5�type structure. It followsfrom the thermodynamic calculations of the disor�der–order transformations [1, 34–37] that theM6C5�type superstructure can be the only orderedphase of the TaCy carbide.

On the whole, the analysis of the experimental dataon the M6C5 superstructures of nonstoichiometricvanadium MCy , niobium, and tantalum carbides indi�cates the possibility of forming trigonal (P31) and twomonoclinic (C2, C2/m) superstructures of the M6C5

type (see Fig. 1). The thermodynamic calculations ofthe phase equilibria in the V–C, Nb–C, and Ta–Cbinary systems with the use of the order�parameterfunctional method [1, 36, 37] confirm the formationof ordered M6C5 phases but do not allow one to deter�mine their symmetry and space group. Whether super�structures of the M6C5 type are mutually exclusive ortwo or all three superstructures can arise one afteranother in some sequence with a decrease in the tem�perature remains unknown. The problem is compli�cated by the fact that superstructures of the M6C5 typeare very similar to each other: the short�range order inthe environment of metal atoms by sites of the non�metal sublattice in these superstructures is completelyidentical [1, 38, 39]. Therefore, NMR investigationscannot reveal the difference between them. The super�structures of the M6C5 type differ only in the mutual

arrangement of nonmetal atomic planes (1 1)B1 con�taining vacant sites [40].

In this respect, in the present work, we performedadditional experimental investigations of the structureof the ordered phases M6C5 of nonstoichiometric

1

420


GUSEV

cubic vanadium and niobium carbides, analyzed theelectron diffraction data, and carried out the symmetryanalysis of the structure of these phases in order todetermine a possible sequence of phase transformationsin the formation of superstructures of the M6C5 type.

2. SAMPLE PREPARATIONAND EXPERIMENTAL TECHNIQUES

Samples of nonstoichiometric niobium carbidesNbCy (0.81 ≤ y ≤ 0.88) of different compositionswithin the homogeneity region of the cubic phase witha B1 structure were synthesized by solid�phase sinter�ing of powdered mixtures (Nb + C) under vacuum at aresidual pressure of 0.0013 Pa (10–5 mmHg). In thecourse of synthesis of the niobium carbides, the timeof treatment at a maximum sintering temperature of2270 K was equal to 10 h and the total synthesis timewas 20 h. Samples of vanadium carbides VCy were pre�pared by hot pressing of powdered mixtures (V + C0.87)at a temperature of 2170 K in an argon atmosphere.The pressure and the time of pressing were equal to30–35 MPa and 30 min, respectively. In order to pro�duce the carbides in states with different degrees oforder (from disordered to ordered), the synthesizedsamples were quenched and annealed under differenttemperature conditions.

The disordered NbCy samples were prepared byquenching from the maximum synthesis temperatureto T = 300 K at a cooling rate of 100 K min–1. Theordered NbCy samples were produced by annealing ata temperature of 1300 K for 10 h, followed by coolingto 300 K at a rate of 0.5 K min–1. The quenching andannealing of the NbCy samples were performeddirectly in a vacuum furnace.

The disordered VCy samples were prepared byquenching in silica tubes, which were evacuated to aresidual pressure of 1.3 × 10–4 Pa (approximately10⎯6 mmHg). The hermetically sealed tubes with thesamples were annealed for 3 h at a temperature of1450 K and then were thrown into water; the quench�ing rate was equal to approximately 200 K/s. Theordered samples of the VCy carbides were prepared byannealing of the synthesized compact samples undervacuum with a residual pressure of 0.0013 Pa at a tem�perature of 1170 K for 20 h and then at T = 1070 and970 K for 20 and 60 h, respectively.

The compositions of the prepared samples of non�stoichiometric carbides were identified by chemicaland spectral analyses. The phase compositions of thesamples and the lattice parameters of the phases weredetermined using X�ray diffraction analysis (Cu

radiation). The measurements of the niobium carbideswere performed on a DRON�UM1 diffractometer(2θ range, 10°–90°; step ∆2θ = 0.03°; scan time, 2 sper point). The measurements of the vanadium car�bides were carried out on a STOE STADI�P diffracto�

Kα1 2,

meter (2θ range, 14°–120°; step ∆2θ = 0.02°; scantime, 10 s per point).

The structure of the ordered phases of the niobiumcarbides was investigated using neutron diffractionanalysis (neutron beam with the wavelength λ =0.1694 nm). The measurements were performed on aneutron diffractometer installed at the horizontalchannel of the IVV�2M reactor (Zarechnyі, Russia).The neutron beam was monochromatized by reflec�tion of neutrons from the (111) face of a germaniumsingle crystal. The measurements were carried out atroom temperature in the 2θ range from 5°–10° to65°–85° (step scan mode; step ∆2θ = 0.1°; exposuretime, 60 s per point). In the final refinement of thephase structure, the diffraction patterns were mini�mized with the X’Pert Plus program package [41]. Thebackground was described by a fifth�order Chebyshevpolynomial, and the diffraction reflection profile wasrepresented by a Voigt pseudofunction.

3. SYMMETRY ANALYSISOF SUPERSTRUCTURES OF THE M6C5 TYPE

The disorder–order or order–order transforma�tions proceeding with a decrease in the temperatureare transitions from the state with a higher free energyto the state with a lower free energy. The state of matterin the atomic or atomic–vacancy ordering can becharacterized by the Landau thermodynamic poten�tial, which in this case is a functional of the probabili�ties of finding atoms of a particular type at lattice sites,the site coordinates, and the temperature. In turn, theprobabilities are functions of the long�range orderparameters. The Landau potential exhibits severalminima corresponding to high�symmetry disorderedand low�symmetry ordered phases. With a decrease inthe temperature, the transition from the disorderedphase to any one of the ordered phases or from oneordered phase to another ordered phase is accompaniedby reduction of symmetry. The symmetry analysismakes it possible to quantitatively determine the reduc�tion of symmetry with the formation of a particularsuperstructure and to reveal the physically admissiblesequence of the formation of these superstructures.

Let us perform the symmetry analysis of possiblesuperstructures M6C5, i.e., let us find the correspond�ing channels of the disorder–order transition, calcu�late the distribution functions of carbon atoms, anddetermine the change in the symmetry upon transitionfrom one phase to another phase. The procedure forcalculating the superstructure vectors of the reciprocallattice, transition channels, and distribution functionsis described in detail in [1, 37, 42].

The trigonal (P31) unit cell of the M6C5 superstruc�ture in the basic cubic lattice is shown in Fig. 2. Thetranslation vectors of the unit cell, as well as the atomicand vacancy coordinates in the ideal trigonal super�structure M6C5, are listed in Table 1. The general formof the coordinates of the atoms occupying the 3(a)



positions in the trigonal superstructure is also given inTable 1. As follows from Fig. 2, the changeover fromthe cubic coordinates xI, yI, and zI to the trigonal coor�dinates is determined by the relationships

The unit cell of the trigonal superstructure involvesthree formula units M6C5. The basis vectors of thereciprocal lattice of the trigonal superstructure M6C5

are as follows:

The translation of the superstructure sites of thereciprocal lattice of the trigonal superstructure M6C5indicates that the first Brillouin zone of the disorderedface�centered cubic lattice contains 13 rays written inthe form

xtr 2xI/3 2yI/3 1/9,+ +=

ytr 2yI/3 2zI/3 8/9,+ +=

ztr xI/6 yI/6– zI/6 1/6.+ +=

atr*23�� 110⟨ ⟩ , btr*

23�� 011⟨ ⟩ ,= =

ctr*16�� 111⟨ ⟩ .=

k93( ) b2

2��, k4

1( ) b1 b2 2b3+ +3

��, k42( ) k4

1( ),–= = =

k47( ) b3 b1–

3��, k4

8( ) k47( )

,–= =

k49( ) 2b1 b2 b3+ +

3��, k4

10( ) k49( )

,–= =

k33( ) 4b1 b2 2b3+ +( )–

6��, k3

4( ) k33( )

,–= =

k39( ) 2b1 3b2 4b3+ +

6��, k3

10( ) k39( )

,–= =

k323( ) 2b1 b2 2b3–+

6��, k3

24( ) k323( )

,–= =

b

a

c

[001]B1

[010]B1

[100]B1

Fig. 2. Position of the trigonal (P31) unit cell of the M6C5superstructure in the lattice with a B1 structure (the originof the coordinates (0, 0, 0)tr of the trigonal unit cell has thecubic coordinates (0, –1/6, –7/6)B1): (�) interstitialatoms, (�) metal atoms, and (�) vacancies.

Table 1. Trigonal (space group P31 ( )) superstructure M6C5 (Z = 3, atr = , btr = , ctr = )

Atom* Position andmultiplicity

Atomic coordinates in the ideal ordered structure** Values of the distribution function n(xI, yI, zI)x/atr y/btr z/ctr

C1 vacancy (C6) General 3(a) 1/9 8/9 1/6 n1 = y – η9/6 – η4/3 – η3/3

C2 (C5) General 3(a) 4/9 5/9 1/6 n3 = y – η9/6 + η4/6 + η3/6

C3 (C4) General 3(a) 7/9 2/9 1/6 n3 = y – η9/6 + η4/6 + η3/6

C4 (C1) General 3(a) 1/9 5/9 1/3 n4 = y + η9/6 + η4/6 – η3/6

C5 (C3) General 3(a) 4/9 2/9 1/3 n4 = y + η9/6 + η4/6 – η3/6

C6 (C2) General 3(a) 7/9 8/9 1/3 n2 = y + η9/6 – η4/3 + η3/3

M1 (M6) General 3(a) 1/9 5/9 1/12

M2 (M5) General 3(a) 4/9 2/9 1/12

M3 (M4) General 3(a) 7/9 8/9 1/12

M4 (M3) General 3(a) 1/9 2/9 1/4

M5 (M1) General 3(a) 4/9 8/9 1/4

M6 (M2) General 3(a) 7/9 5/9 1/4

* The designations of the atomic positions used in [2, 25, 26] are given in parentheses.** In the trigonal (space group P31) structure, the coordinates of the atoms occupying the 3(a) positions have the general form (x, y, z),

(–y, x – y, z + 1/3), (–x + y, –x, z + 2/3). Correspondingly, the initial coordinates of the M4 atom can be chosen in the form (7/9,8/9, 7/12) as in [37]. Similarly, the other initial coordinates (1/9, 5/9, 7/12) and (4/9, 2/9, 7/12) can be chosen for the M5 and M6atoms, respectively.

C32 1

2�� 211⟨ ⟩B1

12�� 112⟨ ⟩B1 2 111⟨ ⟩B1

422


GUSEV

which belong to the three stars {k9}, {k4}, and {k3}(hereinafter, the numbering and description of the stars

{ks}, wave vectors, and their rays are given accord�

ing to [1, 37, 42–44]; b1 = ( 1 1), b2 = (1 1), and

b3 = (1 1 ) are the structure vectors of the reciprocallattice of the basic face�centered cubic lattice in unitsof 2π/a). These 13 nonequivalent superstructure vec�tors are involved in the phase transition channel asso�ciated with the formation of the trigonal superstruc�ture M6C5 under consideration.

The structure of ordered phases is convenientlydescribed using the distribution function n(r), whichrepresents the probability of finding an atom of a par�ticular type at the site r = (xI, yI, zI) in the orderingIsing lattice. In the MCy carbides with the B1 basicstructure, the Ising lattice with atomic–vacancyordering is a nonmetal face�centered cubic sublattice.The deviation of the probability n(r) from its value inthe case of a disordered (statistical) distribution can berepresented as a superposition of several plane con�centration waves [45]. The wave vectors of these wavesare the superstructure vectors forming a disorder–order transition channel [1, 37, 42]. In the method ofstatic concentration waves [45], the distribution func�tion n(r) has the form

(1)

where y is the fraction of sites occupied by atoms of aparticular type in the ordering sublattice; the quantity

is the standing plane static concentration wave gener�

ated by the superstructure vector of the star {ks}; ηs

is the long�range order parameter corresponding to

the star {ks}; and ηsγs and are the amplitude andthe phase shift of the concentration wave, respectively.The function n(r) takes on the same value at the sites rlocated at crystallographically equivalent positions.The total number of values taken by the distributionfunction is larger than the number of long�range orderparameters by unity. The summation in formula (1)should be performed only over nonequivalent super�structure vectors of the first Brillouin zone.

In view of expression (1), the distribution functionof carbon atoms in the trigonal superstructure M6C5depends on the three long�range order parameters η9,η4, and η3 corresponding to the stars {k9}, {k4}, and{k3}, respectively, and has the form

ksj( )

1 1

1

n r( ) y 12�� ηsγs iϕs

j( )( )exp[j s∈

∑s

∑+=

× iksj( ) r⋅( )exp iϕs

j( )–( )exp+

× iksj( )– r⋅( ) ],exp

12��ηsγs iϕs

j( )( ) iksj( ) r⋅( )expexp[

+ iϕsj( )–( ) iks

j( )– r⋅( ) ]expexp ∆ ksj( ) r⋅( )≡

ksj( )

ϕsj( )

(2)

At all sites of the basic nonmetal face�centeredcubic sublattice, the distribution function (2), whichdescribes the trigonal superstructure M6C5, takes onfour different values, namely, n1, n2, n3, and n4(Table 1). This means that the nonmetal sublattice ofthe disordered nonstoichiometric carbide MCy in theordering under consideration is divided into four non�equivalent sublattices. The probability of occupationof sites of the first sublattice by C atoms is equal to n1,the probability of occupation of sites of the secondsublattice is equal to n2, etc.

The symmetry point group 3 (C3) of the trigonalcarbide M6C5 includes three symmetry elements h1,

h5, and h9, and the symmetry point group m m (Oh) ofthe basic disordered cubic phase MCy involves 48 ele�ments h1–h48 [1, 37, 42–44]. Therefore, the rotationalreduction of symmetry is equal to 16. The reduction oftranslational symmetry is equal to the ratio betweenthe volumes of the unit cells of the ordered and disor�dered phases and amounts to 4.5 upon transition fromthe disordered carbide MCy to the trigonal carbideM6C5 under consideration. The total reduction ofsymmetry is the product of the rotational and transla�tional reductions of symmetry. Therefore, the transi�

tion MCy (Fm m) M6C5 (P31) is attended by thetotal reduction of symmetry N = n(G)/n(GD) = 72,where n(G) and n(GD) are the orders of the space groupG of the high�symmetry disordered phase and the

n xI yI zI, ,( ) yη9

6�� π xI yI– zI+( )[ ]cos–

η4

6��–=

× 4π3

�� xI yI+( )cos 33

�� 4π3

�� xI yI+( )sin–⎩⎨⎧

– 2 33

�� 4π3

�� xI zI–( )sin 4π3

�� yI zI+( )cos+

+ 33

�� 4π3

�� yI zI+( )sin⎭⎬⎫

–η3

6�� π

3�� xI 5yI– 3zI–( )cos

⎩⎨⎧

– 33

�� π3�� xI 5yI– 3zI–( )sin

+ 2 33

�� π3�� 5xI 3yI zI+ +( )sin

+ π3�� 3xI yI 5zI–+( )cos

+ 33

�� π3�� 3xI yI 5zI–+( )sin

⎭⎬⎫

.

3

3



space group GD of the low�symmetry ordered phase,respectively.

The unit cell of the monoclinic superstructureM6C5 (Fig. 3) includes four formula units M6C5. Thetranslation vectors of this unit cell and the coordinatesof atoms and vacancies in the ideal monoclinic super�structure M6C5 are presented in Table 2. The basis vec�tors of the reciprocal lattice of the monoclinic super�structure M6C5 are as follows:

The ordered monoclinic structure M6C5 is formedthrough the channel of the transition involving thenine superstructure vectors of the reciprocal lattice,

which belong to the four stars {k9}, {k4}, {k3}, and {k0}.The monoclinic superstructure M6C5 is described

by the distribution function dependent on the fourlong�range order parameters η9, η4, η3, and η0; that is,

aC2* 12�� 111⟨ ⟩ ,=

bC2* 13�� 110⟨ ⟩ , cC2* 1

4�� 111⟨ ⟩ .= =

k93( ) k4

1( ), 13�� b1 b2 2b3+ +( ), k4

2( ) k41( )

,–= =

k33( ) 1

6�� 4b1 b2 2b3+ +( ), k3

4( )– k33( )

,–= =

k04( ) 1

12�� 4b1 b2 4b3–+( ), k0

28( ) k04( )

,–= =

k013( ) 1

12�� 8b1 5b2 4b3+ +( ), k0

37( )– k013( )

,–= =


6�� π xI yI– zI+( )[ ]cos–=

–η4

12�� 4π

3�� xI yI+( )cos

⎩⎨⎧

– 3 4π3

�� xI yI+( )⎭⎬⎫

sin

a

c

b

[010]B1

[001]B1

[100]B1

Fig. 3. Position of the monoclinic (C2) unit cell of theM6C5 superstructure in the lattice with a B1 structure: (�)interstitial atoms, (�) metal atoms, and (�) vacancies.

Table 2. Monoclinic (space group C2 (C121)– ) superstructure M6C5 (Z = 4, am = , bm = , cm = )

Atom Position andmultiplicity

Atomic coordinates in the ideal ordered structure Values of the distributionfunction n(xI, yI, zI)x/am y/bm z/cm

C1 vacancy Special 2(a) 0 0 0 n1 = y – η9/6 – η4/12 – η3/12 – η0/2

C2 vacancy Special 2(b) 0 1/3 1/2 n1 = y – η9/6 – η4/12 – η3/12 – η0/2

C3 Special 2(a) 0 1/3 0 n5 = y – η9/6 – η4/12 – η3/12 + η0/2

C4 Special 2(a) 0 2/3 0 n3 = y – η9/6 + η4/6 + η3/6

C5 Special 2(b) 0 0 1/2 n5 = y – η9/6 – η4/12 – η3/12 + η0/2

C6 Special 2(b) 0 2/3 1/2 n3 = y – η9/6 + η4/6 + η3/6

C7 General 4(c) 0 1/6 1/4 n4 = y + η9/6 + η4/6 – η3/6

C8 General 4(c) 0 1/2 1/4 n2 = y + η9/6 – η4/12 + η3/12

C9 General 4(c) 0 5/6 1/4 n4 = y + η9/6 + η4/6 – η3/6

M1 General 4(c) 1/4 1/6 1/8

M2 General 4(c) 1/4 1/2 1/8

M3 General 4(c) 1/4 5/6 1/8

M4 General 4(c) 1/4 0 3/8

M5 General 4(c) 1/4 1/3 3/8

M6 General 4(c) 1/4 2/3 3/8

C23 1

2�� 112⟨ ⟩B1

32�� 110⟨ ⟩B1 112⟨ ⟩B1

424


GUSEV

(3)

–η3

12�� π

3�� xI 5yI– 3zI–( )cos

⎩⎨⎧

– 3 π3�� xI 5yI– 3zI–( )

⎭⎬⎫

sin

–η0

12�� 3 π

6�� xI 7yI 9zI+ +( )cos

⎩⎨⎧

– 3 π6�� xI 7yI 9zI+ +( )sin

+ 3 π6�� 7xI yI 9zI–+( )cos

– 3 π6�� 7xI yI 9zI–+( )

⎭⎬⎫

.sin

At all sites of the basic nonmetal face�centeredcubic sublattice, the distribution function (3) takes onfive values (Table 2). Consequently, in this ordering,the nonmetal sublattice of the disordered carbide isdivided into five nonequivalent sublattices, which dif�fer in the probabilities of occupation of their sites byinterstitial atoms.

The symmetry point group 2 (C2) of the monocliniccarbide M6C5 includes two symmetry elements h1 andh4. Therefore, the rotational reduction of symmetry inthe formation of this superstructure is equal to 24. Theunit cell volume of the monoclinic phase M6C5 underconsideration is six times larger than the unit cell vol�ume of the disordered carbide with the basic structureB1, and, hence, the reduction of translational symme�try in this case is equal to six. As a result, the transition

of the disordered carbide MCy (Fm m) to the mono�clinic (C2) superstructure M6C5 is accompanied by thereduction of symmetry by a factor of 144.

The position of the unit cell of the monoclinic(C2/m) superstructure M6C5 in the basic cubic latticeis shown in Fig. 4. This superstructure includes twoformula units M6C5. The translation vectors of the unitcell and the coordinates of atoms and vacancies in itare listed in Table 3. The basis vectors of the reciprocallattice of the monoclinic (C2/m) superstructure M6C5are as follows:

The disorder–order transition channel associatedwith the formation of the monoclinic (C2/m) super�structure M6C5 involves five nonequivalent super�structure vectors

3

aC2/m* 12�� 111⟨ ⟩ , bC2/m* 1

3�� 110⟨ ⟩ ,= =

cC2/m* 12�� 111⟨ ⟩ .=

k93( ) k4

1( ), 13�� b1 b2 2b3+ +( ), k4

2( ) k41( )

,–= =

b

a

c

[001]B1

[010]B1

[100]B1

Fig. 4. Position of the monoclinic (C2/m) unit cell of theM6C5 superstructure in the lattice with a B1 structure: (�)interstitial atoms, (�) metal atoms, and (�) vacancies.

Table 3. Monoclinic (space group C2/m (C12/m1) – ) superstructure M6C5 (Z = 2, am = , bm = ,

cm = )


Atomic coordinates in the ideal ordered structure Values of the distribution function n(x1, y1, z1)x/am y/bm z/cm

C1 vacancy Special 2(a) 0 0 0 n1 = y – η9/6 – η4/3 – η3/3

C2 Special 2(d) 0 1/2 1/2 n2 = y + η9/6 – η4/3 + η3/3

C3 Special 4(g) 0 1/3 0 n3 = y – η9/6 + η4/6 + η3/6

C4 Special 4(h) 0 1/6 1/2 n4 = y + η9/6 + η4/6 – η3/6

M1 Special 4(i) 1/4 0 3/4

M2 General 8(j) 1/4 2/3 3/4

C2h3 1

2�� 112⟨ ⟩B1

32�� 110⟨ ⟩B1

12�� 112⟨ ⟩B1



which belong to three stars {k9}, {k4}, and {k3}. Thecorresponding distribution function has the form

(4)

Like the distribution function (2) describing thetrigonal superstructure M6C5, the distribution function(4) at all sites of the basic nonmetal face�centeredcubic sublattice takes on four values (Table 3). How�ever, the relative arrangement of sites of four sublatticesin the ordered monoclinic structure differs from that inthe trigonal superstructure. The long�range order inthe distribution of carbon atoms in the monoclinic(C2/m) superstructure M6C5 under consideration alsodiffers from that in the superstructure M6C5 (C2).

The symmetry point group 2/m (C2h) of the mono�clinic (C2/m) carbide M6C5 includes four symmetryelements h1, h4, h25, and h28. Therefore, the rotationalreduction of symmetry is equal to 12. The reduction oftranslational symmetry upon transition from the dis�ordered carbide MCy to the monoclinic carbide M6C5

is equal to three. The total reduction of symmetryupon transition “disordered nonstoichiometric car�

bide MCy (Fm m) ordered monoclinic (C2/m)carbide M6C5” is equal to 36.

Three superstructures of the M6C5 type under con�sideration are formed with distortion of symmetryaccording to three or four irreducible representations.Hence, it follows that the phase transitions MCy M6C5 do not obey the Landau group�theoretical crite�rion for second�order phase transitions and occurthrough the mechanism of first�order transitions.

The disorder–order or order–order transforma�tions in substitutional solid solutions or nonstoichio�metric compounds proceed with a reduction of thepoint symmetry of the crystal. Actually, a number ofsymmetry transformations of the high�symmetryphase that bring mutually substituted atoms of thesolid solution (or the occupied and unoccupied sites ofthe nonstoichiometric compound) into coincidenceare not involved in the group of symmetry elements ofthe low�symmetry ordered crystal, because these sitesbecome crystallographically nonequivalent.

The symmetry point groups of the trigonal (P31),monoclinic (C2), and monoclinic (C2/m) superstruc�tures M6C5 include three (h1, h5, h9), two (h1, h4), andfour (h1, h4, h25, h28) symmetry elements and are sub�groups of the point group of the basic disordered cubic

k33( ) 1

6�� 4b1 b2 2b3+ +( ), k3

4( )– k33( )

,–= =


6�� π xI yI– zI+( )[ ]cos–=

–η4

3�� 4π

3�� xI yI+( )cos

–η3

3�� π

3�� xI 5yI– 3zI–( ) .cos

3

(Fm m) phase MCy. Therefore, the transition of thedisordered carbide to any one of these phases is a dis�order–order transformation. As regards the transitionsbetween particular superstructures M6C5, it is clearfrom the relationships between the elements hi that thetrigonal superstructure is not symmetry�related to themonoclinic superstructures M6C5, because its pointgroup is not a group or subgroup of the point groups ofmonoclinic superstructures. Consequently, the transi�tion between the trigonal and either of the two mono�clinic phases M6C5 cannot be an order–order trans�formation but can proceed as a polymorphic transfor�mation. The order–order transformation is possibleonly for the monoclinic superstructures, because thesymmetry point group of the monoclinic (C2) phaseM6C5 is a subgroup of higher symmetry monoclinic(C2/m) phase of the same type. The order–ordertransformation of the monoclinic (C2/m) phase M6C5

into the monoclinic (C2) phase M6C5 occurs in such away that the total reduction of symmetry is equal tofour. As a result of the transformation, the unit cell ofthe monoclinic (C2/m) phase M6C5 is doubled alongthe c axis and the C atoms located in the crystallo�graphic positions g with the coordinates (0, 1/3, 1),(1, 1/3, 1), and (1/2, 5/6, 1) in the initial unit cell(see Fig. 4) are displaced along the direction[0, ⎯1/3, 0]C2/m ≡ [–1/2, –1/2, 0]B1 to the nearestvacant sites.

Therefore, a decrease in the temperature can leadto two sequences of transformations associated withthe phases M6C5. The first sequence consists of trans�

formations of the disordered cubic (Fm m) phaseMCy into the ordered monoclinic (C2/m) phase M6C5

and, then, into the ordered monoclinic (C2) phaseM6C5 and involves only the disorder–order andorder–order transformations. The alternativesequence consists of transitions from the disordered

cubic (Fm m) phase MCy to the ordered trigonal(P31) phase M6C5 and, then, to the monoclinic (C2)phase and includes the disorder–order transformationand the polymorphic transformation.

The revealed channels of the disorder–order tran�sition for the superstructures of the M6C5 type underconsideration allow us to calculate the electron dif�fraction patterns and to compare them with the exper�imental diffraction patterns obtained in [2, 10, 13, 15,23], which were used for determining the symmetry ofcarbide phases of the M6C5 type. Figure 5 shows thecalculated positions of the structure and superstruc�ture reflections in six cross sections of the reciprocallattice of the trigonal (P31) and monoclinic

(C2, C2/m) superstructures M6C5 by the planes of the reciprocal face�centered cubic lattice,i.e., model electron diffraction patterns for differentcrystal orientations. The majority of the experimental

3

3

3

hkl( )fcc*

426


GUSEV

Fig. 5. Calculated positions of (�) structure and (�) superstructure reflections in the cross sections of the reciprocal lattice of thetrigonal (P31) and monoclinic (C2, C2/m) superstructures M6C5 by the planes of the reciprocal face�centered cubic lat�

tice. Designations of the cross�sectional planes are also given in the , , and coordinates of each

superstructure M6C5. The specified indices of all structure reflections correspond to the basic cubic lattice. For some structureand superstructure reflections, the indices corresponding to the superstructure under consideration are also additionally specifiedand denoted by the subscript s.

hkl( )fcc*

HKL( )trig* HKL( )C2* HKL( )C2/m*

C2 C2/m

(4 0 1)trig*

(1 0 0)fcc*

P31(hkl)fcc*

(1 1 0)fcc*

(2 1 0)trig*

(0 8 1)trig*

(0 1 0)trig*

(1 1 2)fcc*

(0 0 1)trig*

(1 1 0)trig*

(1 2 1)fcc*

−

(1 1 2)fcc*− −

(1 1 1)fcc*− −

−

−



electron diffraction patterns were measured for theseorientations in [2, 10, 13, 15, 23]. In order to changeover from the indices of the planes of thereciprocal face�centered cubic lattice to the indices ofthe planes of the reciprocal lattice of a par�ticular superstructure M6C5, it is possible to use thefollowing relationships:

for the trigonal (P31) superstructure,

for the monoclinic (C2) superstructure,

and for the monoclinic (C2/m) superstructure,

The factor n is normalizing and chosen so that theindices H, K, and L would be the smallest (in magni�tude) and integer numbers.

It can be seen from Fig. 5 that the positions of the

superstructure reflections in the cross sec�tions of the superstructures are completely identical.

The same holds true for the and cross sections (Fig. 5) given in [2]. This additionallyconfirms a close similarity of the superstructuresM6C5, on the one hand, and the inadequacy of thesecross sections for the determination of the symmetryof the superstructures M6C5 with the use of electrondiffraction analysis, on the other hand. These diffrac�tion patterns can correspond to any one of the threesuperstructures under consideration. However, among

hkl( )fcc*

HKL( )s*

Htrig2n3

�� h k+( ), Ktrig2n3

�� k l+( ),= =

Ltrign6�� h k– l+( );=

HC2n2�� h k– l–( ), KC2

n3�� h k+( ),= =

LC2n4�� h k– l+( );=

HC2/mn2�� h k– l–( ), KC2/m

n3�� h k+( ),= =

LC2/mn2�� h k– l+( ).=

11 2( )fcc*

1 1 0( )fcc* 111( )fcc*

all the diffraction patterns obtained in [2], there is onediffraction pattern observed in the cross section

≡ of the reciprocal lattice (Fig. 5)that uniquely corresponds to the trigonal (P31) sym�metry of the M6C5 (V6C5) phase.

Among the five diffraction patterns reported in[10], one diffraction pattern (the cross section

≡ corresponds to the monoclinic(C2) superstructure M6C5 (V6C5), one more diffrac�

tion pattern (the cross section ≡ isattributed to the monoclinic (C2/m) superstructure,and the third diffraction pattern (the cross section

(not shown in Fig. 5) can be assigned toeither of the two monoclinic superstructures. Two dif�fraction patterns (the cross sections and

can belong to any one of the three super�structures of the M6C5 type.

In [13], the electron diffraction pattern observed in

the cross section can correspond to any oneof the superstructures of the M6C5 type and the dif�

fraction patterns in the cross sections ≡

and ≡ of the reciprocallattice are characteristic only of the monoclinic (C2)superstructure M6C5 (Fig. 5). The other electron dif�fraction patterns are interpreted as diffraction from theorthorhombic superstructure V6C5.

Among the several electron diffraction patternsobtained by Kesri and Hamar�Thibault [15], two dif�

fraction patterns (the cross sections ≡

and ≡ correspondonly to the monoclinic (C2/m) superstructure, the dif�fraction pattern in the cross section ≡

is characteristic of the monoclinic super�structure M6C5 with space group C2, and the electron

diffraction pattern in the cross sections and

1 0 0( )fcc* 4 0 1( )trig*

1 2 1( )fcc* 1 1 0( )C2*

112( )fcc* 1 0 0( )C2/m*

1 0 1( )fcc*

1 1 0( )fcc*

11 2( )fcc*

11 2( )fcc*

112( )fcc*

1 0 0( )C2* 1 2 1( )fcc* 1 1 0( )C2*

112( )fcc*

1 0 0( )C2/m* 1 2 1( )fcc* 1 1 0( )C2/m*

1 2 1( )fcc*

1 1 0( )C2*

111( )fcc*

Fig. 6. The same as in Fig. 5 for the cross�sectional plane.110( )fcc*

P31 C2 C2/m

(0 2 1)trig* (2 0 1)C2* (1 0 1)C2/m*

(1 1 0)fcc*−

−

428


GUSEV

can be attributed to any one of the super�structures of the M6C5 type (Fig. 5). However, in [15],

the electron diffraction in the cross section (Fig. 7 in [15]) was erroneously interpreted as the dif�fraction from the trigonal superstructure. It should benoted that, in [15, Figs. 8, 17], there are errors in theindexing of the superstructure reflections.

The generalization of the experimental dataobtained in [2, 10, 12, 13, 15] on the electron diffrac�tion with due regard for the calculations and simula�tion performed in our work allows us to make the fol�lowing inferences. The electron diffraction character�istic of the trigonal (P31) phase V6C5 (cross section

≡ was observed only by Venableset al. [2]. No diffraction patterns that can be uniquelyassigned to the trigonal superstructure V6C5 wereobtained in other works [10, 12, 13, 15], even thoughthe authors of [10, 12, 15] stated that they observed thetrigonal superstructure.

The electron diffraction specific only of the mono�clinic (C2) superstructure V6C5 was observed in [10,

13, 15]: the cross sections ≡ and

≡ (Fig. 5).

The experimental electron diffraction patternscharacteristic only of the monoclinic superstructurewith space group C2/m were recorded in [10, 15]: the

11 2( )fcc*

111( )fcc*

1 0 0( )fcc* 4 0 1( )trig*

1 2 1( )fcc* 1 1 0( )C2*

112( )fcc* 1 0 0( )C2*

cross sections ≡ and ≡

. Finally, in [2, 10, 13, 15], the authorsobtained the electron diffraction patterns in the cross

sections , , and of the recip�rocal face�centered cubic lattice (Fig. 5) that can beattributed to any one of the superstructures of theM6C5 type under consideration.

A certain ambiguity of the electron diffraction datais associated with the presence of the domain structure[2, 10, 12, 13] (four equivalent domain orientationsare possible for the trigonal superstructure M6C5, andthree equivalent domain orientations are possible forthe monoclinic superstructures M6C5), the existenceof twins [12], and the appearance of additional reflec�tions due to the double diffraction [12, 15].

Thus, the electron diffraction investigations of thevanadium carbide have reliably proved the formationof two monoclinic (C2, C2/m) superstructures M6C5,whereas the evidence for the existence of the trigonalsuperstructure M6C5 is less reliable.

According to the other data available in the litera�ture, the monoclinic (C2/m) [15, 17, 18] and trigonal(P31) [5, 6, 14] phases V6C5 are formed at close tem�peratures of 1350–1520 K and the monoclinic (C2)carbide V6C5 is a low�temperature phase. This is inagreement with the inference made from the symme�try analysis that two alternative sequences of transfor�mations are possible: the transformation from the dis�

112( )fcc* 1 0 0( )C2/m* 1 2 1( )fcc*

1 1 0( )C2/m*

1 1 0( )fcc* 111( )fcc* 112( )fcc

*

602θ, deg

20 40 120

VC0.79 (quenched)

80 100

VC0.79 (annealed)

VC0.83 (quenched)

VC0.83 (annealed)

(111

) B1

(200

) B1

(220

) B1

(222

) B1

(311

) B1

(400

) B1

(331

) B1

(420

) B1

Fig. 7. X�ray diffraction patterns of the quenched andannealed samples VC0.83 and VC0.79 of nonstoichiometricvanadium carbide. Vertical tick marks indicate the diffrac�tion reflections of the ordered monoclinic (C2/m) phaseV6C5. The annealed VC0.83 carbide contains the orderedcubic phase V8C7 in addition to the V6C5 phase. CuK

α1, 2radiation.

Fig. 8. Experimental (crosses) and calculated (solid line)X�ray diffraction patterns of the VC0.83 sample orderedusing long�term annealing. The annealed sample containsapproximately 70 wt % of the monoclinic (C2/m) phaseV6C5 and approximately 30 wt % of the cubic (P4332)phase V8C7. Long and short vertical tick marks indicate thediffraction reflections of the ordered monoclinic phaseV6C5 and the ordered cubic phase V8C7, respectively. Thedifference (Iobs – Icalc) between the experimental and cal�culated X�ray diffraction patterns of the ordered vanadiumcarbide VC0.83 is shown at the bottom. CuK

α1, 2 radiation.

−2000

0

Counts

602θ, deg

−4000

2000

20 40 120

(420

) B1

80 100

4000

6000

8000

(111

) B1

(200

) B1

(220

) B1

(222

) B1

(311

) B1

(400

) B1

(331

) B1

VC0.83 (annealed)

Iobs − Icalc

RI = 0.0637 (V6C5 phase)



ordered cubic phase MCy into the ordered monoclinic(C2/m) phase M6C5 and, then, into the ordered mon�oclinic (C2) phase M6C5 and the transformation fromthe disordered cubic phase MCy into the orderedtrigonal (P31) phase M6C5 and, then, into the mono�clinic (C2) phase M6C5.

The electron diffraction in ordered nonstoichio�metric niobium carbide NbCy was studied in [12, 15,23]. According to Lewis et al. [12], the trigonal (P31)superstructure Nb6C5 similar to the ordered trigonalphase V6C5 is formed in the NbC0.83 carbide annealedat temperatures below 1313 K. However, diffractionpatterns clearly corresponding to the trigonal carbideNb6C5 are absent in [12]. Kesri and Hamar�Thibault[15] studied the Fe–Cr–Nb–C system in the eutecticregion between the austenite and the vanadium car�bide and established that the disorder–order transitionNbC0.83–Nb6C5 is observed at T ≤ 1298 K. Accordingto [15], the Nb6C5 superstructure is hexagonal (trigo�nal), which is confirmed by the electron diffraction

pattern in the cross section of the reciprocalface�centered cubic lattice. However, the performedcalculations demonstrate that the electron diffraction

11 0( )fcc*

pattern in the cross section can correspond toany one of the superstructures of the M6C5 type(Fig. 6). The electron diffraction pattern in the

cross section for the annealed NbC0.84 singlecrystal was obtained in [23]. The annealing was carriedout under vacuum with a decrease in the temperaturefrom 1403 to 1213 K at a rate of 8 K h–1. The afore�mentioned diffraction pattern is similar to thatobtained by Venables et al. [2] for the V6C5 carbide butdiffers in the presence of twin reflections of the M6C5

superstructure and diffuse fringes. However, as wasnoted above, the electron diffraction pattern in the

cross section of the reciprocal face�centeredcubic lattice (Fig. 5) is not specific and can be attrib�uted to any one of the M6C5�type superstructures.

The subsequent investigations of the ordering in theniobium carbide with the use of neutron powder diffrac�tion revealed the formation of trigonal (P31) [25] andmonoclinic (C2/m) [16–19] superstructures of the M6C5type. The X�ray diffraction study [29] confirmed themonoclinic (C2/m) symmetry of the Nb6C5 superstruc�ture. Thus, the experimental investigations of the nio�bium carbide (like the vanadium carbide) have reliablyproved only the formation of M6C5�type superstructureswhich can have monoclinic or trigonal symmetry.

Taking into account the performed symmetry anal�ysis of the M6C5�type superstructures and the electrondiffraction data, we consider the results of the experi�

11 0( )fcc*

1 1 0( )fcc*

1 1 0( )fcc*

602θ, deg

20 40 80

NbC0.83 (annealed)

NbC0.81 (annealed)

(111

) B1

(200

) B1

(220

) B1

(311

) B1

NbC0.81 (quenched)

NbC0.83 (quenched)

NbC0.88 (annealed)

NbC0.88 (quenched)

Fig. 9. Neutron diffraction patterns (λ = 0.1694 nm) of thequenched disordered and annealed ordered samplesNbC0.81, NbC0.83, and NbC0.88 of nonstoichiometric nio�bium carbide. Neutron diffraction patterns of the annealedsamples contain weak superstructure reflections. Verticaltick marks indicate the diffraction reflections of theordered monoclinic (C2/m) phase Nb6C5.

0

Counts

60 2θ, deg

−2 × 104

2 × 104

20 40 70

4 × 104

6 × 104

8 × 104

(111

) B1

(200

) B1

(220

) B1

NbC0.83 (annealed)

Iobs − Icalc

RI = 0.018

5030

λ = 0.1694 nm

Fig. 10. Experimental (crosses) and calculated (solid line)neutron diffraction patterns (λ = 0.1694 nm) of theNbC0.83 sample ordered using long�term annealing. Verti�cal tick marks indicate the diffraction reflections of theordered monoclinic (C2/m) phase Nb6C5. The difference(Iobs – Icalc) between the experimental and calculatedneutron diffraction patterns of the ordered niobium car�bide NbC0.83 is shown at the bottom.

430


GUSEV

mental study of the ordered phases V6C5 and Nb6C5with the use of X�ray and neutron diffraction.

4. REAL STRUCTURE OF M6C5�TYPE ORDERED PHASES

OF NONSTOICHIOMETRICCUBIC CARBIDES NbCy AND VCy

According to the X�ray diffraction data, thequenched VC0.83 and VC0.79 samples contain only the

disordered cubic (Fm m) vanadium carbide VCy withthe B1 structure (Fig. 7). The long�term annealing ofthe VC0.83 and VC0.79 samples with a decrease in thetemperature from 1170 to 970 K leads to the appear�ance of weak superstructure reflections in the X�raydiffraction patterns.

The minimization of the X�ray diffraction patternof the VC0.83 sample (Fig. 8) annealed for 100 h at tem�peratures from 1170 to 970 K revealed that this samplecontains approximately 70 wt % of the phase V6C5 andapproximately 30 wt % of the cubic (P4332) phaseV8C7 with the lattice parameter a = 0.83402 nm. Thepresence of two ordered phases in the annealed VC0.83

carbide was previously noted by Lipatnikov et al. [7]. Itshould be noted that the minimization revealed largedisplacements of all the vanadium atoms and the car�bon atoms occupying the 12(d) positions in theordered cubic phase V8C7. These displacements resultin the appearance of additional weak diffractionreflections , , and at 2θ =

30.30°, 32.12°, and 44.70°, which are absent in the the�oretical X�ray diffraction pattern of the ideal (withoutatomic displacements) superstructure V8C7 due toextinction. Previously, large atomic displacements inthe V8C7 superstructure were found by Rafaja et al. [46].

The refinement of the structure of the V6C5 phasecontained in the annealed VC0.83 sample was per�formed with the X’Pert Plus program package [41].The calculations were carried out for the trigonal (P31)and monoclinic (C2, C2/m) models of the M6C5superstructure with allowance made for the presenceof the cubic phase V8C7. The calculations within themonoclinic (C2) model did not make it possible toachieve a satisfactory convergence due the absence anumber of diffraction reflections (at 2θ = 22.1°, 25.4°,etc.) characteristic of this model in the experimentaldiffraction pattern.

The minimization of the X�ray diffraction patternwith the use of the trigonal (P31) model of the M6C5superstructure leads to the unit cell parameters atr =0.51031(1) nm and ctr = 1.44597(6) nm and the factorRI(RB) = 0.0722. The final convergence factors for thecalculated and experimental X�ray diffraction patternsare Rp = 0.1084 and ωRp = 0.1425.

The use of the monoclinic (C2/m) model for theV6C5 superstructure results in the monoclinic unit cellparameters am = 0.51093(2) nm, bm = 0.88454(3) nm,

3

220( )V8C7221( )V8C7

410( )V8C7

cm = 0.50897(2) nm, and β = 109.436(2)° and the fac�tor RI(RB) = 0.0637 (the final convergence factors areRp = 0.1029 and ωRp = 0.1287). It can be seen that theuse of the monoclinic (C2/m) model slightly improvesagreement between the experiment and the calcula�tion as compared to the trigonal (P31) model. Theresults of the calculation for the monoclinic (C2/m)model of the V6C5 phase are presented in Fig. 8 andTable 4. The occupancies of the 2(a), 2(d), 4(g), and4(h) positions by carbon atoms coincide with theiroccupancies in the ideal ordered phase V6C5. Thismeans that the long�order parameters η9, η4, and η3are equal to unity.

Therefore, the performed experimental investiga�tion of the vanadium carbide VC0.83 and a part of thedata available in the literature indicate that the firstphase formed in the ordering of this carbide is themonoclinic (C2/m) superstructure V6C5. However, theexperimental data obtained for the vanadium carbidedo not allow us to exclude that the trigonal (P31)superstructure V6C5 is first formed upon the disorder–order transition. Actually, the convergences betweenthe experiment and the calculations carried out forboth models are rather close to each other.

The preliminary structural characterization of thesamples of the niobium carbides NbCy was carried outusing X�ray diffraction analysis. The X�ray diffractionpatterns of the quenched samples NbC0.81, NbC0.83,and NbC0.88 contain only the diffraction reflections of

the disordered cubic (Fm m) phase, whereas theX�ray diffraction patterns of the annealed samples inthe small�angle range 2θ < 56° involve additional veryweak reflections with intensities lower than 0.005 ofthe intensity of the strongest structure reflections(111)B1 and (200)B1. The diffraction measurements ofthe quenched disordered samples NbC0.81, NbC0.83,and NbC0.88 revealed that the lattice spacings aB1 of thebasic phase with the B1 structure are equal to 0.44245,0.44276, and 0.44360 nm, respectively. After anneal�ing, the spacings of the basic cubic lattice for the samesamples somewhat increase and amount to 0.44260,0.44308, and 0.44370 nm.

The structure of the annealed NbCy samples wasthoroughly analyzed using neutron diffraction(Fig. 9). It can be seen from Fig. 9 that, apart from thestructure reflections, the neutron diffraction patternscontain identical sets of additional reflections withpositions that can correspond to one of the superstruc�tures M6C5.

The final refinement of the phase structure wasperformed with the X’Pert Plus program package [41].The neutron diffraction pattern of the annealedNbC0.83 sample with the composition most similar tothe ideal composition of the Nb6C5 superstructure wasused for the refinement. The calculations were carriedout for the trigonal (P31) and monoclinic (C2, C2/m)models of the M6C5 superstructure.

3



In the angle ranges 2θ ≈ 22.7°, 26.1°, and 30.4°,the experimental neutron diffraction pattern of theannealed NbC0.83 carbide (Fig. 10) does not containreflections characteristic of the monoclinic (C2)model of the M6C5 superstructure. This allows us toimmediately exclude this superstructure from consid�eration. Indeed, the minimization of the neutron dif�fraction pattern with the use of the monoclinic (C2)model even with allowance made for the atomic dis�placements and different occupancies of the positionsof the nonmetal sublattice by carbon atoms leads to alow convergence: RI(RB) > 0.12. The use of the trigo�nal (P31) model results in an increase in the conver�gence (Rp = 0.0597, ωRp = 0.0815, RI = 0.0344). How�ever, the best convergence (Rp = 0.0348, ωRp = 0.0489,RI = 0.0180) is achieved using the monoclinic modelof the Nb6C5 superstructure with space group C2/m.

The results of the minimization are presented inFig. 10 and Table 5. The occupancies of the 2(d), 4(g),and 4(h) positions by carbon atoms appear to beslightly lower than unity, whereas the 2(a) positions,which in the ideal ordered phase are completelyvacant, are occupied by a small number of carbonatoms (n1 = 0.302). The difference between the occu�pancies of the positions of the nonmetal sublattice andthe ideal values equal to zero and unity is associatedwith the fact that the degree of long�range order in thecarbide under investigation is lower than the maxi�mum degree.

The long�range order parameters η9, η4, and η3 canbe determined from the obtained occupancies n1, n2,n3, and n4 (Table 5) with due regard for the values takenby the distribution function (4) describing the mono�clinic (C2/m) superstructure M6C5 (Table 3). The esti�mated long�range order parameters are as follows:η9 = 0.712, η4 = 0.688, and η3 = 0.550.

Landesman et al. [25] performed a neutron diffrac�tion investigation of the ordering of the niobium car�bide NbC0.83 and, from a comparison of the trigonal(P31) and monoclinic (C2) models of the M6C5 super�structure, drew the conclusion that their experimentaldata correspond to the trigonal symmetry of the Nb6C5phase. They did not discuss the model of the mono�clinic (C2/m) superstructure M6C5. It follows from theperformed symmetry analysis (Table 1) that, in thetrigonal phase of the M6C5 type, the occupancies ofthe C2 and C3 positions should be identical and equalto n3 = y – η9/6 + η4/6 + η3/6 and that the occupan�cies of the C4 and C5 positions should also be identicaland equal to n4 = y + η9/6 + η4/6 – η3/6. However, in[25, 26], the physically inadmissible result wasobtained in the refinement of the structural parame�ters: different occupancies of the C2, C3, C4, and C5positions by carbon atoms. Therefore, the conclusiondrawn in [25, 26] that the Nb6C5 phase has the trigonal(P31) symmetry casts some doubt.

The monoclinic (C2/m) structure of the orderedNb6C5 phase was previously established in [15, 19, 23,29]. Therefore, the experimental investigation of the

Table 4. Monoclinic (space group C2/m (C12/m1)) superstructure V6C5 (VC0.83) (am = 0.51093(2) nm, bm = 0.88454(3) nm,cm = 0.50897(2) nm, β = 109.436(2)°)


Atomic coordinatesOccupancy

x/am y/bm z/cm

C1 vacancy Special 2(a) 0 0 0 0C2 Special 2(d) 0 0.5 0.5 1.0C3 Special 4(g) 0 0.3333 0 1.0C4 Special 4(h) 0 0.1667 0.5 1.0V1 Special 4(i) 0.255(2) 0 0.750(1) 1.0V2 General 8(j) 0.243(2) 0.6701(6) 0.7364(8) 1.0

Table 5. Monoclinic (space group C2/m (C12/m1)) superstructure Nb6C5 (NbC0.83) (am = 0.54466(6) nm, bm = 0.94821(5) nm,cm = 0.54661(1) nm, β = 109.514(6)°)


Atomic coordinatesOccupancy ± 0.012

x/am y/bm z/cm

C1 vacancy Special 2(a) 0 0 0 0.302C2 Special 2(d) 0 0.5 0.5 0.906C3 Special 4(g) 0 0.334(1) 0 0.921C4 Special 4(h) 0 0.181(1) 0.5 0.975Nb1 Special 4(i) 0.26699 0 0.746(2) 1.0Nb2 General 8(j) 0.245(2) 0.6718(9) 0.748(2) 1.0

432


GUSEV

niobium carbide in combination with the data avail�able in the literature indicates the formation of themonoclinic (C2/m) superstructure Nb6C5.

5. CONCLUSIONS

Thus, the performed symmetry analysis of M6C5�type superstructures, the critical review of electrondiffraction data, and the experimental study of thestructure of ordered vanadium and niobium carbidesin comparison with the data available in the literaturegive grounds to believe that a decrease in temperaturecan lead to two sequences of transformations associ�ated with the formation of M6C5 phases. The firstsequence, i.e., the transformation from the disorderedcubic (Fm m) phase MCy into the ordered mono�clinic (C2/m) phase M6C5 and, then, into the orderedmonoclinic (C2) phase M6C5, involves only the disor�der–order and order–order transformations whichoccur with reduction of the symmetry by factors of 36and 4, respectively. An alternative sequence, namely,the transformation from the disordered cubic (Fm m)phase MCy into the ordered trigonal (P31) phase M6C5and, then, into the monoclinic (C2) phase M6C5,includes the disorder–order transformation, whichoccurs with reduction of the symmetry by a factor of72, and the polymorphic transformation of the trigo�nal phase into the monoclinic phase. As can be judgedfrom the experimental data, the first sequence is moreprobable. It can be assumed that the factors responsi�ble for the occurrence of a particular sequence arerelated to the macroscopic state of nonstoichiometriccarbides, namely, with the size and morphology ofgrains of the disordered phase and with the onset of theformation of the primary ordered phase on a specificcrystallographic surface.

ACKNOWLEDGMENTS

I would like to thank A.A. Rempel, V.N. Lipatni�kov, and A.S. Kurlov for their assistance in performingthe experiments and helpful discussions.

REFERENCES

1. A. I. Gusev, A. A. Rempel, and A. J. Magerl, Disorderand Order in Strongly Nonstoichiometric Compounds:Transition Metal Carbides, Nitrides, and Oxides(Springer, Berlin, 2001).

2. J. D. Venables, D. Kahn, and R. G. Lye, Philos. Mag.18, 177 (1968).

3. J. D. Venables and R. G. Lye, Philos. Mag. 19, 565(1969).

4. D. Kahn and R. G. Lye, Bull. Am. Phys. Soc. 14, 332(1969).

5. I. Karimov, F. Faіzullaev, M. Kalanov, A. Émiraliev,A. S. Rakhimov, L. Slepoі, and V. S. Polishchuk, Dokl.Akad. Nauk Uzb. SSR, No. 2, 32 (1976).

3

3

6. I. Karimov, F. Faіzullaev, M. Kalanov, A. Émiraliev, andV. S. Polishchuk, Izv. Akad. Nauk Uzb. SSR. Ser. Fiz.–Mat. Nauk, No. 4, 87 (1978).

7. V. N. Lipatnikov, W. Lengauer, P. Ettmayer, E. Keil,G. Groboth, and E. Kny, J. Alloys Compd. 261, 192(1997).

8. B. V. Khaenko and V. V. Kukol’, Dokl. Akad. Nauk Ukr.SSR, Ser. A: Fiz.–Mat. Tekh. Nauki, No. 1, 78 (1987).

9. B. V. Khaenko, V. V. Kukol’, and L. S. Ershova, Izv.Akad. Nauk SSSR, Neorg. Mater. 25, 263 (1989).

10. J. Billingham, P. S. Bell, and M. H. Lewis, Philos. Mag.25, 661 (1972).

11. J. Billingham, P. S. Bell, and M. H. Lewis, Acta Crys�tallogr., Sect A: Cryst. Phys., Diffr., Theor. Gen. Crys�tallogr. 28, 602 (1972).

12. M. H. Lewis, J. Billingham, and P. S. Bell, in Solid StateChemistry: Proceedings of the Fifth International Materi�als Research Symposium, Berkley, CA, United States,1972 (Berkley, 1972), p. 1084 (NBS Special Publ. (US),No. 364 (1972)).

13. K. Hiraga, Philos. Mag. 27, 1301 (1973).14. I. Karimov, F. Faіzullaev, M. Kalanov, A. Émiraliev, and

V. S. Polishchuk, Izv. Akad. Nauk Uzb. SSR, Ser. Fiz.–Mat. Nauk, No. 4, 74 (1976).

15. R. Kesri and S. Hamar�Thibault, Acta Metall. 36, 149(1988).

16. A. A. Rempel’, A. I. Gusev, V. G. Zubkov, andG. P. Shveіkin, Dokl. Akad. Nauk SSSR 275 (4), 883(1984) [Sov. Phys. Dokl. 29 (4), 257 (1984)].

17. A. I. Gusev and A. A. Rempel’, Fiz. Tverd. Tela (Lenin�grad) 26 (12), 3622 (1984) [Sov. Phys. Solid State 26(12), 2178 (1984)].

18. A. A. Rempel’ and A. I. Gusev, Kristallografiya 30 (6),1112 (1985) [Sov. Phys. Crystallogr. 30 (6), 648 (1985)].

19. A. I. Gusev and A. A. Rempel, Phys. Status Solidi A 93,71 (1986).

20. V. N. Lipatnikov, A. I. Gusev, P. Ettmayer, and W. Len�gauer, J. Phys.: Condens. Matter 11, 163 (1999).

21. V. N. Lipatnikov, A. I. Gusev, P. Ettmeier, andW. Lengauer, Fiz. Tverd. Tela (St. Petersburg) 41 (3),529 (1999) [Phys. Solid State 41 (3), 474 (1999)].

22. A. I. Gusev and A. A. Rempel, Phys. Status Solidi A135, 15 (1993).

23. A. A. Rempel’ and A. I. Gusev, Ordering in Nonstoichi�ometric Niobium Monocarbide (Ural Scientific Centerof the Academy of Sciences of the Soviet Union, Sver�dlovsk, 1983) [in Russian].

24. J. D. Venables and M. H. Meyerhoff, in Solid StateChemistry: Proceedings of the Fifth International Materi�als Research Symposium, Berkley, CA, United States,1972 (Berkley, 1972), p. 583 (NBS Special Publ. (US),No. 364 (1972)).

25. J. P. Landesman, A. N. Christensen, C. H. de Novion,C. H. Lorenzelli, and P. Convert, J. Phys. C: Solid StatePhys. 18, 809 (1985).

26. A. N. Christensen, Acta Chem. Scand., Ser. A 39, 803(1985).

27. M. P. Arbuzov, B. V. Khaenko, and O. P. Sivak, Dokl.Akad. Nauk USSR, Ser. A: Fiz.–Mat. Tekh. Nauki,No. 10, 86 (1984).



28. B. V. Khaenko, O. P. Sivak, and V. S. Sinel’nikova, Izv.Akad. Nauk SSSR, Neorg. Mater. 20, 1825 (1984).

29. B. V. Khaenko and O. P. Sivak, Kristallografiya 35 (5),1110 (1990) [Sov. Phys. Crystallogr. 35 (5), 653 (1990)].

30. A. I. Gusev, A. A. Rempel, and V. N. Lipatnikov,J. Phys.: Condens. Matter 43, 8277 (1996).

31. A. A. Rempel’, V. N. Lipatnikov, and A. I. Gusev, Dokl.Akad. Nauk SSSR 310, 878 (1990) [Sov. Phys. Dokl.35, 103 (1990)].

32. A. I. Gusev, A. A. Rempel’, and V. N. Lipatnikov, Fiz.Tverd. Tela (Leningrad) 33 (8), 2298 (1991) [Sov. Phys.Solid State 33 (8), 1295 (1991)].

33. V. N. Lipatnikov and A. A. Rempel’, Pis’ma Zh. Éksp.Teor. Fiz. 81 (7), 410 (2005) [JETP Lett. 81 (7), 326(2005)].

34. A. I. Gusev and A. A. Rempel, Phys. Status Solidi A163, 273 (1997).

35. A. I. Gusev and A. A. Rempel, in Materials Science ofCarbides, Nitrides, and Borides, Ed. by Y. G. Gogotsiand R. A. Andrievski (Kluwer, Dordrecht, The Nether�lands, 1999), p. 47.

36. A. I. Gusev, Usp. Fiz. Nauk 170 (1), 3 (2000) [Phys.—Usp. 43 (1), 1 (2000)].

37. A. I. Gusev, Nonstoichiometry, Disorder, Short�Rangeand Long�Range Orders in Solids (Fizmatlit, Moscow,2007) [in Russian].

38. A. A. Rempel and A. I. Gusev, Phys. Status Solidi B130, 413 (1985).

39. A. A. Rempel’ and A. I. Gusev, Fiz. Met. Metalloved.60, 847 (1985).

40. A. I. Gusev and A. A. Rempel, J. Phys. C: Solid StatePhys. 20, 5011 (1987).

41. X’Pert Plus Version 1.0: Program for Crystallography andRietveld Analysis (Philips Analytical B. V., KoninklijkePhilips Electronics N. V., Almelo, The Netherlands,1999).

42. A. I. Gusev and A. A. Rempel’, Nonstoichiometry, Dis�order, and Order in Solids (Ural Branch of the RussianAcademy of Sciences, Yekaterinburg, 2001) [in Rus�sian].

43. O. V. Kovalev, Irreducible Representations of the SpaceGroups (Naukova Dumka, Kiev, 1961; Gordon andBreach, New York, 1965).

44. O. V. Kovalev, Representation of Crystallographic SpaceGroups: Irreducible Representations, Induced Represen�tation, and Corepresentations (Nauka, Moscow, 1986;Gordon and Breach, Amsterdam, 1993).

45. A. G. Khachaturyan, Theory of Phase Transformationsand the Structure of Solid Solutions (Nauka, Moscow,1974) [in Russian].

46. D. Rafaja, W. Lengauer, P. Ettmayer, and V. N. Lipatni�kov, J. Alloys Compd. 269, 60 (1998).

Translated by O. Borovik�Romanova

Documents

Sequence of phase transformations in the formation of superstructures of the M6C5 type in nonstoichiometric carbides