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Elastic diffuse neutron scattering study of the defect structure of TiC0.76 and NbC0.73
0038-1098/81/290661-05$02.00/O
%Solid State Communications, Vol. 39, pp. 661-665. Pergamon Press Ltd. 1981. Printed in Great Britain.
ELASTIC DIFFUSE
NEUTRON SCATTERING
STUDY OF THE DEFECT STRUCTURE
OF TiC0.76 AND NbC,,73
V. Moisy-Maurice and C.H. de Novion SESI, CEN, BP 6,92260
Fontenay-aux-Roses,
France
A.N. Christensen Department
of Chemistry, Aarhus University,
Aarhus, Denmark
and W. Just ILL, 156 X Centre de Tri, 38042 Grenoble Cedex, France (Received 19 January 1981; in revised form 20 March 1981 by E. F. Bertaut) The elastic diffuse neutron scattering of the non-stoichiometric carbides TiC0.76 and NbC0.73 has been measured at room temperature, in the (110) reciprocal lattice plane, for 0.2 < 4 = 4?r sin B/h < 4 A-‘. The CowleyWarren short-range order parameters were obtained for the eight first shells of neighbours of the metalloid f.c.c. sublattice. The degree of short-range ordering was found much larger in NbCe.,a than in TiC0.76. Strong displacements of the metal atoms towards their first neighbour carbons occur in both cases.
sublattice, and the atomic relaxations effect of vacancies.
1. INTRODUCTION REFRACTORY TRANSITION METAL CARBIDES with rocksalt crystal structure present many fascinating properties, such as extreme hardness and superconductivity, associated to a complex chemical bonding [ 11. Some of these properties, in particular the plastic and transport properties, are controlled by a departure from the stoichiometric composition MC; one knows indeed that these compounds may be obtained at room temperature in a wide range of C/M ratio: for example, the “Tic” phase extends from TiCe.s,, to TiCe.a, and the “NbC” phase from NbCe,re to NbCo.99, the defects responsible for non-stoichiometry being carbon vacancies [I]. Several superlattice structures corresponding to long range ordering of carbon vacancies were found in these compounds [2]. Short-range ordering of carbon vacancies was also observed, especially in the TiC,,.--o., range and in all the NbCr_, carbides, by the existence of diffuse streaks in the electron diffraction spectra [3]. A quantitative interpretation of this short-range ordering was only made for a VCo.,5 single crystal [4] and for NbCr-, powders [S] , from neutron diffuse scattering data; but, in both studies, the atomic relaxations were not taken into account. Here, we present an elastic diffuse neutron scattering study of the non-stoichiometric carbides TiC0.76 and NbCo.n, where vacancies do not show long-range order. It allowed to obtain, using the Sparks and Borie method [6], the short-range order parameters of the carbon
due to the size
2. EXPERIMENTAL Single crystals TiCe.,e and Nb(&a were grown from sintered specimens (with nominal purities 99.9%) by a zone melting procedure described elsewhere [7]. The crystals were grown on seeds parallel to the ( li0 > axis. The NbCe.,a sample was slowly cooled (65 hr) from the melting point down to 400°C and TiC0.76 was annealed 4 days at 600°C under vacuum. The samples were subsequently cut in cylindrical form (2: 1 cm3) using spark erosion, and their orientation checked by Laue back reflexion method. Their carbon content was determined chemically. The neutron scattering experiment was performed on the two-axis D7 spectrometer of the HFR (ILL, Grenoble) (described in [S]), at room temperature and under vacuum. The incident wavelength was 3.14 A, and time-of-flight analysis allowed to select only the elastically scattered neutrons. The data were obtained for 0.2 < 4 = 4n sin 0/h <4A-’ and calibrated to absolute units by comparison with a vanadium standard, after correction for background and absorption.
661
3. RESULTS Figure 1 shows the elastic scattering differential cross-section in the (li0) reciprocal lattice plane for The data are given in mbarns per Ti(&e and Nb&s.
662
DEFECT STRUCTURE TIC
w
Vol. 39, No. 5
OF TiC0.76 AND NbC0.73
were corrected for a Debye-Waller factor exp (- 2B sin2B/h2) with BM = Bc = 0.25 A2; this correction did not change significantly the displacements and OLevaluation. 4. ANALYSIS OF THE DATA The data, which extend roughly in the two first Brillouin zones, were analysed by a generalisation of the Sparks and Borie method [6], extended to take into account the displacements of the metal atoms. The approximation was restricted to the first order in displacements, the elastic diffuse cross-section being written: do/da)(q)
= C bi bj eiq’(Ri-Rj)( 1 + q
* 6 ii),
i
where q is the scattering vector, Ri and Rj lattice sites, 6ij the relative displacement of atoms i and Z of scattering lengths bi and bj. Let us define in the rocksalt lattice (with Rj taken as origin): Ri = R~mn = Z(aJ2) + m(a2/2) + n(aJ2) (I, m, n, integers), Bi = B Imn = &,,(aJ2) q = 2n]hr(2br)
Fig. 1. Elastic diffuse neutron scattering cross-section_ do/da (in mbarn per metal atom) measured in the (110) reciprocal lattice plane at 300 K. (a) TiC0.76; (b) NbCo.73. metal atom and are not corrected for incoherent scatterthe incoherent scattering of Nb and C are ing do/dfl)i,: in principle weak, but the value for titanium (2: 220 mbarns [9]) corresponds to more than half of the scattering measured for TiC0.76. The diffuse scattering figures are symmetric relative to the (001) and ( 110 ) reciprocal axis, as predicted for a cubic crystal. But the intensity is far from periodic in the reciprocal lattice plane, indicating that atomic displacements are important. The diffuse bands observed in NbCo.73, roughly parallel to the ( 001) reciprocal lattice direction, are similar to those observed by electron diffraction in NbCr_, [4], but the intensity of the second band (when going away in the ( 110 ) direction) is much weaker than that of the first one. The measured data
+ Mr,,(a2/2)
+ M2bz)
+
+ Nr,,(aJ2),
@WI.
(Here, the first Bragg reflexion (111) corresponds to hr = h2 = h3 = l/2.) If I + m + n is even, Rlmn joins two atoms of the same nature (C-C or M-M); if I + m + n is odd, Rlmn joins two atoms of different nature (C-M). One obtains: da/da = (do/da)sno + (do/dn),n with, taking into account that the crystal is centrosymmetric: da da (-1
SRO
- Nx(1 -x)b& -
x
cm (2nh 1I) cos (2rh2 m) cos (2nh,n)] hmn eYen (short-range order cross-section, periodic in the reciprocal space and independent of displacements). l+h
x sin 2n(h,Z + h,m
+ h3n)
(displacement cross-section). For Z+ m + n even:
DEFECTS STRUCTURE
Vol. 39, No. 5
663
OF TiC0.76 AND NbCe.,s
Z+m+nodd: B,L,, = 4n(l
-x)bob&rz”;
(respBr%,, Bk,).
(1 -x is the carbon content; the average interatomic metal displacements are zero: Syi, = 0 for I+ m + n even.) In the (1iO) reciprocal lattice plane, hz = h 1 and f(h 1, h2, h3) will be written f(h,, hs). According to the Sparks and Borie method [6], one may write: R(h,, hs) = do/d.n(h,,
hs) - do/da(hr,
Q(hr, ha) = do/dWrr,
W - dc/dW
(doldW,,(&W
1 -hs); -hr,
hs);
= ~IQ + h,R; = (du/dfi)
(ddd%to
- (du/dQ)*n.
R, Q, and therefore (do/da),,, and (du/dfi)ano + may be obtained by linear combination of (dolda)i, the cross-sections measured in different cells of the (iTo)reciprocal plane. By Fourier transforms of (du/da)aRO, R and Q, one obtains a set of coefficients _$ p and x: 112 112 -%jk
=
I 0
d
x cos (2nh,i)
Nx(l
Ix)b6(%)~no
cos (2nhj)
cos (2nh3k)dh,dh3,
112 112 pijk
=
5
1
0
0
R(hl,ha)
cos(2nhJ)cos
(2nhlj)
0
O&6
050
Fig. 2. Comparison of calculated (- - - - -) and measured ( -) (du/da)An for TiC0.,6 (in mbarn per Ti atom).
x sin (2nh3k) dhr dh3, Table 1
112 112
i Xijk
=
51
0
Q(h,, h3) sin (2nhIi)
cos (2nhJ)
0
x cos (2nh3k)dhIdh3. The pi jk are linear combinations of the short-range order coefficients crrmn; gjk=Oifi+j+kisodd,and ~jk = ~jik. Epperson and Fiirnrohr [lo] showed that the recovery of the a! from the _w’ by resolution of a set of linear equations is good, if the correlations do not extend too far in the lattice, and if the chosen reciprocal lattice plane contains the extrema of diffuse scattering; this is probably’ the case here, as the (1 TO) reciprocal plane contains the three directions of symmetry ( 001 >, ( 110 >, ( 111 >, and as in TiCoeM and NbCo.so, sharp maxima of diffuse scattering are clearly seen in l/2 l/2 l/2,3/2 l/2 l/2, . . . [ll]. !hhdy, the pi jk and X& are linear combinations of the BIL,,, and therefore of the relative displacement components Lf$ and LF;F. These displacements were also obtained from the p and x, by resolution of a set of linear equations; here the displacements were limited to four shells of neighbours (two C-M and two C-C), in
almn a1
=
TiCo.76
NbCo.73
0.005(0.005) 0.080(0.005) + 0.013(0.005) + 0.006 0.003 + 0.025 0.007 + 0.003
+ + + +
a110
-
-
a2
=
a200
a3
=
a211
a4
=
Q220
% =
a310
0L6 =
&222
OL7= ff321 % = @4r!tI
0.095(0.010) 0.275(0.010) 0.05 l(O.005) 0.072 0.044 0.030 0.020 0.030
which particular case,&,, on the average must be parallel to the interatomic vector to maintain statistically the cubic symmetry [6]. The detailed formalism used and the complete set of equations will be presented elsewhere [ 111. 4. DISCUSSION AND CONCLUSION The cr coefficients obtained, up to 8 shells of neighbours (with error bars for the three first ai) are given in Table 1 for the two studied samples, TiC0,76 and
664
DEFECTS STRUCTURE
OS0
Fig. 3. Comparison of calculated (- - ~ - -) and + (du/da) for: (a) -) (dcldn) ~~~~$!) NbC0.73. (in m&r%per metal a;“o”m) Dotted line (--------): interse\tion of the (li0) reciprocal lattice plane with the surface: cos n(2h,) + cos n(2h,) + cos rr(2h3) = 0. Table 2 fi lmn
CA>
&y&M &C-M 111
%c s%?
TiCo.76
NbCo.73
- 0.030(0.010) + 0.006(0.004) + 0.01 l(O.004) + O.OOS(O.004)
+ + +
0.033(0.010) 0.01 l(O.006) 0.006(0.004) 0.004(0.004)
NbC0.73. The first CYterm (aooo) is found equal to the theoretical value 1 if one takes incoherent cross-sections of 226 and 32 mbarn respectively in TiC0.,6 and values which are near the ones given in the mco.73, literature [9]. The relative average radial displacements 6zmn of C-C and C-M pairs are given (in Angstroms) in Table 2. From the obtained parameters, the (du/dQ),o and cross-sections may be reconstructed. On (do/d%,, Fig. 2, one compares the calculated to the experimental (da/da),, for TiCo.76: the agreement is good. On Fig. 3, the calculated and experimental (du/dQ)ano are compared for both samples. For NbC0.73, the calculated figure does not show such a sharp maximum as the
OF Ti&e
AND Nb(&a
Vol. 39, No. 5
measured, which is clearly due to the fact that correlations extend further than nine shells of neighbours: from the half-width of the diffuse bands, one infers in this sample a correlation length of the order of 30 8. For show a broad TiCe.y6, the calculated curves (do/dSI)sno maximum around l/2 l/2 l/2 which is not observed: we attribute this discrepancy to an imperfect separation of the short-range order and displacement contributions to (do/dQ), as we applied the Sparks and Borie method to the first order in displacements, and as the second order displacement terms are known to reduce the intensity of short-range order maxima. Although the short-range ordering is found much weaker in TiC0,76 than in NbC0.73, two common conclusions may be deduced for both samples from the observation of Tables 1 and 2: the negative values of (Ye (meaning that vacancies avoid to put themselves as second neighbours), and the shortening (2 0.03 A) of the metal-carbon first neighbour distances. Both facts suggest a strong value of the second neighbour pair interaction potential between a vacancy and a carbon, via the metal atom which is at half distance [ 121. The fact that this metal atom moves away from the vacancy and goes nearer the carbon is also found in the T&C and VsC7 superlattices [3]. A strengthening of the metal-carbon bonding due to a redistribution of the metal d electrons originating from the dangling bonds may be suggested. On the other hand, striking differences are observed between the two samples: cyr is found strongly negative and (~a positive in NbC&, showing that in this compound vacancies prefer the third neighbour positions as in the long-range ordered superlattice Nb6CS (for which OCR = a, = - 0.2, a!a = + 0.2) (qualitatively similar results were obtained from the study of NbC,_, powders [5]). On the contrary, in TiC0,76, (or z a3 = 0 (in T&C: (Yr = 63 = 0, a!* = - 1). These results may also be discussed in terms of the cluster models developed first by Brunel et al. [ 131 for ionic compounds with rocksalt structure of formula A3’B+Xi-, then generalized by Sauvage and Parthe [ 141 and de Ridder et al. [ 151 to different compositions, crystal structures and chemical bonds. If one assumes a maximum short-range ordering in an f.c.c. alloy MCeT3, with the smallest building blocks of the structure having as far as possible a composition identical with the overall composition of the compound (i.e. a mixture of randomly oriented CJII, and CsO, octahedra, 0 = carbon vacancy), one should have 401r + (Y*= - 0.80; this is not far from the value measured for Nb&a: - 0.65(0.05). The diffuse scattering (da/da)sno should then be distributed near the surface cos n(2ht) + cos 7r(2h,) + cos n(2h3) = 0, which is effectively the case [see Fig. 3(b)]. Therefore the cluster model (or the so-called “transition state” [ 151) describes reasonably well the
Vol. 39, No. 5
DEFECTS STRUCTURE
short-range ordering in NbCe.,a. On the contrary, TiC0,76 for which 4o, + (Ye= - O.lO(O.02) and is very diffuse in the reciprocal space, is (dc/d%Ro very largely disordered. Acknowledgements - Carlsbergfonden is acknowledged for the spark erosion machine. The Department of Geology at Aarhus University is acknowledged for the carbon analysis of the specimens.
OF TiC0.76 AND NbCe.,a
7. 8. 9. 10. 11 II.
REFERENCES 1. 2. 3.
4. 5.
6.
L.E. Toth, Transition Metal Carbides and Nitrides. Academic Press, London (1971). See for example, C.H. de Novion & V. Maurice, J. Phys. CoZZ.38, C7-211 (1977). J. Billingham, P.S. Bell & M.H. Lewis, Acta Cyst. A28,602 (1972); M. Sauvage & E. ParthC,Acta 0yst. A28,607 (1972). M. Sauvage, E. Parthe & W.B. Yelon,Acta Cryst. A30,597 (1974). B.E.F. Fender, Chemical Applications of Thermal Neutron Scattering (Edited by B.T.M. Willis), p. 250. Oxford University Press, Oxford (1973). B. Borie, Acta Cryst. 14,472 (1961); C.J. Sparks & B. Borie, Local Atomic Arrangements Studied
12.
13. 14. 15.
665
by X-ray Diffraction, p. 5. Gordon and Breach, New York (1965). A.N. Christensen, J. Crystal Growth 33,99 (1976). G. Bauer, E. Seitz & W. Just, J. Appl. Ciystallogr. 8, 162 (1975). G.E. Bacon, “Neutron Diffraction”, Clarendon Press, Oxford (1975). J.E. Epperson & P. Fiirnrohr, J. Appl. Crystallogr. 8,115 (1975). V. Moisy-Maurice, C.H. de Novion, A.N. Christensen & W. Just (to be published); V. Moisy-Maurice, Thesis, Universite de Strasbourg, France (1981). A preliminary fit of the observed short-range ordering cross-section of TiC0.76 in terms of two pair interaction potentials on the f.c.c. metalloid sublattice, using the method of Moss and Clapp [Phys. Rev. 171,764 (1968)] gives VI 2: 3.2meV and V, 2 7.1 meV between first and second neighbours respectively. M. Brunel, F. de Bergevin & M. Gondrand,J. Phys. Chem. Solids 33, 1927 (1972). M. Sauvage & E. Parthe, Acta Cryst. A30,239 (1974). R. de Ridder, G. van Tendeloo, D. van Dyck & S. Amelinckx, Phys. Status Solidi (a) 38,663 (1976).
%Solid State Communications, Vol. 39, pp. 661-665. Pergamon Press Ltd. 1981. Printed in Great Britain.
ELASTIC DIFFUSE
NEUTRON SCATTERING
STUDY OF THE DEFECT STRUCTURE
OF TiC0.76 AND NbC,,73
V. Moisy-Maurice and C.H. de Novion SESI, CEN, BP 6,92260
Fontenay-aux-Roses,
France
A.N. Christensen Department
of Chemistry, Aarhus University,
Aarhus, Denmark
and W. Just ILL, 156 X Centre de Tri, 38042 Grenoble Cedex, France (Received 19 January 1981; in revised form 20 March 1981 by E. F. Bertaut) The elastic diffuse neutron scattering of the non-stoichiometric carbides TiC0.76 and NbC0.73 has been measured at room temperature, in the (110) reciprocal lattice plane, for 0.2 < 4 = 4?r sin B/h < 4 A-‘. The CowleyWarren short-range order parameters were obtained for the eight first shells of neighbours of the metalloid f.c.c. sublattice. The degree of short-range ordering was found much larger in NbCe.,a than in TiC0.76. Strong displacements of the metal atoms towards their first neighbour carbons occur in both cases.
sublattice, and the atomic relaxations effect of vacancies.
1. INTRODUCTION REFRACTORY TRANSITION METAL CARBIDES with rocksalt crystal structure present many fascinating properties, such as extreme hardness and superconductivity, associated to a complex chemical bonding [ 11. Some of these properties, in particular the plastic and transport properties, are controlled by a departure from the stoichiometric composition MC; one knows indeed that these compounds may be obtained at room temperature in a wide range of C/M ratio: for example, the “Tic” phase extends from TiCe.s,, to TiCe.a, and the “NbC” phase from NbCe,re to NbCo.99, the defects responsible for non-stoichiometry being carbon vacancies [I]. Several superlattice structures corresponding to long range ordering of carbon vacancies were found in these compounds [2]. Short-range ordering of carbon vacancies was also observed, especially in the TiC,,.--o., range and in all the NbCr_, carbides, by the existence of diffuse streaks in the electron diffraction spectra [3]. A quantitative interpretation of this short-range ordering was only made for a VCo.,5 single crystal [4] and for NbCr-, powders [S] , from neutron diffuse scattering data; but, in both studies, the atomic relaxations were not taken into account. Here, we present an elastic diffuse neutron scattering study of the non-stoichiometric carbides TiC0.76 and NbCo.n, where vacancies do not show long-range order. It allowed to obtain, using the Sparks and Borie method [6], the short-range order parameters of the carbon
due to the size
2. EXPERIMENTAL Single crystals TiCe.,e and Nb(&a were grown from sintered specimens (with nominal purities 99.9%) by a zone melting procedure described elsewhere [7]. The crystals were grown on seeds parallel to the ( li0 > axis. The NbCe.,a sample was slowly cooled (65 hr) from the melting point down to 400°C and TiC0.76 was annealed 4 days at 600°C under vacuum. The samples were subsequently cut in cylindrical form (2: 1 cm3) using spark erosion, and their orientation checked by Laue back reflexion method. Their carbon content was determined chemically. The neutron scattering experiment was performed on the two-axis D7 spectrometer of the HFR (ILL, Grenoble) (described in [S]), at room temperature and under vacuum. The incident wavelength was 3.14 A, and time-of-flight analysis allowed to select only the elastically scattered neutrons. The data were obtained for 0.2 < 4 = 4n sin 0/h <4A-’ and calibrated to absolute units by comparison with a vanadium standard, after correction for background and absorption.
661
3. RESULTS Figure 1 shows the elastic scattering differential cross-section in the (li0) reciprocal lattice plane for The data are given in mbarns per Ti(&e and Nb&s.
662
DEFECT STRUCTURE TIC
w
Vol. 39, No. 5
OF TiC0.76 AND NbC0.73
were corrected for a Debye-Waller factor exp (- 2B sin2B/h2) with BM = Bc = 0.25 A2; this correction did not change significantly the displacements and OLevaluation. 4. ANALYSIS OF THE DATA The data, which extend roughly in the two first Brillouin zones, were analysed by a generalisation of the Sparks and Borie method [6], extended to take into account the displacements of the metal atoms. The approximation was restricted to the first order in displacements, the elastic diffuse cross-section being written: do/da)(q)
= C bi bj eiq’(Ri-Rj)( 1 + q
* 6 ii),
i
where q is the scattering vector, Ri and Rj lattice sites, 6ij the relative displacement of atoms i and Z of scattering lengths bi and bj. Let us define in the rocksalt lattice (with Rj taken as origin): Ri = R~mn = Z(aJ2) + m(a2/2) + n(aJ2) (I, m, n, integers), Bi = B Imn = &,,(aJ2) q = 2n]hr(2br)
Fig. 1. Elastic diffuse neutron scattering cross-section_ do/da (in mbarn per metal atom) measured in the (110) reciprocal lattice plane at 300 K. (a) TiC0.76; (b) NbCo.73. metal atom and are not corrected for incoherent scatterthe incoherent scattering of Nb and C are ing do/dfl)i,: in principle weak, but the value for titanium (2: 220 mbarns [9]) corresponds to more than half of the scattering measured for TiC0.76. The diffuse scattering figures are symmetric relative to the (001) and ( 110 ) reciprocal axis, as predicted for a cubic crystal. But the intensity is far from periodic in the reciprocal lattice plane, indicating that atomic displacements are important. The diffuse bands observed in NbCo.73, roughly parallel to the ( 001) reciprocal lattice direction, are similar to those observed by electron diffraction in NbCr_, [4], but the intensity of the second band (when going away in the ( 110 ) direction) is much weaker than that of the first one. The measured data
+ Mr,,(a2/2)
+ M2bz)
+
+ Nr,,(aJ2),
@WI.
(Here, the first Bragg reflexion (111) corresponds to hr = h2 = h3 = l/2.) If I + m + n is even, Rlmn joins two atoms of the same nature (C-C or M-M); if I + m + n is odd, Rlmn joins two atoms of different nature (C-M). One obtains: da/da = (do/da)sno + (do/dn),n with, taking into account that the crystal is centrosymmetric: da da (-1
SRO
- Nx(1 -x)b& -
x
cm (2nh 1I) cos (2rh2 m) cos (2nh,n)] hmn eYen (short-range order cross-section, periodic in the reciprocal space and independent of displacements). l+h
x sin 2n(h,Z + h,m
+ h3n)
(displacement cross-section). For Z+ m + n even:
DEFECTS STRUCTURE
Vol. 39, No. 5
663
OF TiC0.76 AND NbCe.,s
Z+m+nodd: B,L,, = 4n(l
-x)bob&rz”;
(respBr%,, Bk,).
(1 -x is the carbon content; the average interatomic metal displacements are zero: Syi, = 0 for I+ m + n even.) In the (1iO) reciprocal lattice plane, hz = h 1 and f(h 1, h2, h3) will be written f(h,, hs). According to the Sparks and Borie method [6], one may write: R(h,, hs) = do/d.n(h,,
hs) - do/da(hr,
Q(hr, ha) = do/dWrr,
W - dc/dW
(doldW,,(&W
1 -hs); -hr,
hs);
= ~IQ + h,R; = (du/dfi)
(ddd%to
- (du/dQ)*n.
R, Q, and therefore (do/da),,, and (du/dfi)ano + may be obtained by linear combination of (dolda)i, the cross-sections measured in different cells of the (iTo)reciprocal plane. By Fourier transforms of (du/da)aRO, R and Q, one obtains a set of coefficients _$ p and x: 112 112 -%jk
=
I 0
d
x cos (2nh,i)
Nx(l
Ix)b6(%)~no
cos (2nhj)
cos (2nh3k)dh,dh3,
112 112 pijk
=
5
1
0
0
R(hl,ha)
cos(2nhJ)cos
(2nhlj)
0
O&6
050
Fig. 2. Comparison of calculated (- - - - -) and measured ( -) (du/da)An for TiC0.,6 (in mbarn per Ti atom).
x sin (2nh3k) dhr dh3, Table 1
112 112
i Xijk
=
51
0
Q(h,, h3) sin (2nhIi)
cos (2nhJ)
0
x cos (2nh3k)dhIdh3. The pi jk are linear combinations of the short-range order coefficients crrmn; gjk=Oifi+j+kisodd,and ~jk = ~jik. Epperson and Fiirnrohr [lo] showed that the recovery of the a! from the _w’ by resolution of a set of linear equations is good, if the correlations do not extend too far in the lattice, and if the chosen reciprocal lattice plane contains the extrema of diffuse scattering; this is probably’ the case here, as the (1 TO) reciprocal plane contains the three directions of symmetry ( 001 >, ( 110 >, ( 111 >, and as in TiCoeM and NbCo.so, sharp maxima of diffuse scattering are clearly seen in l/2 l/2 l/2,3/2 l/2 l/2, . . . [ll]. !hhdy, the pi jk and X& are linear combinations of the BIL,,, and therefore of the relative displacement components Lf$ and LF;F. These displacements were also obtained from the p and x, by resolution of a set of linear equations; here the displacements were limited to four shells of neighbours (two C-M and two C-C), in
almn a1
=
TiCo.76
NbCo.73
0.005(0.005) 0.080(0.005) + 0.013(0.005) + 0.006 0.003 + 0.025 0.007 + 0.003
+ + + +
a110
-
-
a2
=
a200
a3
=
a211
a4
=
Q220
% =
a310
0L6 =
&222
OL7= ff321 % = @4r!tI
0.095(0.010) 0.275(0.010) 0.05 l(O.005) 0.072 0.044 0.030 0.020 0.030
which particular case,&,, on the average must be parallel to the interatomic vector to maintain statistically the cubic symmetry [6]. The detailed formalism used and the complete set of equations will be presented elsewhere [ 111. 4. DISCUSSION AND CONCLUSION The cr coefficients obtained, up to 8 shells of neighbours (with error bars for the three first ai) are given in Table 1 for the two studied samples, TiC0,76 and
664
DEFECTS STRUCTURE
OS0
Fig. 3. Comparison of calculated (- - ~ - -) and + (du/da) for: (a) -) (dcldn) ~~~~$!) NbC0.73. (in m&r%per metal a;“o”m) Dotted line (--------): interse\tion of the (li0) reciprocal lattice plane with the surface: cos n(2h,) + cos n(2h,) + cos rr(2h3) = 0. Table 2 fi lmn
CA>
&y&M &C-M 111
%c s%?
TiCo.76
NbCo.73
- 0.030(0.010) + 0.006(0.004) + 0.01 l(O.004) + O.OOS(O.004)
+ + +
0.033(0.010) 0.01 l(O.006) 0.006(0.004) 0.004(0.004)
NbC0.73. The first CYterm (aooo) is found equal to the theoretical value 1 if one takes incoherent cross-sections of 226 and 32 mbarn respectively in TiC0.,6 and values which are near the ones given in the mco.73, literature [9]. The relative average radial displacements 6zmn of C-C and C-M pairs are given (in Angstroms) in Table 2. From the obtained parameters, the (du/dQ),o and cross-sections may be reconstructed. On (do/d%,, Fig. 2, one compares the calculated to the experimental (da/da),, for TiCo.76: the agreement is good. On Fig. 3, the calculated and experimental (du/dQ)ano are compared for both samples. For NbC0.73, the calculated figure does not show such a sharp maximum as the
OF Ti&e
AND Nb(&a
Vol. 39, No. 5
measured, which is clearly due to the fact that correlations extend further than nine shells of neighbours: from the half-width of the diffuse bands, one infers in this sample a correlation length of the order of 30 8. For show a broad TiCe.y6, the calculated curves (do/dSI)sno maximum around l/2 l/2 l/2 which is not observed: we attribute this discrepancy to an imperfect separation of the short-range order and displacement contributions to (do/dQ), as we applied the Sparks and Borie method to the first order in displacements, and as the second order displacement terms are known to reduce the intensity of short-range order maxima. Although the short-range ordering is found much weaker in TiC0,76 than in NbC0.73, two common conclusions may be deduced for both samples from the observation of Tables 1 and 2: the negative values of (Ye (meaning that vacancies avoid to put themselves as second neighbours), and the shortening (2 0.03 A) of the metal-carbon first neighbour distances. Both facts suggest a strong value of the second neighbour pair interaction potential between a vacancy and a carbon, via the metal atom which is at half distance [ 121. The fact that this metal atom moves away from the vacancy and goes nearer the carbon is also found in the T&C and VsC7 superlattices [3]. A strengthening of the metal-carbon bonding due to a redistribution of the metal d electrons originating from the dangling bonds may be suggested. On the other hand, striking differences are observed between the two samples: cyr is found strongly negative and (~a positive in NbC&, showing that in this compound vacancies prefer the third neighbour positions as in the long-range ordered superlattice Nb6CS (for which OCR = a, = - 0.2, a!a = + 0.2) (qualitatively similar results were obtained from the study of NbC,_, powders [5]). On the contrary, in TiC0,76, (or z a3 = 0 (in T&C: (Yr = 63 = 0, a!* = - 1). These results may also be discussed in terms of the cluster models developed first by Brunel et al. [ 131 for ionic compounds with rocksalt structure of formula A3’B+Xi-, then generalized by Sauvage and Parthe [ 141 and de Ridder et al. [ 151 to different compositions, crystal structures and chemical bonds. If one assumes a maximum short-range ordering in an f.c.c. alloy MCeT3, with the smallest building blocks of the structure having as far as possible a composition identical with the overall composition of the compound (i.e. a mixture of randomly oriented CJII, and CsO, octahedra, 0 = carbon vacancy), one should have 401r + (Y*= - 0.80; this is not far from the value measured for Nb&a: - 0.65(0.05). The diffuse scattering (da/da)sno should then be distributed near the surface cos n(2ht) + cos 7r(2h,) + cos n(2h3) = 0, which is effectively the case [see Fig. 3(b)]. Therefore the cluster model (or the so-called “transition state” [ 151) describes reasonably well the
Vol. 39, No. 5
DEFECTS STRUCTURE
short-range ordering in NbCe.,a. On the contrary, TiC0,76 for which 4o, + (Ye= - O.lO(O.02) and is very diffuse in the reciprocal space, is (dc/d%Ro very largely disordered. Acknowledgements - Carlsbergfonden is acknowledged for the spark erosion machine. The Department of Geology at Aarhus University is acknowledged for the carbon analysis of the specimens.
OF TiC0.76 AND NbCe.,a
7. 8. 9. 10. 11 II.
REFERENCES 1. 2. 3.
4. 5.
6.
L.E. Toth, Transition Metal Carbides and Nitrides. Academic Press, London (1971). See for example, C.H. de Novion & V. Maurice, J. Phys. CoZZ.38, C7-211 (1977). J. Billingham, P.S. Bell & M.H. Lewis, Acta Cyst. A28,602 (1972); M. Sauvage & E. ParthC,Acta 0yst. A28,607 (1972). M. Sauvage, E. Parthe & W.B. Yelon,Acta Cryst. A30,597 (1974). B.E.F. Fender, Chemical Applications of Thermal Neutron Scattering (Edited by B.T.M. Willis), p. 250. Oxford University Press, Oxford (1973). B. Borie, Acta Cryst. 14,472 (1961); C.J. Sparks & B. Borie, Local Atomic Arrangements Studied
12.
13. 14. 15.
665
by X-ray Diffraction, p. 5. Gordon and Breach, New York (1965). A.N. Christensen, J. Crystal Growth 33,99 (1976). G. Bauer, E. Seitz & W. Just, J. Appl. Ciystallogr. 8, 162 (1975). G.E. Bacon, “Neutron Diffraction”, Clarendon Press, Oxford (1975). J.E. Epperson & P. Fiirnrohr, J. Appl. Crystallogr. 8,115 (1975). V. Moisy-Maurice, C.H. de Novion, A.N. Christensen & W. Just (to be published); V. Moisy-Maurice, Thesis, Universite de Strasbourg, France (1981). A preliminary fit of the observed short-range ordering cross-section of TiC0.76 in terms of two pair interaction potentials on the f.c.c. metalloid sublattice, using the method of Moss and Clapp [Phys. Rev. 171,764 (1968)] gives VI 2: 3.2meV and V, 2 7.1 meV between first and second neighbours respectively. M. Brunel, F. de Bergevin & M. Gondrand,J. Phys. Chem. Solids 33, 1927 (1972). M. Sauvage & E. Parthe, Acta Cryst. A30,239 (1974). R. de Ridder, G. van Tendeloo, D. van Dyck & S. Amelinckx, Phys. Status Solidi (a) 38,663 (1976).