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Partial pair distribution functions in icosahedral Al-Mn studied by contrast variation in neutron diffraction
J.-M. Dubois, Ch. Janot
To cite this version:
J.-M. Dubois, Ch. Janot. Partial pair distribution functions in icosahedral Al-Mn studied by contrast variation in neutron diffraction. Journal de Physique, 1987, 48 (11), pp.1981-1989.
�10.1051/jphys:0198700480110198100�. �jpa-00210641�
Partial
pair
distribution functions in icosahedral Al-Mn studiedby
contrast variation in neutron diffractionJ.-M. Dubois (*) and Ch. Janot (**)
(*) Unité de Sciences et Génie des Matériaux Métalliques, Ecole des Mines, parc de Saurupt,
54042 Nancy Cedex, France
(**) Institut Laue-Langevin, 156X, 38042 Grenoble Cedex, France (Requ le 14 mai 1987, accept6 le 8 juillet 1987)
Résumé. - Des phases icosaédriques des systèmes Al-Mn et Al-(MnFeCr) ont été étudiées par diffraction des neutrons, jusqu’à des vecteurs de diffusion de 16 Å-1. Les trois fonctions de distribution de paires partielles
ont été calculées. On a pu ainsi confirmer sans ambiguïté le caractère parfaitement isomorphe de la
substitution du Mn par 03C3-FeCr. Les corrélations de paires atomiques apparaissent très semblables à celles de la
phase cristalline 03B1(AlMnSi). En particulier, les plus courtes distances Mn-Mn sont trouvées à 4,47 et 4,97 A.
Abstract. - Icosahedral Al-Mn and Al-(MnFeCr) alloys were studied by neutron diffraction with scattering
vectors up to 16 Å-1. The three partial pair distribution functions (PPDF) were obtained and the actual
isomorphous substitution of the 03C3-FeCr mixture for Mn was confirmed. The PPDF for Mn-Mn indicates the existence of a manganese subnetwork rather similar to that found in the 03B1(AlMnSi) crystalline phase. In particular the shortest Mn-Mn distances were found at 4.47 and 4.97 A.
Classification Physics Abstracts
61.55H - 61.50E - 64.70P - 78.70C
1. Introduction.
The discovery of icosahedral phases of Al-Mn-Si and other alloys [1-4] has generated intense interest
because of the possibility that these phases may be
quasiperiodic crystals. Models ranging from perfect quasiperiodic ordering [5, 6] to interference maxima
arising from close-packed icosahedral clusters [7-10]
have been proposed to explain the observation of
sharp diffraction peaks.
The experimental characterization of the icosahed- ral order in Al-Mn alloys, or other systems, mostly
relies on evidence of five-, three- and twofold axes in electron diffraction patterns [10-16]. Recently, typi-
cal Laue patterns have been obtained, using X-rays [17] or neutrons [18], with millimeter and centimeter sized single domains of quasicrystal of the Al6Li3Cu
system [4]. Four-circle measurements have been also carried out with the same samples [19]. In principle,
the complete structure of a system can be determined from such single crystal data using calculation of 3D- Patterson functions. In fact, such an approach might
be irrelevant to a quasiperiodic structure whose peculiarity is to produce scattering intensity
everywhere in the reciprocal space and not only at Bragg peaks which are the only features entered into Patterson calculations. The same remark applies to high resolution powder diffraction data [20-23], even
when careful contrast variation measurements with neutrons allow the determination of the amplitudes
and phase shifts of the partial structure factors for a
large number of pseudo-Bragg reflexions [24].
Consequently, most of the information regarding microscopic atomic packing in icosahedral phases
has been derived from other probes. Mossbauer [25],
NMR [26, 27] and EXAFS [28-34] measurements have produced some descriptions of the Al and Mn-
atom coordination shells around Mn atoms.
An alternative approach to the conventional crys-
tallography method for diffraction data reduction that may be thought of is to use methods commonly employed to study non-crystalline solids, namely
calculation of pair distribution functions (PDF) by a
direct Fourier transformation of the structure factor.
Obviously, the weak point here is information loss since only average pair distances and coordination numbers can be expected. On the other hand, the
calculation procedure involves isotropic regrouping
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480110198100
1982
in both reciprocal and real spaces which may result in very spurious features in the obtained PDF if the
investigated structure is neither homogeneously dis-
ordered (liquid, amorphous materials) nor described by space group symmetries with inversion centres.
Additionally, it requires structure factors S(Q) to be
determined over a wide range of scattering vector Q
to allow an acceptable spatial resolution of the PDF.
The positive aspect is that not only the speudo-Bragg peaks but also diffuse scattering between them are
Fourier transformed into real space features. The
method, already tested by others [35], was carefully
used in the present work to obtain the partial pair
distribution function (PPDF) Mn-Mn, Al-Al and Al- Mn (Mn-Al) in an Al-Mn icosahedral phase, doped
with 1 at % Si.
2. Experimental methods.
The use of neutrons is particularly attractive in
studying the atomic structure of Al-Mn-Si icosahed- ral compounds, because the neutron scattering length of Mn is negative (bMn = - 0.373 x 10-12 cm)
and because Mn atoms are expected to be substituted
by Cr or Fe which are positive neutron scat- terers (ber = + 0.364 x 10- 12 cm ; bFe = + 0.954 x 10-12 cm) without too much disturbance of the structure [1]. In fact, it has been shown [4, 24, 36, 37] that the substitution of Cr or Fe for Mn is not
perfectly isomorphous. In other words, even if the phases containing Cr or Fe remain apparently icosahedral, they might have atomic structures diffe- rent from that of Al-(pure Mn)-Si compounds.
Fortunately enough, the substitution of an
equiatomic FeCr mixture (the so-called o-phase) for
Mn happens to be isomorphous and random [36, 37],
a point further advocated and confirmed in the next
section of this paper.
Thus, rapidly quenched ribbons of A182Sil [Mnx (FeCr )1 - x ]17 alloys were produced by melt spinning as described in details elsewhere [21, 38], so
that the average scattering length bT of the transition
metal atoms takes values between - 0.373 x
10-12 cm and + 0.658 x 10-12 cm (« pure» u-FeCr mixture) as listed below :
1) A182Si,Mnl7 with
2) Als2Si1 [Mno.64 (FeCr )0.36117 with
3) Als2Si1 (FeCr)17 with
4) Ats2Sii[Mno.s2(FeCr)oig]t7 with
The neutron scattering measurements were carried
out at the High Flux Reactor of the Institut Laue-
Langevin (Grenoble), on the D4 diffractometer set on a hot source beamline (A = 0.5 and 0.7 A). The
Q resolution of this diffractometer, currently used
for structural investigation in liquids and amorphous materials, depends upon the scattering angle but
increases roughly from about 0.06 A-1 to 0.20 A-1
over the measured Q range (FWHM). The raw data,
as shown in figure 1, were corrected for absorption, multiple scattering, inelastic scattering (Placzek cor- rection), incoherent scattering (both nuclear and chemical) and backgrounds such as the scattering
from the vanadium container and ambient noise, to obtain the structure factor up to 16 A-1 1 (spatial
resolution thus limited to about 0.35 A FWHM).
All the samples contained, in addition to icosahed-
ral phase, a dispersed fraction of fcc-Al whose concentration has been determined elsewhere [37].
The diffraction peaks due to the icosahedral phase
can be indexed after Cahn et al. [39], as already
done with high resolution powder diffraction data
[24, 36]. The peak positions in the four measured
Fig. 1. - Neutron diffraction patterns measured with
samples of icosahedral phases of nominal compositions A182Sil [Mn, (FeCr), -.,, 117 with
Fig. 2. - High resolution powder diffraction patterns measured with sample 3 using neutron (bAl/bT = 0.527) and with
an equivalent of sample 1 (no substitution) using X-rays (I Alii Mn = 0.521). (From Ref. [20]).
icosahedral phases exactly match each others and those of the X-ray structure factor [20]. Of course,
intensities change with contrast variation. As a
qualitative evidence for good random isomorphous
substitution of a-(FeCr) for Mn, the neutron diffrac-
tion pattern of sample 3, A182Sil (FeCr )17’ and the X-
ray diffraction pattern of a material [20] not much
different from sample 1, AIs2Si1Mn17, look very similar (see Fig. 2), as expected from similar contrast
effects between Al and T atoms in the two measure-
ments. The diffraction peaks due to crystalline fcc-
Al were subtracted from the raw data patterns using
an experimental Al diffraction pattern measured and corrected in the same conditions as those used for the icosahedral alloys (Fig. 3). From this subtraction
procedure the molecular fractions of fcc-Al in the
alloys were found in the range 5 to 7 %. The total structure factors S(Q) resulting from the Al subtrac-
tion and correction procedure as described above
are shown in figure 4 ; they are almost featureless
beyond 10 Å - 1, as currently observed in very faulted structures, and they correcly converge to unity which
makes easy and accurate the PDF calculation using
direct Fourier transformation, using the equation :
p o is the atomic number density 0.065 at A-3 deter- mined from physical density measurements [38]. The
four corresponding total PDF are shown in figure 5.
Peaks due to Al-Mn (or Al-T) pairs appear as
negative features (see region around R = 2.5 A for
instance) as long as bT is negative (samples 1 and 4) ;
these features disappear in the PDF of sample 2 (bT = 0 ) and become positive in the PDF of sample 3 (bT :> 0) giving a much broader first peak for
instance.
1984
Fig. 3. - Structure factor of fcc-Al measured as for quasi crystalline specimens and the deduced PDF.
Fig. 4. - Total structure factors obtained from the neutron diffraction patterns presented in figure 1 after subtracting fcc-Al and corrections as explained in the text.
Fig. 5. - Total pair distribution functions for the measured
samples as obtained by direct Fourier transformation of the structure factors shown in figure 4.
3. Partial pair distribution functions (PPDF).
The total structure factors are usually expressed in
terms of the compositionally resolved partial struc-
ture factors, as :
where a and /3 denote elements, ca and ba are the
concentration and the neutron coherent scattering length of element a, and S,,p (Q) is the so-called
partial structure factor involving elements a and /3.
Partial pair distribution functions g,,p (R) can be
calculated by direct Fourier transformation of the
SaJ3 (Q), using equation (1), in which po has to be
changed into cp p o. The number of a-,B pairs sepa- rated by a distance R is then given by
4 -,,r R2C. 9 aP (R ) which is called a partial radial
distribution function (PRDF). Forgetting the pre-
sence of Si atoms in a first approximation approach,
the Al-T icosahedral phases can be considered as
pseudo-binary systems in which there are three independent partial structure factors to be deter-
mined, namely SAIAI, SAIT - STAI and STT with the
three corresponding PPDF gAIAI(R), gTAI(R) and
gTT (R). Formally, equation (2) has to be rewritten
as :
with St (Q), i = 1, 2 or 3, the three total structure factors measured for sample number 1, 2 and 3, respectively. The corresponding three linear equation system has to be solved within experimental
accuracy on the Si(Q). As the contrast variations on
the transition metal atoms covered a reasonably
wide range (bT (1 ) = - 0.373, bT (2 ) = 0 and bT(3) = + 0.658 in 10-12 cm), the standard errors
on the three PPDF are only 16, 1 and 3 times the experimental error bar (less than 0.5 %) for T-T, Al-
Al and T-Al pairs, respectively. A critical parameter in the determination of the partial structure factors,
and the corresponding PPDF, from equation (3), is
the true composition of the icosahedral phase, namely the concentration cAl or cT : if cT is overes-
timated, spurious negative minima appear in the
gTI(R) PPDF ; an underestimation of cT, on the contrary, results in these spurious negative minima being observed in the 9AIAI(R). Thus, CT must be introduced as an adjustable parameter whose best value is chosen to minimize the presence of negative
minima in both gTT and 9AIAl PPDF. A by product of
the PPDF calculation is thus the most probable value
of CT = 0.18 which defines the icosahedral phase composition as AI8o.94Si1.06 T 18 or A14.50S.06T, in good agreement with our previous estimations [37, 38].
Such a composition of the icosahedral phase, when compared to the nominal compositions of the as- prepared alloys, leads to a molecular fraction of the residual fcc-Al equal to 5.9 %, in good consistency
with the range of values estimated from the subtrac- tion procedure described in section 2. The best determined PPDF are shown in figure 6.
Now, the fourth sample, not used in the PPDF
calculation, with bT = - 0.187 x 10-12 cm, can pro- vide a very accurate quality check of the procedure.
Using the calculated partial structure factors Saf3 (Q )
and the corresponding PPDF, a total structure factor
S4"(Q) and a total PDF g4cal(R) can be calculated
with equations (1) and (2). These calculated S4cal(Q)
and gcal(R) have to reproduce at best the exper- imental S4exp ( Q ) and the deduced gexP(R) if we have
to be confident in the whole procedure. Looking at figures 7 and 8, everything seems to have worked
more than satisfactorily, including Al subtraction, composition estimation, and, last but not least, isomorphous substitution. A last interesting set of
Fig. 6. - Partial pair distribution functions calculated with data measured on samples 1, 2 and 3.
Fig. 7. - Total structure factor of sample 4 as obtained
from experimental data superimposed on the one calcu-
lated with the partials.
curves is presented in figure 9, giving the three
PRDF from which partial pair distances and inte-
grated partial coordination numbers can be deter- mined.
1986
Fig. 8. - g (R ) for sample 4 as obtained from experimental
data (---) and calculated with the partials (-).
Fig. 9. - Partial radial distribution functions as obtained from the PPDF shown in figure 6. Insert : details of first two coordination shells showing in particular a long tailing
shoulder on the first T-A1 distance.
4. Discussion.
One salient result of the present work is the unam-
biguous confirmation of structural and chemical
isomorphism within the whole range of existence of the AI4.50Sio.o5 [Mnx (FeCr )1 - x] icosahedral system.
Indeed, the total experimental structure factor, as
well as the corresponding total PDF, of a given alloy
of the series, corresponding to a given value of x, can be almost exactly reconstructed by the properly weighted superposition of the three partials deduced
from data obtained with three other alloys of the
series. On the contrary, these three partials are
unable to reproduce total structure factors of PDF
measured on Al-Si- (MnxFel - x ) or Al-Si- (MnxCrl-x) icosahedral alloys in the whole ranges of x values, which demonstrates that substitution of Cr or Fe (each of them alone) for Mn is not perfectly isomorphous.
Looking at the PPDF in figure 6 and/or at the
PRDF in figure 9 leads also to very interesting
conclusions both qualitatively and quantitatively.
First of all, the T-T correlation function exhibits very sharp peaks, with spatial FWHM of 0.40 A (or less) (i.e. about the expected spatial resolution) and
very deep valleys between them, over the whole set
of the measured coordination shells (about 10).
Thus, the PPDF for the Mn-Mn (or T-T) distances
does not look very different from what is expected
for a perfectly ordered crystallized material. For the sake of illustration of this statement, the PDF for fcc Al, as obtained by Fourier transformation of the measured structure factor is shown in figure 3. It
appears clearly that peak width, deepness of the valleys, coherency length... of the Al-PDF and the MnMn-PPDF in icosahedral phases are of very
comparable quality.
In a classical periodic crystal, the pair distances
are very well defined and correspond to the lengths
of crystal vectors. Whatever the chosen origin in
space, periodicity implies that a lattice point sits at a position - R if there is one at a position + R, which makes valid isotropic regrouping for powder of
material with not necessarily isotropic structure.
Thus, the maximum distance for the resolution of the PPDF of a crystal into sharp features will only be
limited by large imperfections in the crystalline
structure and/or the restriction in available scattering
vectors.
In a quasiperiodic structure of the sort described by a Penrose tiling of space, the lattice points are no longer centres of inversion (except for one in special circumstances). Thus, they generally cannot be sim- ultaneously found at positions ± R, with respect to
an origin arbitrarily chosen in space. In addition,
due to tiling using two different units cells with incommensurate sizes, the pair distances are ex- pected to cover a quasi-continuous set of values beyond relatively short distances.
Conclusively, the observation of narrow and gen-
erally unsplit peaks up to quite large R values in the measured PPDF suggest that each atomic site might
be a centre of inversion for the whole corresponding
subnetwork. Among other interests, such a result is
a post-justification of the validity of the isotropic
regrouping operations included in the data reduction
procedure. Close contact between T atoms are not
observed and the smaller coordination T-T shell is
split into two T-T distances at 4.47 A (3.3 atoms)
and 4.97 A (6.0 atoms). Obviously enough, the T
subnetwork seems to behave as the true framework of the whole structure as previously suggested [24, 38].
Comparatively, the Al-Al and T-Al correlations
are less ordered. With the exception of their first peak, the corresponding PPDF or PRDF exhibits
rather broad features which vanish obviously at
much shorter distances than their T-T partner. It appears clearly that the first coordination shell around a T atom contains only Al atoms, 9.45 atoms
at 2.55 A plus 2.35 atoms at 3.05 A, while Al atoms
have 10 other Al nearest-neighbours at 2.82 A and 2.6 Mn at 2.55 A (2 atoms) and 3.05 A (0.6 atom).
Some of the quantitative parameters of a few coordination shells of these pair distribution func- tions are gathered in table I, namely average radius R, maximum RM and minimum Rm values of dis-
tances covered by the shell, FWHM = r and coordi-
nation numbers Z (note : ZAIT = 0.219 ZTAI).
Table I.
The general trends, already obtained by EXAFS
measurements [29-31] or icosahedral cluster models
[9, 10], concerning the first two coordination shells around Mn atoms are confirmed here : a given Mn
atoms has only Al nearest-neighbours distributed on
a shell centred at 2.55 A with a shoulder at 3.05 A (9.45 + 2.35 = 11.80 atoms) ; Mn-Mn shortest dis-
tances are split into two peaks (4.47 and 4.97 A ;
3.3 + 6.0 = 9.3 atoms). As additional information, the first coordination shell around Al atoms
(10.0 Al + 2.6 Mn = 12.6 nearest neighbour atoms)
and of course further coordination shells around Al and Mn, are also described in the present work.
Absence of close contact between Mn-Mn atoms is of great interest but must be taken with care. In the
procedure leading to the extraction of the PPDF from total pair correlation functions, we had to face
the problem of adjusting the icosahedral phase composition, as previously explained, and we ob-
served that this parameter is also critical for the appearance or the absence of short nearest-neigh-
bours distances in gTI(R). The second Al-Al coordi- nation shell shows also quite an interesting feature in
the form of a well defined shoulder at about 4.0 A
which is in the ratio of J-Z with respet to the nearest neighbour distance and, thus, gives an evidence for
the existence of square shaped clusters of Al atoms.
Globally, the observations presented in this paper and summarized by table I and figure 9 tend to
demonstrate that the studied icosahedral phases depart from cubic a-AlMnSi [40] compound only at
detail level. This is illustrated by the PPDF (Fig. 10)
calculated by isotropic regrouping of the pair dis-
tances in this compound convoluted with a Gaussian
broadening function. A comparison of these data with the icosahedral phase PPDF in figure 6 shows
that the two structures must be very similar : oscilla- tions with nearly the same amplitudes, shoulders,
etc..., are found close to the same positions in each
Fig. 10. - Computer simulation of the PPDF correspond- ing to crystalline a-AIMnSi.
1988
Table II. - Coordination shell parameters of the a- and j3-AIMnSi crystalline phases : R : positions of the
successive maxima of the PPDF, Z : coordination numbers, Rm and RM : lower and upper limits of the shell (note that these values are not really significant as they depend on the specific choice of the Gaussian
broadening function defined herein by a variance 0’2 = 0.014 + 0.0035 (R-2 )1.35 to reproduce at best the experimental data of the icosahedral phase).
set of functions. Coordination numbers reported in
table II are also very comparable to those given in
table I. The main differences are observed with the Mn-Mn function : the amplitude ratio of the two first sub-peaks is inverted and maxima positions are slightly shifted towards larger values. But never-
theless, the structure of the icosahedral phase ap- pears as a minor distortion of the «-AIMnSi network.
It is however worth pointing out that - to some
extent - the same conclusion may be reached when
comparing the PPDF in figure 6 with the PPDF calculated for hexagonal B-AlMnSi [41] (Fig. 11 and
Tab. II). The only significant difference with the
previous compound is the existence of a close
contact Mn-Mn distance at R = 2.75 A. With this sole exception, the isotropic structure of icosahedral Al-Mn appears as an intermediate between cubic a-
and hexagonal 13-(isotropic) structures. This close resemblance of the icosahedral structure with the AlMnSi compounds as well as the sharp definition of the Mn-Mn shells raises some questions about the
validity of the 3D-Penrose tiling picture. Fig. 11. - Same as figure 10 but for f3-AIMnSi.
