Simulation of the extended fine structure of K-shell edges in intermetallic ordered alloys

ultramicroscopy ELSEVIER

Ultramicroscopy59 (1995) 121-136

Simulation of the extended fine structure of K-shell edges in intermetallic ordered alloys G. Hug

a

G. Blanche

a

M. Jaouen

b A.-M.

F l a n k c, j . j . R e h r

a

a Laboratoire d'l~tudes des Microstructures, ONERA-CNRS, B.P. 72, 17-92322 Chdtillon, France b Laboratoire de Mdtallurgie Physique, URA 131 du CNRS, 40 avenue du Recteur Pineau, F-86022 Poitiers Cedex, France c Laboratoire d'Utilisation du Rayonnement F~lectromagndtique, CNRS-CEA-MEN, Centre Universitaire de Paris Sud, B~timent 209D, F-91405 Orsay Cedex, France d University of Washington, Seattle, WA 98195, USA

Received 27 November 1994;in final form 16 January 1995

Abstract

The extended fine structure of K-edges ranging from A1-K to Cr-K in metals (A1, Ti, V, Cr) and binary intermetallic compounds (TiA1) has been recorded by X-ray absorption spectrometry (XAS) and/or electron energy-loss spectrometry (EELS). It is found that EEL spectrometry can be used reliably up to energy losses of 7 keV which have generally not been considered with this technique. Simulation of the experimental data have been performed through the so-called standard EXAFS method and using the "ab-initio" code FEFF which allows a high-order multiple scattering calculation.

1. Introduction

Several ordered intermetallic alloys, NiAI, Ni3AI, Fe3A1 or TiA1, are currently investigated for aerospace applications at high stress and high temperature. Their structures result in the ordering of the different atomic species on a simple host lattice (fcc, hcp, bcc . . . . ). In these kind of alloys, the aluminium provides lightness and oxidation resistance whereas the transition element gives the strength. TiA1 is probably one of the most promising compounds in the near future because of a good balance between lightness, strength and oxidation resistance. It suffers, however, of an intrinsic brittleness at low temperature. A lot of development is now being done to enhance the mechanical properties by different

means. Therefore, beyond the binary compositions listed above, the addition of third and fourth elements are used to modify the alloy and enhance its mechanical properties. However, upon alloying, changes can be produced at several different scales, from modifications of the microstructure up to changes at the atomic scale. The understanding of the physical and mechanical properties of these alloys and the relation between these properties and the addition of other elements, requires more detailed analysis. For example, one can expect that at the atomic scale the following phenomena would have a strong importance: • the order parameters in binary alloys (the fact that the alloy is fully ordered, disordered or partially ordered);

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G: Hug et al. / Ultramicroscopy 59 (1995) 121-136

122

• the strength of the bonding between atoms; • if solute atoms are present (ternary alloy) in a low concentration they may develop a strain field because their average size is different from the host matrix. This generally results in hardening due to the interaction of the stress field of dislocation with the solute atom, an effect known as "solid solution strengthening" (SSD) [11. All these effects can be very complex to analyze, knowing that these materials are generally multiphased and therefore only the electron microscope allows one to have access to one phase. From these arguments, there are clearly new needs for characterization at the local atomic level. The Extended X-ray Absorption Fine Structure (EXAFS) and the EXtended electron Energy Loss Fine Structure (EXELFS) are potentially powerful techniques. The principle of the EXA F S / E X E L F S relies on the behavior of the electron ejected from a central atom after an X-ray photon has been absorbed (XAS) or after the inelastic scattering of a fast electron (EELS). The electron ejected from the central atom with a kinetic energy is then reflected by the surrounding atoms. The outgoing wave function of this electron interferes with the reflected waves, giving rise to a system of constructing or destroying interferences depending on the wavelength of the electron and the distance to the neighboring atoms [2]. The E X A F S / E X E L F S effect is detected as a modulation of the absorption probability beyond threshold. In the plane wave and single scattering approximations, the EXA F S / E X E L F S signal x ( k ) is expressed as [3]:

k x ( k ) = -$3~_, i

i rff

k) l e - 2 r i / A ( k )

× e -2~k2 sin(2kr i + 4~i(k)),

(1)

where k is the magnitude of the photoelectron wave number, [fi(Tr, k)l is the backscattering amplitude from each of the N/neighboring atoms of the ith type at a distance r i from the excited atom, o-i is the D e b y e - W a l l e r factor, ffgi(k) is the total phase shift experienced by the photoelectron and e - 2 r i / x ( k ) is a term taking into account the inelastic losses with A(k) being the electron

mean free path. S02 is a reduction factor that takes into account inelastic losses inside the central absorbing atom [4]. Throughout this paper this parameter has been set equal to 1 in the simulations. By analyzing the E X A F S / E X E L F S signal one can, in principle, determine number, nature and distance of the neighbors, and the D e b y e - W a l l e r factors. Apart from the first shell, for which the analysis is quite straightforward, the extraction of this information is generally complicated by photoelectron multiple scattering processes. Therefore the analysis must be done with a more sophisticated approach that takes into account these multiple scattering effects. In this case the technique promises to be very powerful since it should allow one to determine the correlation function of pair of atoms at rather long distance or even correlations between triplets. In a sense it should be very competitive with other techniques, like the so-called ALC H E M I technique, which is used for site determination of minor elements but requires that an alternation of two planes of different compositions be available in the material. Provided the collection angle of the spectrometer is small (to be consistent with the dipole approximation) it has also been shown that the E X E L F S phenomena can be expressed with the same formalism as EXAFS [5]. The advantage of E X E L F S over EXAFS is that the analyzed volume can be several orders of magnitude lower, since the electron interaction is strong compared to that of photons. It is limited, however, to elements of lower Z for which the K-shells are attainable. In spite of this limitation, some attempts have been made to analyze the L- and M-shells of medium and heavy elements [6-9]. Such experiments may bring some useful qualitative information, but a fully quantitative analysis will be very difficult because of the multiplicity of both the initial and final states. The advantage of working with K-edges is easily understood, since (i) the initial state is unique and has a spherical symmetry, and (ii) because of the selection rules, there is also only one final state (p state). Up to now, only the K-edges lying below 2 keV have been studied by EELS [10], but recently we have demonstrated that the Ti-K edge is attainable in

G. Hug et aL / Ultramicroscopy 59 (1995) 121-136

modern microscopes with parallel detection EELS systems [11]. The access to the K-edge of transition elements is of particular interest in the study of intermetallic compounds. The goals of this study are to determine first whether the EXELFS signal can be usefully recorded from K-edges of transition elements in the TEM, and second whether such signals can be quantitatively analyzed.

2. Experimental procedures The samples are made of either pure metals (C r, V) or intermeta!lic compounds. TiA1 alloys have been processed by an arc melting technique under argon atmosphere using high-purity starting metals. The main impurity is oxygen contained in titanium and was about 4-600 wppm in the titanium we have used. The samples have been subsequently annealed at 1300°C during 48 h for homogenization, quenched and then heattreated again for 100 h at 1000°C. The second part of this process results in a precipitation of impurities (C, N, O) in small particles. These particles have a platelet shape with an average size of 30 nm in thickness and 100 nm in diameter with a very low volume fraction (_< 0.1%). Therefore they do not contribute significantly to the EXAFS signal. For EELS experiments it is easy to select the matrix excluding the precipitates. Z1. E E L S

Specimens for electron microscopy have been cut with a low-speed diamond saw and thinned by conventional electro-chemical techniques. EELS spectra have been acquired in a Jeol 4000FX microscope operating at 400 kV and fitted with a Gatan model 666 parallel detection electron spectrometer. The experiments are routinely carried out at 108 K in order to avoid contamination and to minimize thermal vibrations except when a temperature dependence was specifically sought. In order to record the K-edges, the illuminated area was approximately 2 /xm in diameter with thicknesses below 100 nm. The integration on the photo-diode array (PDA) lies between 25 and 50

123

s, depending on the cross-section of the edge and 5 to 10 read-outs are recorded and summed for better statistics. The total probe current was typically 400 nA during these experiments. This total current is rather high but owing to the large illuminated area, it corresponds to an effective electron dose per atom which is not too prohibitive. No specific study of radiation damage was carried out, but it was visually checked that during the experiment the specimen did not evolve. Spectra are recorded in the diffraction mode with a low-convergence incident beam ( < 1.5 mrad) and a collection semi-angle of approximately 3 mrad. 2.2. X-rays

Because of their brittleness, intermetallic specimens cannot be prepared as thin foils suitable for XAS experiments in transmission mode. Consequently, small square samples of 20 m m × 20 mm and 5 mm have been prepared for experiments in the electron conversion mode (CE-EXAFS). The surfaces have been mechanically polished down to 5 /xm diamond grain in order to remove all surface damage. XAS spectra were recorded at L U RE using the synchrotron radiation emitted from the DCI (Ti-K:Si[311] monochromator on beam line D13) and SuperACO (AI-K:quartz [10-10] monochromator on beam line SA32) storage rings. To reduce the thermal disorder, the Ti-K edge data have been recorded at 77 K using a CE-EXAFS device working at liquid-nitrogen temperature [12]. For the experiments on the A1-K edge, the incident beam coming from the synchrotron ring is monochromatized by two SiO~ crystals. Consequently, there is a strong absorption of the beam at the Si-K edge and above which prevents the recording of the signal in this energy range. This feature limits the study of the aluminium edge to a small energy window corresponding to the energy difference between A1-K (1560 eV) and Si-K (1839 eV). It corresponds to a maximum wave vector attainable of approximately 8.5 A-1. The EELS technique does not suffer from such a limitation.

124

G. Hug et al. / Ultramicroscopy 59 (1995) 121-136

3. Data analysis Prior to analysis of EXELFS oscillations, the spectra are corrected for defects of the detectors (PDA). Dark counts are recorded with 10 times more read-outs than the raw data in order to minimize the statistical noise and to keep only the systematic noise. These dark count are then divided by 10 and subtracted from the raw data. The spectra are then corrected for channel-tochannel gain variation by dividing with a spectrum acquired with uniform illumination which is also corrected for dark counts. A low-loss spectrum was also recorded with each K-edge with the same illuminated area. After extraction of the background with a standard power-law function, the spectra are deconvoluted for plural scattering with the Fourier-ratio method [13]. All these treatments are performed within the E L / P program (Gatan) and the subsequent analyses are done in the same manner for EELS and XAS experiments and will be described below. It has been quoted recently [14] that the E E L spectra should also be renormalized by a factor E2/Eo2 to take into account differences in the energy background. Such a correction has not been applied so far to our data although it has been verified that it is a small correction in the highenergy domain (5-7 keV). XAS raw data are also corrected for different well known artifacts (glitches) and a background intensity is also subtracted. In this case, however, the background is adjusted by a linear function. The following step consists in extracting the oscillatory component x(E) which is done by fitting the atomic background absorption with a sixth-order polynomial. The x(E) signal is then normalized to the threshold step with the Lengeler-Eisenberger method [15]. The x(E) is then converted to k-space through the relation:

k= ~2m(E - Eo) h

'

(2)

where E 0 is the energy loss of the inner shell electron ejected with wave vector k = 0. At this point two different procedures are used: a classical EXAFS analysis of the first shell that

requires the extraction of the signal coming from the shell of interest, and a direct simulation of the whole x(k) signal within a multiple scattering formalism. The complete description of this last method can be found in Ref. [16]. It consists in a four-step calculation: (i) Calculation of the scattering potential and phase shifts of each atomic species as if the atoms were isolated in space. (ii) Enumeration of all scattering paths for the electron and filtering to keep only the multiple-scattering paths of significant weight. (iii) Calculation of the effective scattering amplitude for each path. (iv) Reconstruction of the x(k) spectrum in the curved wave approximation. At this point a correlated Debye-Waller model is used to take into account thermal disorder. In the classical EXAFS method [17] the x ( k ) spectrum is multiplied by k3 in order to enhance the high k values and Fourier-transformed to obtain a radial distribution function through the relation:

IF(r) l = l f+_~oW(k)k3x(k)e2ikndk,

(3)

where W(k) represents a Hamming apodization window with soft edge to zero (cosine functions). The signal from the first shell (that involves only single scattering) is then extracted by filtering the RDF with a suitable window and then Fourierbacktransformed. This signal is used for fitting the spectrum following Eq. (1). In these simulations McKale theoretical phase and backscattering amplitudes have been used [18]. The electron mean free path A was approximated by the equation [19]:

k + (~/k 4) A

F

'

(4)

where ff and F are empirical parameters. EXAFS package codes built by Michalowicz [20] have been used for all the treatments from the extraction of the E X A F S / E X E L F S signal to the simulation. The simulation itself is performed by the CERN code "Around Midnight" adapted by Michalowicz [20]. It consists in a multiple

125

G. Hug et al. / Ultramicroscopy 59 (1995) 121-136 i

i

i - -

i

i--

120x103

0.7 0.6

i ~

,~,~

--

AI-K (EELS)

lO0

0.5

c

X J,

.......

~o

8o

.2 "~

60

0.3

~

40

0.2

2O

0.1

o r-

,J ILl LU

0.4

I

I

I

I

i

1600

1700

1800

1900

2000

Photon

energy (XAS)

ro

0,0

Loss (EELS)

or Energy

Fig. 1. AI-K X A S a n d E E L S s p e c t r a f r o m p u r e a l u m i n i u m .

least-squares fit followed by a simplex and gradient minimization procedure.

two experiments are compared. The cut-off due to the Si-K edge at 1839 eV allows only the extraction of the EXAFS over a window of 279 eV. The corresponding x ( k ) signals extracted both from the X - r a y and EELS data are presented in Fig. 2, It is useful to compare the associated radial distribution functions obtained from these data (Fig. 3). When the same integration window is used for both EXELFS and EXAFS the RDFs are comparable. However, the

4. R e s u l t s

4.1. Pure metals: AI, V, Cr

The advantage of using EELS over X-rays for the A1-K edge is exemplified in Fig. 1, where the

0.4 0.2 ~-~ 0,0 &¢ v •

-0.2

-0.4 -0.6 -0.8

"! ~J

i 2

Fig. 2. A I - K

~ 3

4

x(k) ( E X A F S

I

I

5

6

I

7o ktA.l~,

r

f

I

I

I

8

9

10

11

12

a n d E X E L F S ) e x t r a c t e d f r o m e d g e s in Fig. 1.

13

G. Hug et al. / Ultramicroscopy 59 (1995) 121-136

126

spatial resolution in the RDF can be improved significantly in EELS by using a much larger integration window (Fig. 3). The K-edges of vanadium and chromium lie at 5465.1 and 5989.2 eV, respectively. Despite these high energy values, they have been successfully recorded by EELS from pure metals with sufficient signal-to-noise ratio to allow the extraction of the x(k) oscillatory EXELFS signal [21] (Fig. 4). Chromium and vanadium are both bodycentered cubic (bcc) metals. The dimension of the cell is such that there is only a small difference between the distances of the first nearest neighbouts (lnn) and second nearest neighbours (2nn). As a result the first peak in the RDF contains the contributions from the two first shells. Therefore, when the first peak in the RDF is filtered and Fourier-backtransformed, x(k) exhibits a beat due to these two very close spatial frequencies (lnn and 2nn). The beat gives rise to a periodic modulation of the x(k) at a frequency which is related to the difference in frequency between both waves involved. The positions of the nodes (the minima of x(k)) can be found by the formula:

- - At (EELS)

lO-

[ 2 ~-1.11 A "1] -~- AI (EELS) [ 2 A "1 - 7.5 A "+]

...... AI (XAS)

8 -

[2A

"I -

7.5

A"1]

i,

0 0

1

I

I

I

2

3

4

5

t

I

6

7

8

r (A) Fig. 3. A1-K radial distribution function (EELS and XAS) after Fourier transform of data in Fig~ 2. T h e two k-windows which have been applied two the same EELS data are indicated in the figure.

2k A r + Aq~(Ar, k) = (2n + 1)~-,

Energy Loss for Chromium

5400

6000

6200

6400

6600

6800

I

I

[

I

I

~6100

58100

i 6000

t 6200

6400

Energy Loss for Vanadium

Fig. 4. V - K and Cr-K edges recorded by EELS at low temperature (110 K: displayed temperature on the specimen holder),

(5)

G. Hug et al. / Ultramicroscopy 59 (1995) 121-136

127

Table 1 Results of t h e fits of the first and second shells in V and Cr

V Cr

N1

~r~ (,~)

Rt (A)

N2

o"2 (A)

R 2 (A)

a o (.~) at 300 K

8 8

0.06 + 0.004 0.07 + 0.004

2.58 ± 0.03 2.46 ± 0.03

6 6

0.07 ± 0.004 0.07 ± 0.004

3.03 ± 0.03 2.85 ± 0,03

3.027 2.884

N/ n u m b e r of atoms, R i distance, o"i D e b y e - W a l l e r factor, a 0 is the known lattice p a r a m e t e r from X-ray diffraction at room temperature. (In this case the n u m b e r of atoms was kept at the nominal value.)

0.4

I

I -

I

-

~.- V l n n & 2nn (experimental EELS

--

V l n n & 2nn (theoretical)

0.2

Y

,,

0.0

-0.2

(a) -0.4

I

6

I

I

8

10

k (A ~)

0.6

I

I

I

t 7

l 8

t 9

I

0.4

0.2

"~

0.0

-0.2

-0.4

(b) -O.E

r 6

k

ii0

(A"1)

Fig. 5. Fit of the x ( k ) of the l n n and 2nn in bcc metals extracted from data in Fig. 4: (a) v a n a d i u m and (b) chromium.

G. Hug et al. I Ultramicroscopy 59 (1995) 121-136

128

and the second one is just past the end of the available k-window (Fig. 5). The maxima, however, are located at k = 7.7 ,~-1 for vanadium and at k = 8.1 ,~-1 for chromium. These maxima are clearly observable in Fig. 5. Nevertheless, it must be concluded that it is not possible to separate the contributions from the two first shells and one must analyze both simultaneously. Indeed, such simulations of these two shells shown in Figs. 5a and 5b are in good agreement with the known parameters of these metals (Table 1). At

whereas the maxima will be defined by: 2k A r + A ~ ( A r ,

k ) = 2nTr.

(6)

In the case of a pure metal the term A@(Ar, k) vanishes. From the known lattice parameter of vanadium and chromium it is therefore possible to find these positions. With the values listed in Table 1 one gets the minima at k = 3.4 A71 and k = 11.6 ,~-1 for vanadium and at k = 4 ~ - x and k = 12.2 ~ - i for chromium. Unfortunately, the first node of the beat lies at too low a k value

o8I

Experimental (EELS) Theoretical (FEFF simulation)

....o-..

--

0.6

A

0.4

0.2

0.0

-0.2

-0.4

L

I

I

I

I

2

3

4

5

6

',

I

k

~A

-1)

8

9

,

,

10

11

(a) 12

1.0

I

0.8 0.6 0.4

•

....~ Experimental (EELS) - - Theoretical (FEFF simulation)

0.2 0,0 -0,2 -0.4 2

3

4

5

6

7k

(A8"1)

I-

I

I

I

9

10

11

12

,(b)1 13

44

Fig. 6. F E F F simulation of the x(k) of bcc metals extracted from data in Fig. 4: (a) vanadium and (b) chromium.

G. Hug et al. / Ultramicroscopy 59 (1995) 121-136

129

Table 2 Parameters used for the simulation of x(k) in vanadium and chromium metals with the FEFF code a0(,~) Debye Experiment E0(eV) temperature (K) temperature (K) V 3.027 Cr 2.847

390 460

199 149

6 2

longer distances from the central atom, a supplementary complication arises from the multiple scattering of the photoelectron. Therefore the simulation of higher shells must be done with a more sophisticated formalism that takes into account these additional contributions. Figs. 6a and 6b show the results of the simulation of the experimental x ( k ) of V and Cr using the multiple scattering F E F F [16] code (version 5.05). The different input parameters that must be supplied to this code are the lattice parameter (through the atomic positions), the Debye temperature, the experimental temperature and E 0 which corresponds to the origin of the wave-vector in kspace. In the input, the atomic positions in a cluster of approximately 250 atoms have been set up, and the calculation is performed over paths of approximately 7 A in total length which roughly correspond to the ninth nearest neighbours. The

Fig. 7. Conventional cell of TiAI ° (L10). The lattice constants are a = 3.976 A and c = 4.049 A for a TisoA150 composition and the c / a ratio increases slightly with aluminium content.

other parameters that have been used are listed in Table 2. E 0 is somewhat adjustable because it depends on the Fermi level which is not exactly known. In o r d e r to obtain a correct correspondence between the experimental data and theoretical curves only E o and the experiment temperature T have been adjusted. The best fits are obtained in both cases (V and Cr) for temperatures T that are somewhat higher ( ~ 40°C) than the temperature displayed on the Gatan cold stage. In fact, the t e m p e r a t u r e probe is not located directly on the specimen in the Gatan stage

o

0.2

0, "~

o.o

i

t jiil

/

-o.1

,i -......

Experimental Experimental

(EELS) (XAS)

Ti4eAIs4

Ti4eAIs2

-0.2 7

4

,

_

i

5

6

'7'

I

8

9 k (,&,-1)

In

1,.i

J

1~1

12

~

1,3

Fig. 8. Comparison of XAS and EELS x(k) of Ti-K in TiA1. Data extracted from experiment at low temperature 77 K for XAS and approximately 110 K for EELS.

130

G. Hug et al. / Ultramicroscopy 59 (1995) 121-136

design but closer to the wire linked to the liquidnitrogen tank. A higher specimen temperature than the one displayed can thus be expected, although no detailed calibration has been done so far. The importance of the multiple scattering contributions can be exemplified if one considers the contributions at distances close to the third neighbours.

- - Ti-K (EELS) ....... Ti-K (XAS)

4.2. TiAI The TiA1 intermetallic compound crystallizes under the L10 structure which is based on the ordering upon the fcc lattice. It consist of alternate (001) pure planes of aluminium and titanium (Fig. 7). However, due to a complicated phase diagram it is mostly impossible to obtain an equiatomic composition. Two phases are present at a 50-50 composition: the hexagonal a 2 phase (structure DO19) and the y phase (TiAl). For this reason the alloys studied here are slightly aluminium rich in order to be in the single-phase domain. The fact that the alloy is single-phased insures that good conditions for X-ray experiments can be fulfilled. However, it adds some difficulties in the interpretation since the behavior of the excess aluminum atoms is not properly known. Two composition h a v e b e e n studied, so far: Ti48Al52 and Z i 4 6 A 1 5 4 . For technical reasons, it has not yet been possible to record XAS and EELS spectra for exactly the same composition. The X A spectrum and a E E L spectrum taken on the Ti-K edge in Ti48A152 and Ti46A154 a r e shown in Fig. 8, respectively. Both experiments have been carried out at liquid-nitrogen temperature. It is not known if the differences observed are due to the fact that different techniques are used or to the difference in composition.

4.2.1. TiAk Ti-K edge Fig. 9 shows the RDFs obtained after Fourier transformation of the spectra displayed in Fig. 8. In this case the Fourier transformation has been carried out in the k-space range [3.5-13.6 ~ - 1 ] . T w o remarkable features can be observed on these curves. The first is that between 2 and 3 A, two peaks are clearly separated. This distance correspond roughly to the l n n (2.83 A). The o

#. I,I.

0

1

2

3

4

5

6

7

8

9

r (A)

Fig. 9. Comparison of XAS and EELS radial distribution function from Ti-K in TiAI extracted from data in Fig. 8.

o

second feature is a very strong peak at 5 A which corresponds to the 4nn. Interestingly, when the XAS R D F is extracted from a spectrum recorded at room temperature by taking the same range in k-space as that used for the data collected at 77 K, this splitting in two peaks disappears (Fig. 10). Moreover, if the XAS R D F is extracted from the spectra recorded at 77 K but with a smaller window in k-space [3.5-10.5 R -a] the low r region exhibits only one peak while the peak at 5 remains relatively high (Fig. 10). A similar behavior has been found on EELS data. It has been analyzed as follows: The two peaks at low r are due to a separation of the T i - T i and Ti-A1 pairs in the coordination shell. Such separation can be viewed as a splitting of the first shell because the phase shifts associated with the backseattering parameters of the electron are different for Ti and A! atoms. As usual, in direct-reciprocal space relationships,

G. Hug et al. / Ultrarnicroscopy 59 (1995) 121-136~

7

-

---

TiA177 K

[3.5-13.5A ~] TiAI 77 K [ 3 . 5 - 1 0 . 5 A "1]

...........

--

TiA1300 K [ 3 . 5 - 1 0 . 5 A -~]

ii

:

Ii

0

0

1

2

,,

3

4

5

6

7

r (A)

Fig. 10. Effect of temperature and size of the k-window on the RDF (XAS data).

j

x

131

two close frequencies in real space are viewed as a beating in reciprocal space. Such beating can be observed on the Ti-K ~((k) since it gives rise to an amplitude reduction at approximately 9.5 A-1. This is more clearly visible after filtering the two first peaks in real space and Fourier-back-transforming them. This procedure allows one to obtain the E X A F S / E X E L F S oscillations due only to the l n n (Fig. 11). In effect, it can be seen in Fig. 11 that, in the experimental data, that the fifth positive oscillation is the smallest and that the subsequent ones are higher. However, unlike the pure metals (V and Cr) the difference in the phase shifts (Aq~(k) in Eqs. (5) and (6)) also contributes to the position of the minima and maxima Of the beating. It can be found using the McKale [18] phase shift that the phase difference A(~ = t~TiXi(~TiAI is almost constant and close to 3 rad between 5 and 14 ~ - 1 . Consequently, the position of maxima and minima induced by the beat depends sensitively upon the difference between 7r and Aq~ which are close numbers in this case. As a result, the precision of the calculated phase shifts is nor sufficient to determine the positions of the nodes. In addition, the difference in the backscattering amplitudes for aluminium and titanium also contribute to the over-

'~ ~

4

i

i

/r shell

6

8

10

I

12

0

-0.05

~0~1

I/ ~

~ A

4

,

K I

6

77K (experimental) TiN {theoretical)

8 k (A "1)

10

12

Fig. 11. Fit of the x(k) of the lnn in TiA1 at 77 K (XAS data). The insert shows the relative contribution to the x(k) of the Ti shell and the A1 shell of Inn.

G. Hug et al. / Ultramicroscopy 59 (1995) 121-136

132

Table 3 Results of the fits of the l n n shell from AI-K and Ti-K in Ti48AI52 (EELS data) Ti-Ti pairs

Ti-A1 pairs

N~ Cryst. data Fit (109 K)

3.84 3.84 + 0.1

o-z (A)

R x (A)

N2

0-2 (A)

R 2 (-A)

0.035 _+0.004

2.81 2.83 + 0.03

8.16 8.16 + 0.1

0.064 + 0.004

2.83 2.85 _+ 0.03

A1-AI pairs N, Cryst. data Fit (109 K)

4.32 4.32 ,- 0.1

A1-Ti pairs ~, (£)

R, (A)

N2

o'2 (£)

R2 ( M

0.022 + 0.004

2.83 2.83 + 0.03

7,68 7,68 + 0.1

0.064 + 0.004

2.81 2.82 _+ 0.03

all amplitude of x(k) in a complicated way. The aluminium are mainly backscatters at low k whereas the titanium are backscatters at high k. Nevertheless, an example of the fit of the filtered data through the standard EXAFS procedure described above is also shown in,Fig. 11 and the results are summarized in table 3 and will be commented on hereafter. The insert in Fig. 11 shows the relative contribution of the Ti and AI backscatters, and it can be clearly seen that both contributions are out of phase. In order to clearly separate the two peaks in the R D F the integration window in k-space must at least include the first zero of the beating in the x(k). Ineffect, the width of the k-range used for the Fourier transform defines the resolution in r-space. This is the

reason why the splitting is not resolved when a small window is used (Fig. 10): an artificial loss of resolution has been created. At room temperature, the splitting is also not resolved for two reasons. Firstly, because of the thermal vibration of the atoms, their positions are not as well defined, giving rise to a widening of the peaks of each shell. Secondly, the E X A F S / E X E L F S signal intensity also decreases because of the loss of coherence. At high k values the signal falls below the noise giving rise to a natural reduction of the k window that can be used for the Fourier transform. The second feature of interest is the strong peak at 5 A in the R D F that corresponds to the 4nn. In fcc metals the l n n and 4nn are aligned

0.2

o

o.1 i

0.0

~

o~

-0.1

I

-0.2

40

o --

so

E x p e r i m e n t a l ( EELS d.ata filtered between 2-7 A) T h e o r e t i c a l a = 3.944 A ; c = 3.954 .~

60

k (A' 1)

70

80

90

~oo

Fig. 12. F E F F simulation of the x(k) of TiA1. The experimental EELS data are filtered in r-space between 2 and 7 A.

G. Hug et aL/ Ultramicroscopy 59 (1995) 121-136

along (110) directions. In addition, when an electron hits an atom, the maximum scattering probability is for back-scattering (0 = ~') and for forward-scattering (0 = 0). Therefore when three atoms are aligned the E X A F S signal can be greatly enhanced because the atom in the middle act as a lens to "focus" the electron wave onto the the third atom. This p h e n o m e n o n is well known as "focusing effect" or "shadowing effect". At room temperature, the alignment is not so well realized because of thermal vibrations. Consequently, the effect is dampened and the R D F exhibits a more classical shape for a fcc structure. However, when the R D F is extracted from the low-temperature spectrum but with the short kwindow the focusing effect is preserved. This is because the small window reduced the resolution in real space but does not change the relative intensities of the peaks. It is worth mentioning that this effect is probably a very sensitive way to monitor the temperature. Additional peaks are also visible between the first doublet and the peak of the 4nn but their identification requires some care, since some of them are also numerical artefacts due to the Fourier transform. The peak at 3.5 A can b e unambiguously attributed to the 2nn and the contribution of the 3nn probably appears as a shoulder before the 4nn peak. F E F F simulations have also been tried on the Ti-K edge of TiAl. In this case, for practical reasons, the simulations have been limited so far o to a cluster with distances below 7 A. Therefore, the R D F in Fig. 9 has been filtered between 2 and 7 A and then Fourier back-transformed for best comparison of experimental and theoretical data. It can be seen from Fig. 12 that there is a good match between simulations and experiments, i.e. all oscillations are present in both curves. The amplitudes are not satisfactory but it will not be straightforward to obtain a good match. The reason is that a good simulation receives a cluster that is representative of the alloy. In the present case an important point is that the alloy does not strictly have the equiatomic composition. Therefore antisite defects have to be introduced in the cluster (mainly as Al atoms on Ti sites), but then some kind of averaging over dif-

S

133

-Theoretical (FEFF simulation) ....... TiAI Ti-K (exp. EELS)

4

/'i/i 0 0.0

, 1.0

2.0

3.0

4.0

5.0

6.0

7.0

r (A)

Fig. 13. R D F of the s i m u l a t i o n in Fig. 12 w i t h a k - w i n d o w of

[3.5-11.5 ~-1]: the splitting of the lnn and the strong 4nn peak are reproduced.

ferent configurations is also necessary. Clearly some additional work is now needed and this is the reason why it is not physically meaningful to try to optimize the match with an equiatomic cluster. Another effect that will be important to take into account is the fact that a hybridization of Ti-d and Al-p electrons leading to a build up of an anisotropic charge density distribution should exist in this materials as it has been demonstrated from theoretical calculations and experimental measurements [22-26]. Although it should be a small effect, the F E F F calculations which rely on a muffin-tin approximation clearly do not include such anisotropic potential. Nevertheless, there is already a very promising reproduction of the experimental data which is visible on the R D F in Fig. 13. Actually, the splitting of the first shell and the focusing effect on the fourth shell are correctly reproduced. Interest-

G. Hug et aL / Ultramicroscopy 59 (1995) 121-136

134

0.15 0.10

A

0.05 0.00 -0.05 -0.10

2

3

4

5

6

7

8 9 k (A "1)

10

11

12

13

14

Fig. 14. EELS x(k) of A1-K in TiA1; experiment at 100 K.

ingly, the F E F F simulation allows one to detail the focusing phenomenon. The six paths that have the most important amplitude involve 2, 3 or 4 atoms depending on whether there is scattering on the atom in the middle or not. A path with 2 atoms would be central atom to 4nn and return (no scattering by the atom in the middle). A path with 3 atoms: central, 4nn, l n n and return. And a path with 4 atoms: central, lnn, 4nn, l n n and return. The middle atom can be titanium (<110] directions: T i - T i - T i ) or aluminium ((011] directions: T i - A 1 - T i ) . They are other paths of the same length but with lower amplitude. Due to the fact that they are more numerous, the total contribution of the T i - A 1 - T i paths is more than twice the contribution of the T i - T i - T i paths.

L¢.

4.2.2. TiAl: AI-K edge E X E E L F spectra have also b e e n recorded on the A1-K edge. The x(k) for this edge is shown in Fig. 14. For the reasons mentioned above, there is here a great advantage to use E E L S over XAS since the x(k) can be extracted up to a much higher k value. Thus this allows one to obtain good resolution in the R D F with again the separation in the pairs of the l n n (Fig. 15). As mentioned above, some aluminium atoms have to sit

l 0

vv

o

vI

1

2

3

4 5 r (A)

6

7

8

9

Fig. 15. EELS radial distribution function from A1-K in TiAI after Fourier transform of data in Fig. 14.

G. Hug et al. / Ultramicroscopy 59 (1995) 121-136

135

at an antisite position since the alloy is aluminium rich. Consequently, the E X E L F S signal comes from a mixture of contributions of AI central atoms on different sites which complicates seriously the analysis. The results of the fit are summarized in Table 3 together with the results obtained from the Ti-K edge. The distances deduced from this fitting procedure are in good agreement with the known lattice parameters deduced from crystallographic data. It is also found that the site occupancy is consistent with the values that can be deduced from the composition of the alloy.

example, that some addition of niobium would give rise to a bcc-type structure. Nevertheless, this feature explains also why the strong focusing effect that has been observed on the fourth neighbours is destroyed at room temperature. This effect would be quite sensitive to the correct alignment of the three atoms. Since the main contribution comes from the T i - A I - T i triplets, the fact that the titanium and aluminium planes are decorrelated by the thermal motion decreases the focusing effect. This is consistent with the experimental observations.

4.2.3. Temperature dependence

5. Conclusions

The first neighbour EXAFS signal from the Ti-K edge at 77 and 300 K has been analyzed through the standard method. The main result obtained here (Table 4) can be deduced from the comparison of the D e b y e - W a l l e r factors between liquid-nitrogen temperature and room temperature. The evolution o f the o-ii parameters is very different between homopairs (Ti-Ti) and heteropairs (A1-Ti). The first ones exhibit a small increase with temperature while the latter are twice as large at room temperature than at 77 K. This feature, which is also abnormal, means that titanium and aluminium planes have tendency to be de-correlated at high temperature while the atomic order within an aluminium or a titanium plane remains stable. It can be understood as a precursor of a phonon soft mode. This may explain the lack of amplitude observed on the reconstructed spectrum shown in Fig. 12. Such a behavior is normally expected in materials that are subject to martensitic transformations but it does not exist in TiAI. However, it is possible that the stability of the L10 structure compared to another is only weakly achieved. It is known for

• E X E E L F spectra of the K-edges of transition metals have been successfully recorded in the energy range between 2 and 7 keV both from pure metals (Ti, V, Cr) and intermetallic compounds (TiAI). • The spectra are comparable to EXAFS, although E E L S data suffer from slightly lower energy resolution and lower signal-to-noise ratio at high k. • However, spectra of sufficient quality can be achieved to make useful comparison with theoretical calculations either with a single-scattering formalism (analysis of the first shell) or with an "ab initio" formalism with high-order multiple scattering contributions ( F E F F simulation). • The F E F F simulations are in good agreement with the experimental x(k) and known physical constants (lattice parameter, Debye-Waller) for the pure metals (V, Cr). • Concerning the TiA1 intermetallic compound, the analysis of the first shell has been carried out with the standard EXAFS procedure. The results suggest that a precursor of a phonon

Table 4 Results of the fits of titanium l n n (Ti-K) in Ti46A154 at 80 K and 300 K (XAS data) T i - T i pairs

N1 Cryst. data Fit (80 K) Fit (300 K)

3.68 3.68 _+ 0.1 3.68 _+ 0.1

Ti-A1 pairs

°'1

R1

N2

0"2

R2

0.062 _+ 0.004 0.064 _+ 0.004

2.81 2.86 _+ 0.03 2.84 + 0.03

8.32 8.32 + 0.1 8.32 _+ 0.1

0.060 _+ 0.004 0.082 _+ 0.004

2.83 2.88 + 0.03 2.85 _+ 0.03

136

G. Hug et aL / Ultramicroscopy 59 (1995) 121-136

soft m o d e is p r e s e n t in this alloy. This m o d e c a n b e d e s c r i b e d as a loss of c o r r e l a t i o n of the h e t e r o p a i r s ( T i - A 1 ) relatively to the h o m o p a i r s ( T i - T i , AI-A1). • " A b i n i t i o " c a l c u l a t i o n s of the TiA1 x ( k ) are n o t straightforward b e c a u s e it is necessary to i n c l u d e fine effects (like t h e d e v i a t i o n f r o m stoichiometry). However, they c a n r e p r o d u c e the two m a i n f e a t u r e s t h a t have b e e n identified, a splitting of the T i - A 1 a n d T i - T i pairs in the first shell a n d a very large focusing effect of t h e l n n o n t o the 4nn, showing t h a t these struct u r e s are well u n d e r s t o o d . T h e c o m b i n a t i o n of the access to the K - e d g e s of t r a n s i t i o n e l e m e n t s a n d h i g h - o r d e r m u l t i p l e scatt e r i n g c a l c u l a t i o n s p r o v i d e very s t i m u l a t i n g n e w o p p o r t u n i t i e s for s t r u c t u r a l studies of m a t e r i a l s at s u b - m i c r o n scale.

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[9] LK. Okomoto, C.C. Ahn and B. Fultz, in: Microbeam Analysis - 1991, p. 273. [10] V. Serin, G. Zanchi and J. S~vely, Microsc. Microanal. Microstruct. 3 (1992) 201. [11] G. Blanche, G. Hug, M. Jaouen and A.-M. Flank, Ultramicroscopy 50 (1993) 141. [12] J. Mimault, JJ. Faix, T. Girardeau, M. Jaouen and G. Tourillon, Meas. Sci. Technol. 5 (1994) 482. [13] R.F. Egerton, Electron Energy-Loss Spectroscopy in the Electron Microscope (Plenum, New York, 1986). [14] E.A. Stern, M. Qian and M. Sarikaya, in: Determining Nanoscale Physical Properties of Materials by Microscopy and Spectroscopy, Eds. M. Sarikaya, M. Isaacson and K. Wickramasinghe [Mater. Res. Soc. Symp. Proc. 332 (1994) 3]. [15] B. Lengeler and P. Eisenberger, Phys. Rev. B 21 (1980) 4507. [16] J.J. Rehr, S.I. Zabinsky and R.C. Albers, Phys. Rev. Lett. 69 (1992) 3397. [17] D.C. Koningsberger, Stereo Chemistry and Electronic Structure, XAFS Spectroscopy: Data-analysis and Applications, Neutron and Synchrontron Radiation for Condensed Matter Studies (Les Editions de Physique, Les Ulis, 1994) p. 213. [18] A.G. McKale, J. Am. Chem. Soc. 110 (1988) 3763. [19] B.K. Teo, J. Am. Chem. Soc. 103 (1981) 3990. [20] A. Michalowicz, EXAFS pour le Mac, in: Logiciels pour la Chimie (Soc. Fr. Chimie, Paris, 1991) p. 102. [21] G. Blanche, M. Jaouen, A.-M. Flank and G. Hug, J..de Phys. IV C9, 4 (1994) 145. [22] B. Greenberg, V.I. Anisimov, Yu.N. Gornostirev and G.G. Taluts, Scripta Met. 22 (1988) 859. [23] C.L. Fu and M.H. Yoo, Phil. Mag. Lett 62, 3 (1990) 159. [24] C. Woodward, J.M. McLaren and S. Rao, High-Temperature Ordered Intermetallics Alloys IV, Eds. L.A. Johnson, D.P. Pope and J.O. Stiegler [Mater. Res. Soc. Symp. Proc. 213 (1991) 715]. [25] Z.W. Lu, A. Zunger and A.G. Fox, Acta Metall. Mater. 42 (1994) 3929. [26] R. Holmestad, J.M. Zuo, J.C.H. Spence, R. Hoier and Z. Horita, in: Proc. 13th ICEM, Eds. B. Jouffrey and C. Colliex (Les Editions de Physique, Paris, 1994) Vol. 1, p. 855.