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Determination of long range order in Ni-Base ternary alloys by X-ray anomalous diffraction using synchrotron radiation
Aeta metall, mater. Vol. 38, No. 2, pp. 345-350, 1990
0956-7151/90$3.00+ 0.00 Copyright © 1990PergamonPress plc
Printed in Great Britain. All rights reserved
D E T E R M I N A T I O N OF LONG RANGE ORDER IN Ni-BASE T E R N A R Y ALLOYS BY X-RAY A N O M A L O U S DIFFRACTION USING SYNCHROTRON RADIATION A. M A R T Y l, M. BESSIERE 2, F. BLEY 3, Y. CALVAYRAC t and S. LEFEBVRE L2 ICentre d'~tudes de chimie M&allurgique, CNRS LP 2801, 15 rue Georges Urbain, 94407 Vitry sur Seine Cedex, France, :LURE-CNRS, Bht. 209D 91405 Orsay, France and 3LTPCM. E.N.S.E.E.G.-Domaine Universitaire, B.P. 75, 38402 Saint Martin d'H6res, France (Received 20 March 1989)
Abstract--Long range order (LRO) parameters in three ternary L12 alloys [NiT~AllsTit0,Ni70Al20Crl0 and (Ni3Fe)~.4Cr36] have been measured using a new method. This method takes advantage of the variation in atomic scattering factors for X-rays near the absorption thresholds and opens a new area of research with the development of synchrotron facilities. The results show that Ti substitutes mostly, but not exclusively, for A1 in NiA1Ti. In NiA1Cr, Cr occupies both Ni and A1 sites. In NiFeCr, the order is not perfect; the Cr atoms are distributed on both types of sites, with a slight preference for Fe sites. R6sum~-Les param&res d'ordre/t longue distance dans trois alliages ternaires [Ni7sAltsTito,NivoA120Crt0 and iNi3Fe)964Cr36] ont 6t6 mesur6s en utilisant une nouvelle m6thode. Cette m6thode tire parti de la variation des facteurs de diffusion atomiques des rayons X pr6s des seuils d'absorption et ouvre un nouveau champ de recherche avec le d6veloppement du rayonnement synchrotron. Les r6sultats montrent que le titane se substitue principalement-mais pas exclusivement-~ l'aluminium dans NiAITi. Dans NiAICr, le chrome est r6parti ~i la fois sur les sites du nickel et sur ceux de l'aluminium. Dans NiFeCr, rordre n'est pas parfait; les atomes de chrome sont distribu6s sur les deux types de sites, avec une 16g6re pr6f6rence pour les sites du fer. Zusammenfassung--Die Parameter der weitreichenden Ordnung werden in den tern/iren L12-Legierungen [Ni75AllsTi~o, NiToAl20Crl0und (Ni3Fe)964Cr36] mit einer neuen Methode gemessen. Diese Methode nutzt die .~nderung in den Atomformfaktoren i;iir R6ntgenstreuung in der N/ihe der Absorptionskanten aus und er6ffnet neue Forschungsm6glichkeiten mit Synchrotronstrahlung. Die Ergebnisse zeigen, dab Ti das AI in NiA1Ti meistens, aber nicht ausschliel31ich,substituiert; Cr besetzt Ni- und AI-Pl/itze. In NiFeCr ist die Ordnung nicht perfekt; die Cr-Atome sind auf beide Platzarten verteilt, wobei der Fe-Platz ein wenig bevorzugt ist.
INTRODUCTION The mechanical properties of ordered binary L12 alloys are strongly modified by the addition of a third element [1, 2]. A m o n g these alloys, the intermetallic c o m p o u n d Ni3A1 (~') has received much attention, because of a substantial increase in flow stress with increasing temperature. Although this compound has a narrow composition range, it has been shown that it can accommodate considerable amounts of ternary additions in solution, resulting in significant strengthening [3]. Since in most of the nickel-base "superalloys" the major factor enabling high temperature strength is due to precipitates of an intermetallic compound, based on Ni3AI, dispersed in the nickelbased solid solution (':), it may be expected that the mechanical properties of nickel-base superalloys may be optimized by a proper choice of alloy composition. In order to understand the role that the different elements play in the strengthening mechanism, it is important to know the location of the third element
in the lattice, i.e. to determine the state of long range order. A prediction on the location of the substitutional elements may be obtained from an analysis of ternary phase diagrams, and for example, Ochiai et al. [4] suggest site preferences for several elements (Co, Cr, Ti, Mn) in three alloys (Ni3Ga, Ni3Si, Ni3AI) from the knowledge of the y' solubility lobe direction in the ternary phase diagrams. Experimentally, the site occupation may be determined by a large variety of techniques (e.g. see [5]). A m o n g them, the most promising seem to be atom probe field-ion microscopy (APFIM), M6ssbauer spectroscopy and X-ray diffraction. The atomic spatial resolution of the atom probe field-ion microscopy permits the site occupation probability of any substitutional element to be determined. This technique has no theoretical but experimental limits: there are difficulties in preparing samples and problems of preferential evaporation. Recently, Miller and Horton [6] have analyzed the 345
346
MARTY et al.: LONG RANGE ORDER IN Ni-BASE ALLOYS
position of Hf, Fe, Co in a Ni-A1 alloy doped with boron. Chambreland [7] has studied the position of Ti and Cr in a Ni rich complex industrial alloy NiCoAITiCr; his results will be compared with those we have obtained on ternary alloys. M6ssbauer spectroscopy, which is a local order probe, is limited by the available radioactive sources, among which iron is a favourite. Using this technique, Cranshaw et al. [8] have studied a Ni75Fe23Cr2 single crystal. We will see that their results lead to the same conclusion as ours concerning the substitutional behaviour of Cr in Ni3Fe. X-ray diffraction is used currently to measure long range order parameter in binary alloys. In multicomponent alloys, several independent LRO parameters have to be determined. A first approach to the problem consists in considering the multicomponent alloy as a quasi-binary alloy which is therefore characterized by only one LRO parameter. A trial and error method is then used to give an estimation of the possible positions of the substitutional elements in the lattice. This method has been used by Karg et al. in their study of y'-Ni3A1 based multicomponent alloys [9]. However, as these authors conclude, the attribution of a single LRO parameter to such complex alloy systems is not reasonable. The determination of each LRO parameter in multicomponent alloys is formally possible by taking advantage of the variation in the atomic scattering factors due to the anomalous correction. However, it is in practice very difficult when using the wavelengths available with classical targets. Ferjani et al. [10] have used two different radiations (CUK~, COKe) in order to measure the two long range order parameters in NiFeMo and NiFeCr alloys. The determination was unsuccessful for the NiFeCr alloy because, for the two available wavelengths, there is not a sufficient change in intensity resulting from the different possible assumptions as for the Cr position in the lattice. The development of synchrotron facilities open a new field of investigation, giving the possibility of adjusting the wavelength in order to optimize the use of the anomalous scattering. Using such a technique, we have performed a determination of the state of LRO in three ternary alloys: NivoA120Crjo, Ni75AllsTilo and (Ni3Fe)96.4Cr36. We have chosen compositions with the highest concentration in the third element compatible with LI 2 structure, in order to get the maximum contrast. For NiFeCr we had to choose a smaller Cr concentration, because the ordering kinetics becomes very slow when the Cr concentration increases.
e (dog)
140-
N|A1TI
1301
Ti threshold
120l la 1001 N1 Threshold 90-
2
~(A)
Fig. I. Ellipse orientation as a function of wavelength. the difference of proportions ai and bi of each element i on the two sublattices: r/i = a~-b~. In a ternary alloy, there are only two independent LRO parameters since: Er/i = 0. The concentration of each element being known, it is possible to determine every sublattice occupancy ai and b i according to: c~= 0.25a i + 0.75bi, with the knowledge of the two LRO parameters. The LRO parameters may be obtained from the ratio of integrated intensities of the superlattice reflections Is to those of the fundamental reflections I r as follows Is
LPsmse-2MslFd 2
ff = L P f m f e _ 2 M r iFrl 2
(1)
where L P is the Lorentz-polarization factor, m is the multiplicity factor of each reflection and e x p ( - 2 M ) is the temperature factor. The structure factors are: Fs=Zrhf~ and F r = 4Zcif~, where f~ is the atomic scattering factor corrected for anomalous dispersion
o/sin 0 \
f = f ; + tf:' = f i [ - - ~ - - ) + Af;(2) +/Af;'(2). (2) Then we can write the ratio Ifsl 2 I r t ~ ( f , - A ) + ~ 2 ( A - A ) l R(2) = i--F-~fl2= itfl 2
2
(3)
R(2) is a quadratic form of ~/l, r/2. On the plane ~/l, rh, the locus of the points solution of equation (3) is an ellipse centered on the point r/l = ~/2= 0, the orientation and the eccentricity of which depend only on the atomic scattering factors. The LRO parameters are given by the intersection of at least two ellipses. To increase the accuracy of this intersection, it is necessary for the orientation of the ellipses to be different. We have computed ellipse orientation and eccentricity as a function of the wavelength of the incident X-rays. Figures 1 and 2 give an example
Iog(elb) I "~ 3
NJAITi
MATHEMATICAL CONSIDERATIONS The unit cell of an ordered L12 structure involves two types of sublattices, which are defined as an a-sublattice {0 0 0} and a b-sublattice {1/2 1/2 0}. Let us define the long range order (LRO) parameters as
1
2
~(A)
Fig. 2. Ellipse eccentricity as a function of wavelength.
MARTY et al.:
LONG RANGE ORDER IN Ni-BASE ALLOYS
347
Table 1. Ratios of superlattice to fundamental integrated intensities for each alloy Ni070A10.20Cr0.~0 ~. (,~) R (,~) × 103
Nio.75Alo.15Tio.lo 2 (,~) R (2) X 103
Ni0.723Feo.24~Cro.036 2 (/~) R (2) x 103
1.4896 1.4917 1.4998 2.0780 2.1000
1.4915 1.5161 1.7902 2.3000 2.5068
1.7462 1.7902 2.0707 2.0727 2.0771 2.1000
10.4 12.9 15.7 27.7 25.5
for Ni75Al~sTi10. Obviously, we have to choose wavelengths near the absorption thresholds of the elements of the alloy. These considerations have led us to use synchrotron radiation which provides a continuous spectrum of wavelengths in the domain of interest. In order to obtain physical solutions (at >i 0 and b~ >~ 0) the values of ~/~ must obey the relation: - 4 c ~ / 3 <~ ~l, <~ 4c,. Hence, on the plane r/l , ~/2 the L R O representative points have to be inside a polygon the limits of which are ~/~ = 4 c l , rh = - 4 C l / 3 , r/: =4c2, rh = - 4 c 2 , 3 , rh + ~h = 4 c : / 3 , rh + r/2 = - 4 c 3. EXPERIMENTAL
The alloys were prepared by induction melting in alumina crucibles and casting in vacuum into a water-cooled copper mould. Metals of 99,95% purity were used as the starting materials. The Ni-0A120Crl0 and Ni75AllsTi10 ingots were powdered ( < 125 #m). The NiToA120Crl0 sample was annealed for 2h30 at 7 0 3 K and the Ni75AllsTi10 sample was annealed for 6 h at 723 K. We textured the (Ni3Fe)96.4Cr3.6 sample by rolling and annealing 30 mn at 923 K because the contrast would have been too small with a powder. Then this sample was ordered by annealing at 723 K for 45 days. The X-ray diffraction experiments have been performed on the four circles goniometer set up on the " D 2 3 " beam line at L U R E - D C I (Orsay-France) [11]. The beam line is equipped with a double crystal (Si 111) fixed beam exit monochromator. The second crystal is bent to focus horizontally the beam. The sample maintained under a vacuum beryllium hemisphere is fixed on a goniometer head. The Si-Li solid state detector has an energy resolution of 200 eV. The wavelengths chosen are adjusted by reference to an E X A F S spectrum from a copper foil. The accuracy of energy determination is 2 eV and hence the wavelengths are measured with an accuracy of 2.10 -4 A. The instabilities in the incident monochromatic beam are automatically corrected by a monitor detector which records diffuse scattering from a K a p t o n foil. Table 2. LRO parameters and degree of occupancy of a- and b-type sites for NiAICr alloy Species c rlj a~ bi Ni 0.70 -0.84 0.07 0.91 AI 0.20 0.72 0.74 0.02 Cr 0.10 0.12 0.19 0.07
8.8 14.5 17.4 18.6 23.8
4.6 1.5 0.83 0.83 0.85 0.76
The data are recorded by step scanning in the vertical plane. DATA ANALYSIS Intensities of both fundamental and superlattice reflections are corrected for background, Lorentzpolarization, multiplicity and temperature factors. They are integrated on an angular domain chosen arbitrarily to be five times larger than the full width at half-maximum ( F W H M ) of each reflection. This procedure reasonably takes into account the large tails of the Bragg reflections. F o r (Ni3Fe)96.4Cr3.6, the Debye parameter is taken from ordered N i 3 F e : B = 0.37 fl~2 [12]. In the absence of any data for Ni75AllsTi10 and Ni70Al20Crl0, we have taken a weighted mean of the atomic Debye parameters [13, 14]: B = 0.43/~2 and B = 0.42/~2 respectively. Atomic scattering factors are taken from Doyle and Turner [15] and anomalous dispersion corrections from Sazaki [16]. The lattice parameters are taken from the tables in Pearson's [17]: for Ni75AllsTi10, Ni7oAl20Crz0 and (NiaFe)96.4Cr3. 6 they are respectively a0 = 3.589, 3.559 and 3.552/~. RESULTS The ratios R(2) are given in Table 1. They vary strongly with the wavelength for each alloy, owing to the important effect of the anomalous scattering. As seen before [equation (3)], R(2) is a quadratic form of the two L R O parameters ~/1 and ~h, and hence the locus of the points solution are ellipses. The different ellipses computed are drawn on Figs. 3(a), 4(a) and 5(a). We used more than two wavelengths for each alloy in order to check the consistency of the measurements: the ellipses satisfactorily cross nearly at the same point. This point is inside or near the edge of the polygon of the possible L R O parameter set. We have obtained graphically the values of ~1 and r/2 and computed ai and bi (Tables 2, 3 and 4). The error domain on the L R O parameters values, the calculation of which is given in the Appendix, Table 3. LRO parameters and degree of occupancy of a- and b-type sites for NiA1Tialloy Species ci r/, a~ b~ Ni 0.75 -0.80 0.15 0.95 AI 0.15 0.60 0.60 0.00 Ti 0.10 0.20 0.25 0.05
348
MARTY
et al.:
LONG RANGE ORDER IN Ni-BASE ALLOYS
Table 4. LRO parametersand degreeof occupancyof a- and b-type sites for NiFeCr alloy Species ci rh at b~ Ni 0.723 -0.66 0.23 0.89 Fe 0.241 0.64 0.74 0.08 Cr 0.036 0.02 0.05 0.03 is drawn for each alloy on Figs 3(b), 4(b) and 5(b). The accuracy on the concentration is estimated as follows: Ac = +0.1 at.% for Ni, AI, Fe, Ti and Ac = + 0 . 2 a t . % for Cr, which is a more volatile element. The accuracy on the atomic scattering factors depends mainly on the determination of the wavelength, it is about +0.2 electron. The relative error on the intensity, estimated from reproducibility measurements, is about A I / I = _+2%. In order to ascertain the accuracy on the LRO parameters obtained, we have measured another set of Bragg reflections (1 l0 and 220) for the Ni75AllsTi]0 sample, using CoK~ radiation with a classical diffractometer. The agreement with the results obtained from the 100 and 200 reflections is within 2.5%. Hence, the calculated error domain seems to be overestimated. DISCUSSION For Ni70A120Cr~0, Ni and AI are almost completely ordered [Table 2 and Fig. 3(a)]. The Cr atoms are
distributed on both sublattices to fill up the sites unoccupied by Ni and AI. These results are in good agreement with those of Ochiai e t al. [4]; these authors have found that the solubility lobe of Cr in the Ni3Ai ~' phase extends in a direction almost bisecting the quasibinary sections Ni3A1-Ni3Cr and Ni~AI--Cr3AI on the ternary phase diagram. For Ni70AllsTil0 [Table 3 and Fig. 4(a)] all the A1 atoms are on the a sublattice. Ti has a strong preference for the same sublattice: it is distributed in a ratio of 5 : 1 between a-sites and b-sites. This site preference is again in agreement with the trend predicted by Ochiai e t al.: the solubility lobe of Ti in ~' phase lies in constant nickel content direction. The general trend of the substitution behaviour of Ti and Cr appears as being the same in more complex alloys: Chambreland [7] has analyzed a multicomponent alloy NiCoAITiCr using a time of flight atom probe. This method allows an estimate of the degree of site occupancy in the ~,' precipitates to be made. This author finds that the A1 and Ti atoms are on a-sites and the Cr atoms are on both a- and b-sites. As is shown in Table 4 and on Fig. 5, the long range order parameters of the (Ni3Fe)96.aCr3.6 alloy characterize an imperfectly ordered state: for a perfectly ordered state, the representative point would be on one limit of the polygon. The alloy was annealed for 45 days at 723K i.e. a long time near the .5
(a)
~ x
~
Z
1.
-
(b) Cr
I
~ - 1.
1"1 Ni
-1.
.0
1.
-1 .
T! m
-.5
Fig. 3. LRO parameters graphic determination (a) and error polygon (b) for NiA1Cr. Wavelengths used: 1.4896 A (a), 1.4917 A (b), 1.4998 A (c), 2.0780 A (d), 2.1000 A (e).
,,,q
(a)
1.
.5
(b)
~qTl
Z ~l-rl
1.
1"1 NI
1.
- 1.
1"1 Nt
-.5
Fig. 4. LRO parameters graphic determination (a) and error polygon (b) for NiA1Ti. Wavelengths used: 1.4915 A (a), 1.5161 A (b), 1.7902A (c), 2.3000 A (d), 2.5068 A (e).
MARTY et al.:
LONG RANGE ORDER IN Ni-BASE ALLOYS
349
.25
(a)
(b) ~qcr
11Cr
-.25 ml
.
I] Ni
|"
-.8
1] N1
-.5
Fig. 5. LRO parameters graphic determination (a) and error polygon (b) for NiFeCr. Wavelengths used: 1.7462 A (a), 1.7902 A (b), 2.0707 A (c), 2.0727 A (d), 2.0771 A (e), 2.1000 A (f). order-disorder transition temperature (743 K). For an equivalent ordering treatment, Ni3Fe is totally ordered, but Cr slo~s down the long range ordering kinetics [18]. In this state, Cr atoms occupy both aand b-sites with a slight preference for a sites: the ratio acr/bcr is about 5/3. This result may be compared with the result obtained by Cranshaw et al. by M6ssbauer spectroscopy on a NivsFe23Cr02 single crystal [8]. In this study, the experimental spectra may be fitted assuming that Cr atoms are distributed on a- and b-sites in the ratio 7/1 i.e. mostly on a-sites. The annealing treatment is equivalent for the two samples; the discrepancy between the two results is only apparent and may be explained as follows: Cr occupies both a- and b-sites without disturbing the N i - F e order, which tends to be maximum; a random distribution of Cr intervenes only in an alloy where the Ni and Fe concentrations are in the ratio 3/1. In the alloy studied by Cranshaw et al., the Ni concentration is 75%, hence Cr is distributed mostly on a-sites. As we have seen, Cr has the same substitution behaviour in Ni-A1 alloys: it has no preference for a given sublattice and it is distributed on both of the sites in order to complete the stoichiometry of the phase. Concerning Ti, the substitution behaviour is quite different. Ti is generally considered as substituting exclusiveb for A1. The chosen alloy composition would make possible to have the stoichiometry 3/1 for the ratio: number of Ni atoms/number of (Ti + AI) atoms. However our results show that 37% of the Ti atoms are distributed on the three Ni sublattices. Hence the alloy has to be considered as non-stoichiometric: there are only 71 at.% Ni on the b-sites. This behaviour can explain some features of the mechanical properties. In the 7' ternary alloys, significant strengthening occurs only when the ternary addition substitues for AI. Strengthening is then dependent on the degree of non-stoichiometry. Only a small strengthening occurs for Ni-rich alloys, whereas considerable strengthening occurs for stoichiometric and Al-rich alloys. The present results suggest that in any case the origin of significant strengthening is the same: the presence of AI and/or Ti atoms on the Ni-sublattices. AMM 38/2
N
The partial substitution of Ti on the Ni-sites, in a 75 at.% Ni alloy, also shows that the substitution behaviour is not a simple function of the size of the addition and other factors are involved: from size factor considerations one would expect the large Ti atoms (atomic radius: 1.47A) to substitute exclusively for A1 (atomic radius: 1.43 A).
CONCLUSION We have shown that, owing to X-ray scattering measurement using synchrotron radiation, absolute values can be assigned to the two long range order parameters in ternary alloys. Those give the knowledge of the preferential occupancy by ternary additions in LI 2 alloys, which is crucial to the understanding of the controlling factors in the alloying behaviour. Such information may be predicted but neither formally nor accurately derived from the ternary phase diagrams. In the Ni70Al20Crl0 and (Ni3Fe)964Cr3.6 L12 alloys, the Cr atoms can occupy sites on both sublattices. The Ni-A1 and N i - F e order tends to be maximum and Cr is distributed in order to complete the stoichiometry of the phase. In NiAITi alloys, it is generally thought that Ti substitutes exclusively for AI. Our results for the Ni75Al~sTilo alloy show that actually Ti substitutes mostly for A1; however 37% of the Ti atoms substitutes for Ni in the lattice, thus promoting disorder. This substitution behaviour may depend on the Ti concentration and experiments are in progress in order to investigate the proportion of Ti atoms at the different substitution sites as a function of Ti content in NiAlTi alloys. Anyway, our results show that the substitution behaviour is not a simple function of the size of the addition and other factors are clearly involved. The substitution behaviour is more complex than is generally thought and further accurate determinations of the long range order parameters are clearly needed in order to get a good understanding of the strengthening due to ternary additions in 7' alloys.
350
MARTY et al.:
LONG RANGE ORDER IN Ni-BASE ALLOYS
Acknowledgements--It is a pleasure to thank Dr J. P. Chevalier for helpful comments. This work was done within the framework of the Groupement Scientifique Superalliages Monocristallins grouping CNRS, SNECMA, TURBOMECA and IMPHY S.A.
APPENDIX Error Calculation Let R be the ratio of integrated intensities of the superlattice reflection to the fundamental R = l , oo
REFERENCES 1. D. P. Pope and S. S. Ezz, lnt. Metall. Rev. 29, 136 (1984). 2. T. Suzuki, Y. Oya and D. M. Wee, Acta metall. 28, 301 (1980). 3. R. D. Rawlings and A. E. Staton-Bevan, J. Mater. Sci. 10, 505 (1975). 4. S. Ochiai, Y. Oya and T. Suzuki, Acta metall. 32, 289 (1984). 5. High Temperature Ordered Intermetallic Alloys II, Mater. Res. Soc. Syrup. Proc. 81 (edited by N. Stoloff, C. C. Koch, C. T. Lin and O. Izumi), Pittsburgh (1987). 6. M. K. Miller and J. A. Horton, Mater. Res. Soc. Syrup. Proc. 81, 117 (1987). 7. S. Chambreland, Th6se de rUniversit6 de Rouen, France (1987). 8. T. E. Cranshaw, J. Phys. F, Metal Phys. 18, 43 (1988). 9. A. V. Karg, D. E. Fornwalt and O. H. Kriege, J. Inst. Metals 99, 301 (1971). 10. H. Ferjani, F. Bley and M. Fayard, Scripta metall. 13, 17 (1979). 11. M. Bessiere, G. Bessenay, J. Frouin, M. Jouvin and S. Lefebvre, Nucl. Instr. Meth. A261, 591 (1987). 12. P. Turchi, Y. Calvayrac and F. Plicque, Physica status solidi A45, 229 (1978). 13. International Tables for X-ray Crystallography Vol. 3, p. 237 (1968). 14. T. Paakkari, Acta crystallogr. A30, 83 (1974). 15. P. A. Doyle and P. S. Turner, Acta erystallogr. A24, 390 (1968). 16. S. Sasaki, Anomalous scattering factors for synchrotron radiation users, calculated using Cromer and Liberman's method. National Lab. for Energy Physics oho-machi, Japan (1984). 17. W. B. Pearson, Handbook o f Lattice Spacings and Structures o f Metals (edited by G. V. Raynor). Pergamon Press, Oxford (1967). 18. H. Ferjani, F. Bley and Y. Calvayrac, J. Physique 38, C7-55 (1977).
Let ~i and c~ (i = 1, 2, 3) be the long range order parameters and the concentrations in a ternary alloy. We write (Zq,f;)2 + (Zq~f'/)2 = 16R" {(ZGf ;)2 + ( Z G f I, )2} (A 1) where f~ and fi' are the real and imaginary parts of the atomic scattering factor for the element i. The first-order expansions of the equation (1) with respect to a little variation of ~b on the one hand and c~, f~, f 7 on the other hand are 6E = Z H * fr b
(A2)
6R 6E = - - + ZC*6c, + Y_,F;*6f; + ZFI'*ffI' 2R
(A3)
where H*-
C* =
r~*
(Dlkf'k)f; + (Zrlkf~)f:! (Zqkf ~)2 + (Y~qtf'k)2 (Y-,ckf'Df; + (Zckf';)f7 (Zckf,k) 2 + (Zckf~)2 (Zckf'k) c, (Eckf'k) 2 + (Eckf~) 2
(Zqkf~) qi (Zqkf~) z + (EF/kf~) 2
(Eckf'~ )ci (Eqkf'~ )qi FI'* = (y.ckf~)2 + (Eckf'~) 2 -- (~lkf,k)2 + (Zqkf~) 2' We see that EH*~ b = 1. Hence, bE gives the relative error on the long range order parameters. For each ellipse solution, it is possible to draw an upper limit and a lower limit. These ellipses have the homothetic ratios 1 + bE and 1 - bE. For several wavelengths, we obtain a curvilinear polygon limiting the error domain for long range order parameters in the plane r/~, q2. The coefficients C~*, F~* and FI'* are calculated when the long range order parameters have been found, fiE is the sum of the absolute values of each errors due to 6c~, bf;, 6fi', so it is expected to be overestimated.
0956-7151/90$3.00+ 0.00 Copyright © 1990PergamonPress plc
Printed in Great Britain. All rights reserved
D E T E R M I N A T I O N OF LONG RANGE ORDER IN Ni-BASE T E R N A R Y ALLOYS BY X-RAY A N O M A L O U S DIFFRACTION USING SYNCHROTRON RADIATION A. M A R T Y l, M. BESSIERE 2, F. BLEY 3, Y. CALVAYRAC t and S. LEFEBVRE L2 ICentre d'~tudes de chimie M&allurgique, CNRS LP 2801, 15 rue Georges Urbain, 94407 Vitry sur Seine Cedex, France, :LURE-CNRS, Bht. 209D 91405 Orsay, France and 3LTPCM. E.N.S.E.E.G.-Domaine Universitaire, B.P. 75, 38402 Saint Martin d'H6res, France (Received 20 March 1989)
Abstract--Long range order (LRO) parameters in three ternary L12 alloys [NiT~AllsTit0,Ni70Al20Crl0 and (Ni3Fe)~.4Cr36] have been measured using a new method. This method takes advantage of the variation in atomic scattering factors for X-rays near the absorption thresholds and opens a new area of research with the development of synchrotron facilities. The results show that Ti substitutes mostly, but not exclusively, for A1 in NiA1Ti. In NiA1Cr, Cr occupies both Ni and A1 sites. In NiFeCr, the order is not perfect; the Cr atoms are distributed on both types of sites, with a slight preference for Fe sites. R6sum~-Les param&res d'ordre/t longue distance dans trois alliages ternaires [Ni7sAltsTito,NivoA120Crt0 and iNi3Fe)964Cr36] ont 6t6 mesur6s en utilisant une nouvelle m6thode. Cette m6thode tire parti de la variation des facteurs de diffusion atomiques des rayons X pr6s des seuils d'absorption et ouvre un nouveau champ de recherche avec le d6veloppement du rayonnement synchrotron. Les r6sultats montrent que le titane se substitue principalement-mais pas exclusivement-~ l'aluminium dans NiAITi. Dans NiAICr, le chrome est r6parti ~i la fois sur les sites du nickel et sur ceux de l'aluminium. Dans NiFeCr, rordre n'est pas parfait; les atomes de chrome sont distribu6s sur les deux types de sites, avec une 16g6re pr6f6rence pour les sites du fer. Zusammenfassung--Die Parameter der weitreichenden Ordnung werden in den tern/iren L12-Legierungen [Ni75AllsTi~o, NiToAl20Crl0und (Ni3Fe)964Cr36] mit einer neuen Methode gemessen. Diese Methode nutzt die .~nderung in den Atomformfaktoren i;iir R6ntgenstreuung in der N/ihe der Absorptionskanten aus und er6ffnet neue Forschungsm6glichkeiten mit Synchrotronstrahlung. Die Ergebnisse zeigen, dab Ti das AI in NiA1Ti meistens, aber nicht ausschliel31ich,substituiert; Cr besetzt Ni- und AI-Pl/itze. In NiFeCr ist die Ordnung nicht perfekt; die Cr-Atome sind auf beide Platzarten verteilt, wobei der Fe-Platz ein wenig bevorzugt ist.
INTRODUCTION The mechanical properties of ordered binary L12 alloys are strongly modified by the addition of a third element [1, 2]. A m o n g these alloys, the intermetallic c o m p o u n d Ni3A1 (~') has received much attention, because of a substantial increase in flow stress with increasing temperature. Although this compound has a narrow composition range, it has been shown that it can accommodate considerable amounts of ternary additions in solution, resulting in significant strengthening [3]. Since in most of the nickel-base "superalloys" the major factor enabling high temperature strength is due to precipitates of an intermetallic compound, based on Ni3AI, dispersed in the nickelbased solid solution (':), it may be expected that the mechanical properties of nickel-base superalloys may be optimized by a proper choice of alloy composition. In order to understand the role that the different elements play in the strengthening mechanism, it is important to know the location of the third element
in the lattice, i.e. to determine the state of long range order. A prediction on the location of the substitutional elements may be obtained from an analysis of ternary phase diagrams, and for example, Ochiai et al. [4] suggest site preferences for several elements (Co, Cr, Ti, Mn) in three alloys (Ni3Ga, Ni3Si, Ni3AI) from the knowledge of the y' solubility lobe direction in the ternary phase diagrams. Experimentally, the site occupation may be determined by a large variety of techniques (e.g. see [5]). A m o n g them, the most promising seem to be atom probe field-ion microscopy (APFIM), M6ssbauer spectroscopy and X-ray diffraction. The atomic spatial resolution of the atom probe field-ion microscopy permits the site occupation probability of any substitutional element to be determined. This technique has no theoretical but experimental limits: there are difficulties in preparing samples and problems of preferential evaporation. Recently, Miller and Horton [6] have analyzed the 345
346
MARTY et al.: LONG RANGE ORDER IN Ni-BASE ALLOYS
position of Hf, Fe, Co in a Ni-A1 alloy doped with boron. Chambreland [7] has studied the position of Ti and Cr in a Ni rich complex industrial alloy NiCoAITiCr; his results will be compared with those we have obtained on ternary alloys. M6ssbauer spectroscopy, which is a local order probe, is limited by the available radioactive sources, among which iron is a favourite. Using this technique, Cranshaw et al. [8] have studied a Ni75Fe23Cr2 single crystal. We will see that their results lead to the same conclusion as ours concerning the substitutional behaviour of Cr in Ni3Fe. X-ray diffraction is used currently to measure long range order parameter in binary alloys. In multicomponent alloys, several independent LRO parameters have to be determined. A first approach to the problem consists in considering the multicomponent alloy as a quasi-binary alloy which is therefore characterized by only one LRO parameter. A trial and error method is then used to give an estimation of the possible positions of the substitutional elements in the lattice. This method has been used by Karg et al. in their study of y'-Ni3A1 based multicomponent alloys [9]. However, as these authors conclude, the attribution of a single LRO parameter to such complex alloy systems is not reasonable. The determination of each LRO parameter in multicomponent alloys is formally possible by taking advantage of the variation in the atomic scattering factors due to the anomalous correction. However, it is in practice very difficult when using the wavelengths available with classical targets. Ferjani et al. [10] have used two different radiations (CUK~, COKe) in order to measure the two long range order parameters in NiFeMo and NiFeCr alloys. The determination was unsuccessful for the NiFeCr alloy because, for the two available wavelengths, there is not a sufficient change in intensity resulting from the different possible assumptions as for the Cr position in the lattice. The development of synchrotron facilities open a new field of investigation, giving the possibility of adjusting the wavelength in order to optimize the use of the anomalous scattering. Using such a technique, we have performed a determination of the state of LRO in three ternary alloys: NivoA120Crjo, Ni75AllsTilo and (Ni3Fe)96.4Cr36. We have chosen compositions with the highest concentration in the third element compatible with LI 2 structure, in order to get the maximum contrast. For NiFeCr we had to choose a smaller Cr concentration, because the ordering kinetics becomes very slow when the Cr concentration increases.
e (dog)
140-
N|A1TI
1301
Ti threshold
120l la 1001 N1 Threshold 90-
2
~(A)
Fig. I. Ellipse orientation as a function of wavelength. the difference of proportions ai and bi of each element i on the two sublattices: r/i = a~-b~. In a ternary alloy, there are only two independent LRO parameters since: Er/i = 0. The concentration of each element being known, it is possible to determine every sublattice occupancy ai and b i according to: c~= 0.25a i + 0.75bi, with the knowledge of the two LRO parameters. The LRO parameters may be obtained from the ratio of integrated intensities of the superlattice reflections Is to those of the fundamental reflections I r as follows Is
LPsmse-2MslFd 2
ff = L P f m f e _ 2 M r iFrl 2
(1)
where L P is the Lorentz-polarization factor, m is the multiplicity factor of each reflection and e x p ( - 2 M ) is the temperature factor. The structure factors are: Fs=Zrhf~ and F r = 4Zcif~, where f~ is the atomic scattering factor corrected for anomalous dispersion
o/sin 0 \
f = f ; + tf:' = f i [ - - ~ - - ) + Af;(2) +/Af;'(2). (2) Then we can write the ratio Ifsl 2 I r t ~ ( f , - A ) + ~ 2 ( A - A ) l R(2) = i--F-~fl2= itfl 2
2
(3)
R(2) is a quadratic form of ~/l, r/2. On the plane ~/l, rh, the locus of the points solution of equation (3) is an ellipse centered on the point r/l = ~/2= 0, the orientation and the eccentricity of which depend only on the atomic scattering factors. The LRO parameters are given by the intersection of at least two ellipses. To increase the accuracy of this intersection, it is necessary for the orientation of the ellipses to be different. We have computed ellipse orientation and eccentricity as a function of the wavelength of the incident X-rays. Figures 1 and 2 give an example
Iog(elb) I "~ 3
NJAITi
MATHEMATICAL CONSIDERATIONS The unit cell of an ordered L12 structure involves two types of sublattices, which are defined as an a-sublattice {0 0 0} and a b-sublattice {1/2 1/2 0}. Let us define the long range order (LRO) parameters as
1
2
~(A)
Fig. 2. Ellipse eccentricity as a function of wavelength.
MARTY et al.:
LONG RANGE ORDER IN Ni-BASE ALLOYS
347
Table 1. Ratios of superlattice to fundamental integrated intensities for each alloy Ni070A10.20Cr0.~0 ~. (,~) R (,~) × 103
Nio.75Alo.15Tio.lo 2 (,~) R (2) X 103
Ni0.723Feo.24~Cro.036 2 (/~) R (2) x 103
1.4896 1.4917 1.4998 2.0780 2.1000
1.4915 1.5161 1.7902 2.3000 2.5068
1.7462 1.7902 2.0707 2.0727 2.0771 2.1000
10.4 12.9 15.7 27.7 25.5
for Ni75Al~sTi10. Obviously, we have to choose wavelengths near the absorption thresholds of the elements of the alloy. These considerations have led us to use synchrotron radiation which provides a continuous spectrum of wavelengths in the domain of interest. In order to obtain physical solutions (at >i 0 and b~ >~ 0) the values of ~/~ must obey the relation: - 4 c ~ / 3 <~ ~l, <~ 4c,. Hence, on the plane r/l , ~/2 the L R O representative points have to be inside a polygon the limits of which are ~/~ = 4 c l , rh = - 4 C l / 3 , r/: =4c2, rh = - 4 c 2 , 3 , rh + ~h = 4 c : / 3 , rh + r/2 = - 4 c 3. EXPERIMENTAL
The alloys were prepared by induction melting in alumina crucibles and casting in vacuum into a water-cooled copper mould. Metals of 99,95% purity were used as the starting materials. The Ni-0A120Crl0 and Ni75AllsTi10 ingots were powdered ( < 125 #m). The NiToA120Crl0 sample was annealed for 2h30 at 7 0 3 K and the Ni75AllsTi10 sample was annealed for 6 h at 723 K. We textured the (Ni3Fe)96.4Cr3.6 sample by rolling and annealing 30 mn at 923 K because the contrast would have been too small with a powder. Then this sample was ordered by annealing at 723 K for 45 days. The X-ray diffraction experiments have been performed on the four circles goniometer set up on the " D 2 3 " beam line at L U R E - D C I (Orsay-France) [11]. The beam line is equipped with a double crystal (Si 111) fixed beam exit monochromator. The second crystal is bent to focus horizontally the beam. The sample maintained under a vacuum beryllium hemisphere is fixed on a goniometer head. The Si-Li solid state detector has an energy resolution of 200 eV. The wavelengths chosen are adjusted by reference to an E X A F S spectrum from a copper foil. The accuracy of energy determination is 2 eV and hence the wavelengths are measured with an accuracy of 2.10 -4 A. The instabilities in the incident monochromatic beam are automatically corrected by a monitor detector which records diffuse scattering from a K a p t o n foil. Table 2. LRO parameters and degree of occupancy of a- and b-type sites for NiAICr alloy Species c rlj a~ bi Ni 0.70 -0.84 0.07 0.91 AI 0.20 0.72 0.74 0.02 Cr 0.10 0.12 0.19 0.07
8.8 14.5 17.4 18.6 23.8
4.6 1.5 0.83 0.83 0.85 0.76
The data are recorded by step scanning in the vertical plane. DATA ANALYSIS Intensities of both fundamental and superlattice reflections are corrected for background, Lorentzpolarization, multiplicity and temperature factors. They are integrated on an angular domain chosen arbitrarily to be five times larger than the full width at half-maximum ( F W H M ) of each reflection. This procedure reasonably takes into account the large tails of the Bragg reflections. F o r (Ni3Fe)96.4Cr3.6, the Debye parameter is taken from ordered N i 3 F e : B = 0.37 fl~2 [12]. In the absence of any data for Ni75AllsTi10 and Ni70Al20Crl0, we have taken a weighted mean of the atomic Debye parameters [13, 14]: B = 0.43/~2 and B = 0.42/~2 respectively. Atomic scattering factors are taken from Doyle and Turner [15] and anomalous dispersion corrections from Sazaki [16]. The lattice parameters are taken from the tables in Pearson's [17]: for Ni75AllsTi10, Ni7oAl20Crz0 and (NiaFe)96.4Cr3. 6 they are respectively a0 = 3.589, 3.559 and 3.552/~. RESULTS The ratios R(2) are given in Table 1. They vary strongly with the wavelength for each alloy, owing to the important effect of the anomalous scattering. As seen before [equation (3)], R(2) is a quadratic form of the two L R O parameters ~/1 and ~h, and hence the locus of the points solution are ellipses. The different ellipses computed are drawn on Figs. 3(a), 4(a) and 5(a). We used more than two wavelengths for each alloy in order to check the consistency of the measurements: the ellipses satisfactorily cross nearly at the same point. This point is inside or near the edge of the polygon of the possible L R O parameter set. We have obtained graphically the values of ~1 and r/2 and computed ai and bi (Tables 2, 3 and 4). The error domain on the L R O parameters values, the calculation of which is given in the Appendix, Table 3. LRO parameters and degree of occupancy of a- and b-type sites for NiA1Tialloy Species ci r/, a~ b~ Ni 0.75 -0.80 0.15 0.95 AI 0.15 0.60 0.60 0.00 Ti 0.10 0.20 0.25 0.05
348
MARTY
et al.:
LONG RANGE ORDER IN Ni-BASE ALLOYS
Table 4. LRO parametersand degreeof occupancyof a- and b-type sites for NiFeCr alloy Species ci rh at b~ Ni 0.723 -0.66 0.23 0.89 Fe 0.241 0.64 0.74 0.08 Cr 0.036 0.02 0.05 0.03 is drawn for each alloy on Figs 3(b), 4(b) and 5(b). The accuracy on the concentration is estimated as follows: Ac = +0.1 at.% for Ni, AI, Fe, Ti and Ac = + 0 . 2 a t . % for Cr, which is a more volatile element. The accuracy on the atomic scattering factors depends mainly on the determination of the wavelength, it is about +0.2 electron. The relative error on the intensity, estimated from reproducibility measurements, is about A I / I = _+2%. In order to ascertain the accuracy on the LRO parameters obtained, we have measured another set of Bragg reflections (1 l0 and 220) for the Ni75AllsTi]0 sample, using CoK~ radiation with a classical diffractometer. The agreement with the results obtained from the 100 and 200 reflections is within 2.5%. Hence, the calculated error domain seems to be overestimated. DISCUSSION For Ni70A120Cr~0, Ni and AI are almost completely ordered [Table 2 and Fig. 3(a)]. The Cr atoms are
distributed on both sublattices to fill up the sites unoccupied by Ni and AI. These results are in good agreement with those of Ochiai e t al. [4]; these authors have found that the solubility lobe of Cr in the Ni3Ai ~' phase extends in a direction almost bisecting the quasibinary sections Ni3A1-Ni3Cr and Ni~AI--Cr3AI on the ternary phase diagram. For Ni70AllsTil0 [Table 3 and Fig. 4(a)] all the A1 atoms are on the a sublattice. Ti has a strong preference for the same sublattice: it is distributed in a ratio of 5 : 1 between a-sites and b-sites. This site preference is again in agreement with the trend predicted by Ochiai e t al.: the solubility lobe of Ti in ~' phase lies in constant nickel content direction. The general trend of the substitution behaviour of Ti and Cr appears as being the same in more complex alloys: Chambreland [7] has analyzed a multicomponent alloy NiCoAITiCr using a time of flight atom probe. This method allows an estimate of the degree of site occupancy in the ~,' precipitates to be made. This author finds that the A1 and Ti atoms are on a-sites and the Cr atoms are on both a- and b-sites. As is shown in Table 4 and on Fig. 5, the long range order parameters of the (Ni3Fe)96.aCr3.6 alloy characterize an imperfectly ordered state: for a perfectly ordered state, the representative point would be on one limit of the polygon. The alloy was annealed for 45 days at 723K i.e. a long time near the .5
(a)
~ x
~
Z
1.
-
(b) Cr
I
~ - 1.
1"1 Ni
-1.
.0
1.
-1 .
T! m
-.5
Fig. 3. LRO parameters graphic determination (a) and error polygon (b) for NiA1Cr. Wavelengths used: 1.4896 A (a), 1.4917 A (b), 1.4998 A (c), 2.0780 A (d), 2.1000 A (e).
,,,q
(a)
1.
.5
(b)
~qTl
Z ~l-rl
1.
1"1 NI
1.
- 1.
1"1 Nt
-.5
Fig. 4. LRO parameters graphic determination (a) and error polygon (b) for NiA1Ti. Wavelengths used: 1.4915 A (a), 1.5161 A (b), 1.7902A (c), 2.3000 A (d), 2.5068 A (e).
MARTY et al.:
LONG RANGE ORDER IN Ni-BASE ALLOYS
349
.25
(a)
(b) ~qcr
11Cr
-.25 ml
.
I] Ni
|"
-.8
1] N1
-.5
Fig. 5. LRO parameters graphic determination (a) and error polygon (b) for NiFeCr. Wavelengths used: 1.7462 A (a), 1.7902 A (b), 2.0707 A (c), 2.0727 A (d), 2.0771 A (e), 2.1000 A (f). order-disorder transition temperature (743 K). For an equivalent ordering treatment, Ni3Fe is totally ordered, but Cr slo~s down the long range ordering kinetics [18]. In this state, Cr atoms occupy both aand b-sites with a slight preference for a sites: the ratio acr/bcr is about 5/3. This result may be compared with the result obtained by Cranshaw et al. by M6ssbauer spectroscopy on a NivsFe23Cr02 single crystal [8]. In this study, the experimental spectra may be fitted assuming that Cr atoms are distributed on a- and b-sites in the ratio 7/1 i.e. mostly on a-sites. The annealing treatment is equivalent for the two samples; the discrepancy between the two results is only apparent and may be explained as follows: Cr occupies both a- and b-sites without disturbing the N i - F e order, which tends to be maximum; a random distribution of Cr intervenes only in an alloy where the Ni and Fe concentrations are in the ratio 3/1. In the alloy studied by Cranshaw et al., the Ni concentration is 75%, hence Cr is distributed mostly on a-sites. As we have seen, Cr has the same substitution behaviour in Ni-A1 alloys: it has no preference for a given sublattice and it is distributed on both of the sites in order to complete the stoichiometry of the phase. Concerning Ti, the substitution behaviour is quite different. Ti is generally considered as substituting exclusiveb for A1. The chosen alloy composition would make possible to have the stoichiometry 3/1 for the ratio: number of Ni atoms/number of (Ti + AI) atoms. However our results show that 37% of the Ti atoms are distributed on the three Ni sublattices. Hence the alloy has to be considered as non-stoichiometric: there are only 71 at.% Ni on the b-sites. This behaviour can explain some features of the mechanical properties. In the 7' ternary alloys, significant strengthening occurs only when the ternary addition substitues for AI. Strengthening is then dependent on the degree of non-stoichiometry. Only a small strengthening occurs for Ni-rich alloys, whereas considerable strengthening occurs for stoichiometric and Al-rich alloys. The present results suggest that in any case the origin of significant strengthening is the same: the presence of AI and/or Ti atoms on the Ni-sublattices. AMM 38/2
N
The partial substitution of Ti on the Ni-sites, in a 75 at.% Ni alloy, also shows that the substitution behaviour is not a simple function of the size of the addition and other factors are involved: from size factor considerations one would expect the large Ti atoms (atomic radius: 1.47A) to substitute exclusively for A1 (atomic radius: 1.43 A).
CONCLUSION We have shown that, owing to X-ray scattering measurement using synchrotron radiation, absolute values can be assigned to the two long range order parameters in ternary alloys. Those give the knowledge of the preferential occupancy by ternary additions in LI 2 alloys, which is crucial to the understanding of the controlling factors in the alloying behaviour. Such information may be predicted but neither formally nor accurately derived from the ternary phase diagrams. In the Ni70Al20Crl0 and (Ni3Fe)964Cr3.6 L12 alloys, the Cr atoms can occupy sites on both sublattices. The Ni-A1 and N i - F e order tends to be maximum and Cr is distributed in order to complete the stoichiometry of the phase. In NiAITi alloys, it is generally thought that Ti substitutes exclusively for AI. Our results for the Ni75Al~sTilo alloy show that actually Ti substitutes mostly for A1; however 37% of the Ti atoms substitutes for Ni in the lattice, thus promoting disorder. This substitution behaviour may depend on the Ti concentration and experiments are in progress in order to investigate the proportion of Ti atoms at the different substitution sites as a function of Ti content in NiAlTi alloys. Anyway, our results show that the substitution behaviour is not a simple function of the size of the addition and other factors are clearly involved. The substitution behaviour is more complex than is generally thought and further accurate determinations of the long range order parameters are clearly needed in order to get a good understanding of the strengthening due to ternary additions in 7' alloys.
350
MARTY et al.:
LONG RANGE ORDER IN Ni-BASE ALLOYS
Acknowledgements--It is a pleasure to thank Dr J. P. Chevalier for helpful comments. This work was done within the framework of the Groupement Scientifique Superalliages Monocristallins grouping CNRS, SNECMA, TURBOMECA and IMPHY S.A.
APPENDIX Error Calculation Let R be the ratio of integrated intensities of the superlattice reflection to the fundamental R = l , oo
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Let ~i and c~ (i = 1, 2, 3) be the long range order parameters and the concentrations in a ternary alloy. We write (Zq,f;)2 + (Zq~f'/)2 = 16R" {(ZGf ;)2 + ( Z G f I, )2} (A 1) where f~ and fi' are the real and imaginary parts of the atomic scattering factor for the element i. The first-order expansions of the equation (1) with respect to a little variation of ~b on the one hand and c~, f~, f 7 on the other hand are 6E = Z H * fr b
(A2)
6R 6E = - - + ZC*6c, + Y_,F;*6f; + ZFI'*ffI' 2R
(A3)
where H*-
C* =
r~*
(Dlkf'k)f; + (Zrlkf~)f:! (Zqkf ~)2 + (Y~qtf'k)2 (Y-,ckf'Df; + (Zckf';)f7 (Zckf,k) 2 + (Zckf~)2 (Zckf'k) c, (Eckf'k) 2 + (Eckf~) 2
(Zqkf~) qi (Zqkf~) z + (EF/kf~) 2
(Eckf'~ )ci (Eqkf'~ )qi FI'* = (y.ckf~)2 + (Eckf'~) 2 -- (~lkf,k)2 + (Zqkf~) 2' We see that EH*~ b = 1. Hence, bE gives the relative error on the long range order parameters. For each ellipse solution, it is possible to draw an upper limit and a lower limit. These ellipses have the homothetic ratios 1 + bE and 1 - bE. For several wavelengths, we obtain a curvilinear polygon limiting the error domain for long range order parameters in the plane r/~, q2. The coefficients C~*, F~* and FI'* are calculated when the long range order parameters have been found, fiE is the sum of the absolute values of each errors due to 6c~, bf;, 6fi', so it is expected to be overestimated.