Metallurgical and Materials Transactions A

, Volume 43, Issue 5, pp 1413–1422

Stress Annealing Induced Diffuse Scattering from Ni3 (Al, Si) Precipitates

Authors

    • Oak Ridge National Laboratory
  • G. E. Ice
    • Oak Ridge National Laboratory
  • E. A. Karapetrova
    • Materials Science and Technology DivisionAdvanced Photon Source
  • P. Zschack
    • Materials Science and Technology DivisionAdvanced Photon Source
Symposium: Neutron and X-Ray Studies of Advanced Materials IV

DOI: 10.1007/s11661-011-0937-z

Cite this article as:
Barabash, R.I., Ice, G.E., Karapetrova, E.A. et al. Metall and Mat Trans A (2012) 43: 1413. doi:10.1007/s11661-011-0937-z

Abstract

Diffuse scattering caused by L12 type Ni3 (Al, Si) precipitates after stress annealing of Ni-Al-Si alloys is studied. Peculiarities of diffuse scattering in the asymptotic region as compared to the Huang scattering region are discussed. Coupling between the stress annealing direction and the precipitate shape is demonstrated. Experimental reciprocal space maps (RSMs) are compared to theoretical ones. Oscillations of diffuse scattering due to Ni3 (Al, Sc) precipitates are observed. The strengths of the precipitates are estimated from the analysis of the diffuse scattering oscillations.

1 Introduction

X-ray and neutron diffuse scattering was used in many studies to access the fundamental properties of materials and defects in them.[139] The grouping of point defects into clusters, microscopic pores, coherent precipitates, dislocation loops, and other organized structures changes the character of the local strain field and results in a redistribution of diffuse scattering intensity. The diffuse scattering depends on both the local and the average lattice distortions. The average distortion is related to the static Debye–Waller factor (DWF). Peculiarities of diffuse scattering are distinct for defects corresponding to different ranges of static DWF. A qualitative classification of defects with respect to their influence on diffuse scattering and reciprocal space was developed by Krivoglaz.[1] This classification is based on an analysis of the static DWF exponent, e-2W. Defects can be described as belonging to one of two kinds.

1.1 Defects of the First Kind

In crystals with defects of the first kind, the intensity from the Bragg peaks is redistributed into broad diffuse intensity. In addition, the Bragg peaks remain sharp like those of perfect crystals, but become weaker and can be displaced from their reciprocal space location without defects. For this reason, both the Bragg and diffuse scattering can be simultaneously observed in the diffraction pattern. Usually, the intensity of the diffuse component ID increases and the intensity of the Bragg component I0 decreases with defect concentration c.

1.2 Defects of the Second Kind

In crystals with defects of the second kind, long-range spatial correlations of atomic sites are lost and the Bragg term becomes meaningless. Sharp Bragg reflections of perfect crystals are replaced by broad peaks, which can be anisotropic in reciprocal space.

Whether a defect is of the first or second kind depends on the behavior of the DWF exponent, 2W, at large distances from the defect (ρ ⇒ ∞). It should be noted that the DWF has a complicated dependence on the details of the distortion fields near a defect. As a first approximation,[1]
$$ 2W = \text{Re} T_{\infty } \cong c\mathop {\lim }\limits_{\rho \Rightarrow \infty } \sum\limits_{tss'} {\left[ {1 - \cos \left( {{\bf{Qu}}_{ss't} } \right)} \right]} \left[ {1 + \frac{1}{f}\left( {\phi_{st} + \phi_{s't} } \right)} \right] $$
(1)
Here, T is the correlation function for defects at large distances (ρ ⇒ ∞), c is the concentration of defects, f is the structure factor of the average crystal, Q is the momentum transfer for certain (hkl) reflections, usst is the difference between displacements in two scattering cells s and s′ caused by the defect located in position t, and φst and φst describe structure amplitude changes of scattering cells s and s′ caused by the defect located in the position t. We note that for dislocations, changes in structure amplitudes are small and the behavior of the 2W depends mainly of the asymptotic behavior of the displacement field created by the dislocation. With defects there are two possibilities: 2W is either finite at large distances or 2W tends to infinity at large distances. It was shown[16] that if the displacements fall off faster, then \( {1 \mathord{\left/ {\vphantom {1 {r^{3/2} }}} \right. \kern-\nulldelimiterspace} {r^{3/2} }}, \) the value 2W is finite, and the defects belong to the first kind. If the displacements decrease lower than\( {1 \mathord{\left/ {\vphantom {1 {r^{3/2} }}} \right. \kern-\nulldelimiterspace} {r^{3/2} }}, \) the value 2W ⇒ ∞ and these defects are of the second kind.
In this study, we report the diffuse scattering analysis of stress-annealing-induced coherent Ni3(Al,Sc) precipitates, which correspond to the first kind defects (Figure 1). As described previously, these defects cause diffuse scattering, ID(Q), that exists together with regular (strong) Bragg reflections, IBragg(Q).[15] The Bragg positions are displaced relative to the ideal positions of the matrix without precipitates. The analysis of the diffuse scattering differs for the case of small values 2W << 1 (weak distortions) and large values 2W > > 1 (strongly distorted solids). When 2W ≤ 1,  it is possible to use the single defect approximation in the isotropic matrix.[710] When 2W > > 1 or 2W ∼ 1,  the overlapping strain fields from different defects interact and the influence of such overlapping strain fields on scattering should be taken into account.[16] Many examples of the diffuse scattering analysis for crystals with defects of the first kind are described in the literature.[133] Lately, diffuse scattering was applied to study order-disorder transformations in amorphous, disordered, and complex materials.[3439] In this study, we focus on the peculiarities of diffuse scattering caused by nanosize Ni3 (Al, Si) precipitates after stress annealing of Ni-Al-Si alloys.
https://static-content.springer.com/image/art%3A10.1007%2Fs11661-011-0937-z/MediaObjects/11661_2011_937_Fig1_HTML.gif
Fig. 1

Model of the concentration distribution around a precipitate with a (a) sharp boundary and (b) strain field in the vicinity of the precipitate. Region of increased intensity around the (h,k,l) reciprocal lattice point in the precipitated alloy. (c) Vector q is the deviation of the diffraction vector Q from the exact momentum transfer G

2 Experimental Details

Diffuse scattering measurements were made using bending magnet radiation at beamline 33-BM-C of the Advanced Photon Source. 22 keV X-rays were selected using a sagitally focusing double crystal Si (111) monochromator. A pair of Rh-coated mirrors was used to eliminate harmonics and to focus the beam meridianally. The focal spot was ~0.6 mm in diameter.

Thermal diffuse scattering was suppressed by cooling the samples to ~6 K (~–267 °C) using a closed-cycle He refrigerator. Compton scattering and X-ray fluorescence were resolved from elastic scattering using an energy-dispersive VORTEX1 Si drift detector. Incident X-ray flux was monitored using an air-filled ion chamber. Slits were used to define the instrumental resolution. Resolution varies with scattering angle and slit dimensions; at low angles, the resolution was 0.0012 Å−1 in the radial and 0.0024 Å−1 in the transverse direction. Far from the Bragg reflections, the slits were widened to increase the signal. Reciprocal space units are G = 4π sin θ/λ,  where G is the momentum transfer modulus, θ is the Bragg angle, and λ is the wavelength. Titanium filters were used to reduce the beam intensity near Bragg reflections so the detector was not saturated.

The samples were needle-shaped (10 mm × 0.5 mm diameter) ternary Ni-Al-Si alloys with 11.9 pct Al and 2.1 pct Si corresponding to the boundary region between the face-centered-cubic (fcc) γ solid solution and ordered fcc-based Ni3(Al, Si)-based γ′ phase with ordered L12 structure, where Ni atoms occupy the face-center positions and Al/Si atoms occupy the cube corner positions. These precipitates have small misfits with the matrix and are coherent. Similar L12-type coherent precipitates are observed in different Ni-based superalloys.[4,5,40] The sample was quenched and stress annealed, with the sample being heated under load. Stress annealing was performed at a temperature of ~1373 K (~1100 °C) for 5 hours. Tensile stress was applied along [100] directions coinciding with the needle-sample axes. The level of tensile stress was just below the yield stress for these alloys. The structure factor for a disordered fcc matrix does not turn to zero around so-called fundamental reflections with Miller indices of the same parity. In contrast, for L12-type ordered precipitates, the structure factor is not equal to zero for additional so-called superlattice reflections with different parity of Miller indices.[41] The diffuse scattering is distinct around the fundamental and superlattice reflections and was measured and described separately (Sections III and IV).

3 Diffuse scattering by crystals with defect clusters

Defect grouping results in the formation of small clusters, small dislocation loops, and voids. The intensity distribution I(Q) of X-ray (or neutron) scattering due to defects can be computed from the expression[16,30]
$$ I({\mathbf{Q}}) = \left| {\sum\limits_{i} {f_{i} \exp (i{\mathbf{Q}} \cdot ({\mathbf{R}}_{i}^{0} + {\mathbf{u}}_{i} )} )} \right|^{2} $$
(2)
Here, fi is the scattering factor from an individual atom i, with relaxed coordinates RiRi0 + ui due to the presence of defects. Here, ui is the displacement from the equilibrium positions, Ri0, corresponding to the ideal crystal. We calculate the ui using continuum elastic theory. Q = k2 − k1 is the diffraction vector, with k1 and k2 being wave vectors of the incident and scattered X-rays. In the crystals with several defect kinds, an expression for diffuse scattering intensity can be written as follows:
$$ I_{D} ({\mathbf{Q}}) = N_{d} e^{ - 2W} \sum {c_{\alpha } \left| {\Upphi_{\alpha } ({\mathbf{Q}})} \right|^{2} } $$
(3)
$$ \Upphi_{\alpha } ({\mathbf{Q}}) = \Updelta F_{\alpha } ({\mathbf{Q}}) - F{\mathbf{Qu}}_{\alpha } ({\mathbf{q}}) + F\sum\limits_{S} {e^{{i{\mathbf{qR}}_{St} }} } \left( {e^{{i{\mathbf{Qu}}_{St\alpha } }} - i{\mathbf{Qu}}_{St\alpha } - 1} \right) $$
(4)
Here, Nd is a total number of α-type defects in the crystal, ΔFα is related to the change of the structure factor F of the S unit cell caused by an α-type defect in the location t, and uStα is a displacement of the S unit cell caused by an α-type defect in the location t. Vector q = Q − G characterizes the deviation of the diffraction vector from the momentum transfer G in the reciprocal space (Figure 1). In Eq. [4], the first term is related to the change of structure factor due to defects, while the second and third terms are strain induced. Moreover, the last term takes into account cross-terms between strain and the structure factor change due to defects. For small local distortions, this term may be eliminated[13] and expressions [3] and [4] simplified
$$ I_{D} ({\mathbf{Q}}) = Ne^{ - 2W} \sum\limits_{\alpha } {c_{\alpha } \left| {\Updelta F_{\alpha } ({\mathbf{Q}}) - F{\mathbf{Qu}}_{\alpha } ({\mathbf{q}})} \right|}^{2} $$
(5)

3.1 Precipitates with Strong Distortions

When n0 point defects group into a cluster or a dislocation loop, the character of diffuse scattering depends on the strength of the cluster, which in turn determines whether the cluster causes strong or weak distortions in the matrix. In the elastic isotropic continuum approximation, the static displacements in the vicinity of the spherical precipitate (Figure 1) can be written as follows:[13]
$$ {\mathbf{u}}_{St} = \frac{C}{{{R}_{0}^{3} }}{\varvec{R}}_{St} \;{\text{inside}}\;{\text{the}}\;{\text{precipitate}}\; ( {\text{for}}\;R_{St} \;{ < }\;R_{0} ) $$
(6a)
and
$$ {\mathbf{u}}_{St} = C\frac{{{\varvec{R}}_{St} }}{{R_{St}^{3} }}\;{\text{outside}}\;{\text{of}}\;{\text{the}}\;{\text{precipitate}}\; ( {\text{for}}\;R_{St} > R_{0} ) $$
(6b)
$$ {\text{The}}\;{\text{defect}}\;{\text{strength}}\;C = \frac{\Upgamma }{4\pi }n_{0} \Updelta v\;{\text{and}}\;\Upgamma = \frac{1}{3}\frac{1 + \sigma }{1 - \sigma } $$
(7)
Here, n0 is a number of matrix unit cells substituted by the precipitate, σ is the Poisson ratio, Δv is the difference between the precipitate and matrix unit cell volume, and RSt is the distance vector between the defect in the position t and the unit cell S. Parameter C characterizes the “strength” of the defects. Parameter Γ is a function of the Poisson ratio and is introduced for convenience. If the precipitate is coherent and has a similar structure to the matrix, differing mainly in composition, then
$$ \Updelta F = F^{P} - F\;{\text{inside}}\;{\text{the}}\;{\text{precipitate}}\; ( {\text{for}}\;R_{St} < R_{0} ) $$
(8a)
and
$$ \Updelta F = 0\;{\text{outside}}\;{\text{of}}\;{\text{the}}\;{\text{precipitate}}\; ( {\text{for}}\;R_{St} > R_{0} ) $$
(8b)
Here, FP is the precipitate structure factor and ΔF is the structure factor change in the region n0v occupied by the precipitate.
Consider crystal with a small concentration of precipitates causing large local distortions in the matrix. Due to the small concentration of precipitates, the average distortions in the crystal still can be considered small, 2W ≤ 1, and the following condition is valid:[1]
$$ \frac{Q\left| C \right|}{{R_{0}^{2} }} = \frac{\Upgamma }{3}\frac{{\left| {\Updelta v} \right|}}{v}R_{0} Q > > 1 $$
(9)
If the precipitate causes large local distortions in the surrounding matrix, then the last term in Eq. [4] related to the coupling between strain and structure factor change needs to be taken into account. Close to reciprocal lattice points (RLPs), the scattered intensity is still mainly determined by the first terms. With the increase of the distance from the RLPs, the last term in Eq. [4] starts to dominate the intensity distribution. That is why the intensity character is distinct in the regions of reciprocal space with small and large q (Figure 1).[1]

3.1.1 Small q

In the immediate vicinity of the RLPs, when
$$ q < < \frac{1}{{\left( {Q\left| C \right|} \right)^{1/2} }} $$
(10)
the intensity is dominated by Huang scattering. When the precipitates’ sizes increase, they cause larger distortion in the matrix and their strength, C, increases. That is why the size of the Huang scattering region around RLPs shrinks with the growth of precipitates according to Eq. [10].

3.1.2 Large q

When q is in the range of
$$ \frac{1}{{\left( {Q\left| C \right|} \right)^{1/2} }} < < q < < \frac{Q\left| C \right|}{{R_{0}^{3} }} $$
(11)
the intensity is mainly dominated by the last term in Eq. [4]. This region of reciprocal space corresponds to the region of the lattice in real space, where distortions are smoothly varying and the Stokes–Wilson approximation can be used.

3.2 Stokes–Wilson Approximation

In regions with smoothly varying distortions, diffuse scattering is mainly due to static displacements rather than the change of structure factors. If the distortion fields from defects vary smoothly, then relatively large regions of the crystal still possess periodic structure. Certainly, there is essential diversity of lattice parameters in different regions of the crystal. The Stokes–Wilson approximation[21] is a useful method considering Bragg scattering by separate regions and describing total scattering intensity and amplitudes as a sum of such partial amplitudes. In this simplest approximation, Bragg scattering by different crystal regions is analyzed as being independent.

In that approximation, around each defect, which has an inversion center so that u(r) = −u(−r), there are only two symmetrically located unit cells S that give the main input into scattering, and the expression for the scattering intensity ID(Q) can be rewritten as[1]
$$ I_{D} ({\mathbf{Q}}) = 16\pi^{3} \frac{{f^{2} }}{{v^{2} }}N_{d} \frac{1}{{\left| {D({\mathbf{r}})} \right|}};\;D({\mathbf{r}}) = - \sum\limits_{i,j} {\left| {\frac{{\partial^{2} \left( {{\mathbf{Qu}}({\mathbf{r}})} \right)}}{{\partial x_{i} \partial x_{j} }}} \right|} $$
(12)
Equation [12] does not take into account interference of radiation scattered from different regions of the crystal. The more accurate expression for diffuse intensity in this case is
$$ I_{D} ({\mathbf{Q}}) = 16\pi^{3} N_{d} \frac{{f^{2} }}{{v^{2} }}\frac{1}{{\left| {D({\mathbf{r}})} \right|}}\left\{ {1 - \sin \left[ {2{\mathbf{qr}} + 2{\mathbf{Qu}}({\mathbf{r}})} \right]} \right\} $$
(13)
The second oscillation-related term in brackets in Eq. [13] dominates the interference. The interference of scattered intensity from these regions causes oscillations of the term I1q4. To observe oscillations experimentally, the defect strength C and the static DWF exponent 2W should not be large (when 2W > 1, these oscillations wash out due to the superposition of the displacement fields from different defects). For this reason, such kind of oscillations may wash out for reflections with higher Miller indices due to an increase of 2W. In our experiments (Section IV), these oscillations were observed for (200) reflections and they were practically washed out for (600) and (800) reflections due to the increase of 2W.

In this case, intensity contour maps do not peak at RLPs, but become stretched along the diffraction vector G. This is observed experimentally due to precipitation in Ni-Al-Si alloys, as described in Section IV.

3.3 Shape Function of the Precipitate

Symmetry of precipitates, dislocations loops, and clusters is often lower than the matrix symmetry. Hence, there are several possible orientations of the precipitates in the matrix coordinate system. Consider ordered precipitates with an average precipitate size D0 and long-range order parameter η. Assume that the precipitates have sharp but coherent boundaries with the matrix. For simplicity, let the shape of the precipitates (P) be the same.

The following shape function of precipitates can be used (Figure 1):
$$ s_{p} ({\mathbf{r}} - {\mathbf{R}}_{mp} ) = \left\{ \begin{array}{ll} 1 \, ({\text{inside }}P) \hfill \\ 0 \, ({\text{outside }}P) \hfill \\ \end{array} \right. $$
(14)
where Rmp characterizes the position of the mth precipitate center.
Matrix shape function can be defined similarly:
$$ s_{m} ({\mathbf{r}}) = \left\{ \begin{array}{ll} 1 \, ({\text{outside }}P) \hfill \\ 0 \, ({\text{inside }}P) \hfill \\ \end{array} \right. $$
(15)
Fourier transform of the shape function, s(q), can be written:
$$ s({\mathbf{q}}) = \int {s({\mathbf{r}})\exp (i{\mathbf{qr}})d{\mathbf{r}}} $$
(16)
Integration over dk and over dr is performed over the entire space.

3.4 Weakly Distorted Matrix with Precipitates

The exponent of DWF, e−2W, in weakly distorted solids is small, 2W ∼ 1. If the precipitate and matrix have the same structure and are coherent, the difference between their structure factors can be defined as ΔF = Fp − F. Strain-induced changes of the scattered intensity are typically coupled with the changes in structure factor. This is the case for fcc Ni-based alloys with L12 Ni3(Al Si)- based precipitates.

In the vicinity of the matrix RLPs, the diffuse scattering intensity can be written as
$$ I_{D} = \frac{{N_{p} e^{ - 2W} }}{{v^{2} \lambda }}\sum\limits_{\alpha }^{\lambda } {\left| {\Updelta F - F{\mathbf{QA}}_{{{\mathbf{q}}\alpha }} } \right|^{2} \left| {s_{\alpha } ({\mathbf{q}})} \right|^{2} } $$
(17)
Here, ν is a unit cell volume, λ is the number of different possible orientations of the precipitate in the matrix, Np is the number of precipitates in the probed volume, and Aq depends on the Fourier transform of the precipitate-induced strain in the matrix. For cubic precipitates in the cubic matrix,
$$ {\mathbf{A}}_{{{\mathbf{q}}X}} = {{3L\left( {C_{11} + 2C_{12} } \right)\left( {1 + \xi n_{Y} } \right)\left( {1 + \xi n_{Z}^{2} } \right)} \mathord{\left/ {\vphantom {{3L\left( {C_{11} + 2C_{12} } \right)\left( {1 + \xi n_{Y} } \right)\left( {1 + \xi n_{Z}^{2} } \right)} {3Dq}}} \right. \kern-\nulldelimiterspace} {3Dq}} $$
(18)
Here, ij is a self-deformation tensor of the precipitate; D is a function of the matrix anisotropy parameter, ξ, elastic moduli tensor components, C11C12C44; and n = q/q is a direction in the reciprocal space. Other Aq components can be similarly written.

At larger distances from the RLPs, \( q > > {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {R_{0} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${R_{0} }$}}, \) the diffuse scattering intensity drops off faster, for instance, for spherical precipitates, as \( {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {q^{4} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${q^{4} }$}}. \) The change of the q dependence can be used to estimate the size of the coherent precipitate. If the precipitate shape differs from the sphere, the decrease of intensity with q becomes steeper at large distances from the RLPs (q ≥ 1/R0) due to the specific shape function. For example, for precipitates with polyhedron shape, the dependence ID ∼ 1/q6 may take place instead of typical ID ∼ 1/q4 dependence.

3.5 Scattering near the Superlattice Peaks

The Ni3(Al Si) precipitates have an ordered L12 structure and, therefore, produce superlattice peaks. If distortions from precipitates are small and precipitates are not correlated, the diffuse scattering intensity around the superlattice peaks can be written as
$$ I_{D} ({\mathbf{Q}}) = N_{d} \frac{{\left| {F_{P} } \right|^{2} }}{{v^{2} }}\sum\limits_{G'} {\sum\limits_{\alpha = 1}^{\lambda } {N_{p\alpha } } \left| {s_{\alpha } \left( {{\mathbf{Q}} - {\mathbf{G}}'} \right)} \right|}^{2} $$
(19)
where N is a number of α-type precipitates in the probed volume, and FP is a structure factor of the superlattice reflection for precipitate. In that case, ID(Q) peaks at Q = G′ (Q is the diffraction vector and G′ is a reciprocal lattice vector of the superlattice site) and depends only on the shape function of the precipitate, \( \left| {s_{\alpha } \left( {\mathbf{q}} \right)} \right|^{2} . \) Equation [19] for ID(Q) describes an intensity distribution around the superlattice peak with full-width at half-maximum (FWHM) ≈ 2π/R0.

3.6 Scattering near the Fundamental Reflections

Similar intensity distributions appear around fundamental reflections. If precipitates have significantly larger elastic moduli than their matrix (as in the case of ordered L12 type Ni3(Al Si) based precipitates), then they usually create strong distortion fields in the surrounding matrix, while in the precipitates themselves, the strain level is low. For precipitates with large distortions,
$$ I_{D} ({\mathbf{Q}}) = N_{d} \frac{{\left| {F_{P} } \right|^{2} }}{{v^{2} }}\sum\limits_{G'} {\sum\limits_{\alpha = 1}^{\lambda } {N_{p\alpha } } \left| {s_{\alpha } \left( {{\mathbf{Q}} - {\mathbf{G}}' - \delta {\mathbf{G}}_{\alpha } } \right)} \right|}^{2} \;\delta G_{i} = - L_{ij} G_{j}^{\prime } $$
(20)
For example, for disc-shape precipitates parallel to (100), lattice parameters within these planes are the same as in the matrix and tetragonal distortions appear in the orthogonal directions. The latter may be described by the following relations:
$$ \delta G_{\alpha x} = - L_{ll}^{0} G_{x} (c_{11} + 2c_{12} )/3c_{11} ,\;\delta G_{\alpha y} = \delta G_{z} = 0 $$
Here, Lll0 is a sum of diagonal components of the defect self-deformation tensor, \( \hat{L}^{0} . \) Disk shape precipitates parallel to (010) and (001) planes result in the appearance of the satellites displaced in the qx and qy directions.
If a precipitate’s shape is close to sphere or cube and structures of the matrix and precipitates are similar, the scattering by the volume of precipitates themselves can be described as
$$ I_{D} ({\mathbf{Q}}) = N_{d} n_{0}^{2} F_{P}^{2} e^{ - 2W} \eta^{2} (q_{P} R_{0} ) $$
(21)
with
$$ {\varvec{q}}_{P} = {\mathbf{Q}} - {\mathbf{G}}_{P} - \frac{1}{3}(1 - \Upgamma )\frac{\Updelta v}{v}{\mathbf{G}}_{P} $$
(22)
and
$$ \eta (q_{P} R_{0} ) = 3\frac{{\sin (q_{P} R_{0} ) - q_{P} R_{0} \cos (q_{P} R_{0} )}}{{(q_{P} R_{0} )^{3} }} = $$
(23)
Here, qp defines the distance from the RLPs of the precipitate under uniaxial compression/tension, and Gp defines the position of the RLPs of the precipitate not deformed by matrix. We note that at early stages of precipitation, the shape and structure of the precipitate may not coincide with its bulk value.

When the matrix is relatively weakly distorted, the DWF related to distortion fields around precipitates is of the order of unity: exp (−2W) ∼ 1. The presence of coherent precipitates causes local lattice distortions due to the lattice mismatch between the matrix and the precipitate, \( \left| {{\mathbf{u}}({\mathbf{r}})} \right|\sim C/r^{2} . \)

4 Example: diffuse scattering from stress annealed Ni-Al-Si alloys

An experimental reciprocal space map (RSM) of diffuse intensity around the (010) superlattice reflection together with intensity profiles in qX and qY directions is shown in Figure 2. For noncorrelated precipitates, the precipitate’s size was estimated from the analysis of superlattice reflections with small (100) Miller indices. Intensity profiles were measured for four orders of (h00) reflections (100, 300, 500, 700) (<100> type reflections for both [100] and [010] directions and used to determine the size of coherent precipitates in two orthogonal directions).
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Fig. 2

Logarithmic map of intensity around the superlattice (010) reflection. Normalized intensity profiles along and transverse to the diffraction vector are shown on both sides of the map

Figure 3(a) shows the dependence of the FWHM of the <100> and <010> type superlattice reflections on the order of the reflection. The intensity of the superlattice reflection is described by Eq. [19]. The broadening of the superstructure reflections is insensitive to H and is due to the small size of precipitates. In the analyzed sample, the precipitate’s size was ~92 A along and ~77 A perpendicular to the stress annealing direction. The observed coupling between the stress annealing direction and the precipitate shape can be understood in the framework of the continuum model considered in Reference 40.
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Fig. 3

FWHM dependence on H: (a) the <100> type (open squares) and <010> type (open triangles) superstructure reflections; and (b) the fundamental <020> type reflections (filled squares). Dashed lines serve as guidance for eyes

In contrast to the superstructure reflections, the FWHM of the fundamental reflections increases with H (Figure 3(b)), which is typical for precipitation-induced strain broadening. The intensity around the fundamental reflections follows Eqs. [20] through [23]. The structure factor, F, changes in the region occupied by the precipitate. Distortions of the lattice together with the changes of the structure factor in the volume occupied by the precipitate cause asymmetry of the diffuse scattering distribution around the regular Bragg peak positions.

The intensity profile along the diffraction vector of the fundamental (800) reflection is shown in Figure 4. The asymmetry of the intensity profile is more visible in the log/log scale plot (Figure 5) of the positive and negative intensity branches.
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Fig. 4

Radial scan of intensity through (080) fundamental reflection in relative reciprocal lattice units. Left inset shows the enlarged maximum in 2θ; right inset shows lattice parameters for Ni3(AlSi) determined from superlattice reflections of <010> type (triangles) and for matrix (squares) determined from fundamental <020> type reflections

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Fig. 5

Diffuse intensity in log scale for (1) positive and (2) negative q values along the 0h0 direction through the (080) reflection. Inset shows the term ID × q2 as a function of q

To emphasize the q dependence of the intensity, the diffuse intensity is scaled by q2 (inset in Figure 5) and by q4 (Figure 6). The product ID(q) × q4 is shown in Figure 6 for different q values in reciprocal space for (200) and (600) reflections. The intensity distributions for positive and negative q are symmetric except for the narrow central region occupied by the regular Bragg component. This is in contrast to the diffuse scattering by crystals with vacancy or interstitial clusters observed in the References 7 through 10. At higher q values, oscillations of the diffuse scattering intensity are observed. These oscillations (marked by dashed circles in Figure 6) are in line with the theoretical model proposed by Trinkaus[11] and Krivoglaz[23] and observations by Trinkaus et al.[24] The oscillations become narrower for higher Ghkl (compare Figures 6(a) and (b)). Oscillations are caused by nonvanishing contributions from the cross-terms corresponding to scattering by the pairs of points rk = Rt ± Rst that are symmetrically located around each precipitate (Figure 7). Following Trinkaus,[11,22] Larson and co-workers,[79] and Krivoglaz,[23] Eq. [13] for diffuse scattering can be written as follows:
$$ I_{D} ({\mathbf{Q}}) = N_{d} \frac{{F^{2} }}{{v^{2} }}\frac{Q\left| C \right|}{{q^{4} }}A(\phi )\left\{ {1 + \sin \left[ {(Q\left| C \right|q^{2} )^{1/3} B(\phi )} \right]} \right\} $$
(24)
$$ B(\phi ) = \frac{{3^{1/2} 2^{1/6} \left[ {\left( {8 + \cos^{2} \phi } \right)^{1/2} + 3\cos \phi } \right]^{1/2} }}{{\left[ {\left( {8 + \cos^{2} \phi } \right)^{1/2} + \cos \phi } \right]^{1/6} }} $$
(25)
$$ A(\phi ) = \frac{{4\sqrt 6 \pi^{3} }}{9}\frac{{\left[ {\left( {8 + \cos^{2} \phi } \right)^{1/2} + \cos \phi {\text{sign}}C} \right]^{5/2} }}{{\left( {8 + \cos^{2} \phi } \right)^{1/2} \left[ {\left( {8 + \cos^{2} \phi } \right)^{1/2} + 3\cos \phi {\text{sign}}C} \right]^{1/2} }} $$
(26)
Here, ϕ is an angle between vectors q and Q (Figure 1(c)). From Eqs. [24] through [26], it follows that in the region of large q,ID(Q) ∼ Q|C|,  in contrast to the region of small q, where ID(Q) ∼ Q2C2. The last term in Eqs. [24] through [26] describes oscillations of the product IDq4. For positive q value functions, B(φ) = 3.78, while for negative q values, B(φ) = 0. That is why IDq4 oscillations should be observed only for the positive q branch, as observed experimentally (Figure 6). Experimental IDq4 distributions are shown for (200) and (600) reflections at Figures 6(a) and (b). Pronounced oscillations of the IDq4 product are observed at the positive q side for the (200) reflection, while they practically wash out for (600). Positive and negative branches of the IDq4 product for (200) reflections are shown together in Figure 6(c) for comparison. A theoretical analysis of the ID(Q)intensity (Eq. [24]) predicts that with the parameter C or with higher Miller indices of the reflection, the frequency of oscillations increases. That is why such oscillations should be most distinct for reflections with smaller H and are washed out at higher H values, as is experimentally observed in our results (compare IDq4 distributions for (200) and (600) reflections in Figures 6(a) and (b)). There is a narrow interval of the 2W ~ 1 values when such oscillations can be observed. It should be noted that the sign change of the misfit between the matrix and the precipitate reverses this dependence, and oscillations will appear on the negative side of q instead of the positive side.
https://static-content.springer.com/image/art%3A10.1007%2Fs11661-011-0937-z/MediaObjects/11661_2011_937_Fig6_HTML.gif
Fig. 6

(a) and (b) The ID × q4 term for (200) and (600) reflections. (c) Comparison between the positive (open squares) and negative (open triangles) q values along the radial direction for (200) reflection. Dashed circles mark the pronounced oscillations for (200) reflection and washed out oscillations for (600) reflection. (d) Fit of the experimental results with the function described by Eqs. [24] through [26]. Symbols—experimental data, and solid line—theoretical calculations

https://static-content.springer.com/image/art%3A10.1007%2Fs11661-011-0937-z/MediaObjects/11661_2011_937_Fig7_HTML.gif
Fig. 7

Sketch of the distorted matrix around the precipitate. Vectors, ± RSt, show symmetrically located scattering cells around the precipitate; Rt and rK define the center of the precipitate and of the scattering cell correspondingly

Equations [24] through [26] were used further for the quantitative analysis of precipitate strength from IDq4 oscillations. Theoretical IDq4 oscillations described by Eq. [25] in the range q = (0.12 to 0.2) are shown in Figure 6(d) by a solid line. From the fit of the experimentally observed IDq4 oscillations with the theoretically calculated oscillations according to Eq. [25], the estimated strength of defect C is equal to 510.

The size and ellipsoidal shape of the precipitates determined from the analysis of the superstructure reflections were used as an input parameters for simulations of RSM around fundamental reflections. With the increase of lattice mismatch and precipitate size (when ΔF is relatively small), the diffuse scattering is dominated by the matrix distortions. The RSM around the matrix RLP becomes stretched in the Q direction, which is observed experimentally for all fundamental reflections (Figure 8). At very small \( q < < {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {R_{0} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${R_{0} }$}}, \) diffuse intensity follows ID = IH ∼ 1/q2 and RSM symmetry depends on the symmetry of the self-deformation tensor of precipitate, Lij0. At large q, it follows that \( I_{D} \approx {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {q^{4} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${q^{4} }$}} \) dependence and RSM shape changes.
https://static-content.springer.com/image/art%3A10.1007%2Fs11661-011-0937-z/MediaObjects/11661_2011_937_Fig8_HTML.gif
Fig. 8

RSMs around (080) and (800) fundamental reflections in the HK0 plane are stretched along the diffraction vector

In the intermediate range of parameters, |ΔF| ∼ FQR0|L0|,  both the structure factor change and precipitate-induced strain influence diffuse scattering. In that case, the RSM shapes become asymmetric and are distinct from the previous two limiting cases. At strong distortions induced by precipitates, the asymmetry sign depends on the tensor Lij0. If Δv < 0,  the intensity is higher at smaller angles, which is observed experimentally (Figure 4). Similar intensity distributions were observed in References 8, 9, and 26. Strain-induced diffuse scattering increases with the value of momentum transfer G. Sometimes this can change the preceding criterions and the character of diffuse scattering with higher G.

Experimental and simulated RSMs of the intensity around the (800) reflection are shown in Figure 9. Simulations were performed for the model of strain fields caused by ellipsoidal coherent precipitates in the disordered Ni-based matrix.
https://static-content.springer.com/image/art%3A10.1007%2Fs11661-011-0937-z/MediaObjects/11661_2011_937_Fig9_HTML.gif
Fig. 9

Simulated (left) and experimental (right) diffuse scattering map around the fundamental (800) reflection

5 Summary

During stress annealing, tensile stress was applied along the [100] crystallographic direction of the Ni-Al-Si sample at a temperature of ~1373 K (~1000 °C). As a result of stress annealing, the precipitates obtained an ellipsoidal shape instead of a spherical shape with the long axes of the ellipsoid aligned along the stress annealing direction. The shape of the coherent precipitates demonstrates a ~15 pct elongation along the stress annealing direction. Diffuse scattering around both fundamental and superstructure reflections for crystals with strongly distorting L12 type ordered Ni3 (Al, Si) precipitates after stress annealing reveals precipitation-induced anisotropic strain in the matrix. Oscillations of IDq4 product are observed at large q near fundamental <h00> type reflections caused by nonvanishing contributions from scattering by the symmetrically located pairs of points rk = Rt ± Rst around each precipitate. The strength of the precipitates is estimated by fitting the experimentally observed IDq4 oscillations with the theoretically calculated ones.

Footnotes
1

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Acknowledgments

Research supported by the Materials Sciences and Engineering Division, Office of Basic Energy Sciences, United States Department of Energy. X-ray microbeam measurements were performed at 33-ID at the Advanced Photon Source (APS). The use of the APS was supported by the Scientific Users Facilities Division, Office of Basic Energy Sciences, United States Department of Energy.

Copyright information

© The Minerals, Metals & Materials Society and ASM International (outside the USA) 2011