# Stress Annealing Induced Diffuse Scattering from Ni_{3} (Al, Si) Precipitates

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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

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## Abstract

Diffuse scattering caused by L_{1}2 type Ni_{3} (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 Ni_{3} (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.[1–39] 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 *I*_{D} increases and the intensity of the Bragg component *I*_{0} 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.

*W*, 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]

*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,

**u**

_{ss’t}is the difference between displacements in two scattering cells

*s*and

*s′*caused by the defect located in position

*t*, and

*φ*

_{st}and

*φ*

_{s’t}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 2

*W*depends mainly of the asymptotic behavior of the displacement field created by the dislocation. With defects there are two possibilities: 2

*W*is either finite at large distances or 2

*W*tends to infinity at large distances. It was shown[1–6] that if the displacements fall off faster, then \( {1 \mathord{\left/ {\vphantom {1 {r^{3/2} }}} \right. \kern-\nulldelimiterspace} {r^{3/2} }}, \) the value 2

*W*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 2

*W*⇒ ∞ and these defects are of the second kind.

_{3}(Al,Sc) precipitates, which correspond to the first kind defects (Figure 1). As described previously, these defects cause diffuse scattering,

*I*

_{D}(

**Q**), that exists together with regular (strong) Bragg reflections,

*I*

_{Bragg}(

**Q**).[1–5] 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 2

*W*<< 1 (weak distortions) and large values 2

*W*> > 1 (strongly distorted solids). When 2

*W*≤ 1, it is possible to use the single defect approximation in the isotropic matrix.[7–10] When 2

*W*> > 1 or 2

*W*∼ 1, the overlapping strain fields from different defects interact and the influence of such overlapping strain fields on scattering should be taken into account.[1–6] Many examples of the diffuse scattering analysis for crystals with defects of the first kind are described in the literature.[1–33] Lately, diffuse scattering was applied to study order-disorder transformations in amorphous, disordered, and complex materials.[34–39] In this study, we focus on the peculiarities of diffuse scattering caused by nanosize Ni

_{3}(Al, Si) precipitates after stress annealing of Ni-Al-Si alloys.

## 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 VORTEX^{1} 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* = 4*π* 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 Ni_{3}(Al, Si)-based *γ′* phase with ordered L_{1}2 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 L_{1}2-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 L_{1}2-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

*I*(

**Q**) of X-ray (or neutron) scattering due to defects can be computed from the expression[1–6,30]

*f*

_{i}is the scattering factor from an individual atom

*i*, with relaxed coordinates

**R**

_{i}=

**R**

_{i}

^{0}+

**u**

_{i}due to the presence of defects. Here,

**u**

_{i}is the displacement from the equilibrium positions,

**R**

_{i}

^{0}, corresponding to the ideal crystal. We calculate the

**u**

_{i}using continuum elastic theory.

**Q**=

**k**

_{2}−

**k**

_{1}is the diffraction vector, with

**k**

_{1}and

**k**

_{2}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:

*N*

_{d}is a total number of

*α*-type defects in the crystal,

*ΔF*

_{α}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

**u**

_{Stα}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[1–3] and expressions [3] and [4] simplified

### 3.1 Precipitates with Strong Distortions

*n*

_{0}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:[1–3]

*n*

_{0}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

**R**_{St}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

*F*

^{P}is the precipitate structure factor and

*ΔF*is the structure factor change in the region

*n*

_{0}

*v*occupied by the precipitate.

*W*≤ 1, and the following condition is valid:[1]

*q*(Figure 1)

*.*[1]

#### 3.1.1 Small q

*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

*q*is in the range of

### 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.

**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

*I*

_{D}(

**Q**) can be rewritten as[1]

*I*

_{1}

*q*

^{4}. To observe oscillations experimentally, the defect strength

*C*and the static DWF exponent 2

*W*should not be large (when 2

*W*> 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 2

*W.*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 2

*W*.

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 *D*_{0} 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.

**R**

_{mp}characterizes the position of the

*m*th precipitate center.

*s*(

**q**), can be written:

*d*

**k**and over

*d*

**r**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, 2*W* ∼ 1. If the precipitate and matrix have the same structure and are coherent, the difference between their structure factors can be defined as *ΔF* = *F*_{p} − *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 L_{1}2 Ni_{3}(Al Si)- based precipitates.

*ν*is a unit cell volume,

*λ*is the number of different possible orientations of the precipitate in the matrix,

*N*

_{p}is the number of precipitates in the probed volume, and

**A**

_{q}depends on the Fourier transform of the precipitate-induced strain in the matrix. For cubic precipitates in the cubic matrix,

*Lδ*

_{ij}is a self-deformation tensor of the precipitate;

*D*is a function of the matrix anisotropy parameter,

*ξ*, elastic moduli tensor components,

*C*

_{11},

*C*

_{12},

*C*

_{44}; and

**n**=

**q**/

*q*is a direction in the reciprocal space. Other

**A**

_{q}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/*R*_{0}) due to the specific shape function. For example, for precipitates with polyhedron shape, the dependence *I*_{D} ∼ 1/*q*^{6} may take place instead of typical *I*_{D} ∼ 1/*q*^{4} dependence.

### 3.5 Scattering near the Superlattice Peaks

_{3}(Al Si) precipitates have an ordered L

_{1}2 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

*N*

_{pα}is a number of

*α*-type precipitates in the probed volume, and

*F*

_{P}is a structure factor of the superlattice reflection for precipitate. In that case,

*I*

_{D}(

**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

*I*

_{D}(

**Q**) describes an intensity distribution around the superlattice peak with full-width at half-maximum (FWHM) ≈ 2

*π*/

*R*

_{0}.

### 3.6 Scattering near the Fundamental Reflections

_{1}2 type Ni

_{3}(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,

*L*

_{ll}

^{0}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

*q*

_{x}and

*q*

_{y}directions.

**q**_{p}defines the distance from the RLPs of the precipitate under uniaxial compression/tension, and

**G**

_{p}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 (−2*W*) ∼ 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

*q*

_{X}and

*q*

_{Y}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 (

*h*00) 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).

*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.

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.

*q*dependence of the intensity, the diffuse intensity is scaled by

*q*

^{2}(inset in Figure 5) and by

*q*

^{4}(Figure 6). The product

*I*

_{D}(

*q*) ×

*q*

^{4}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

**G**

_{hkl}(compare Figures 6(a) and (b)). Oscillations are caused by nonvanishing contributions from the cross-terms corresponding to scattering by the pairs of points

**r**

_{k}=

**R**

_{t}±

**R**

_{st}that are

*symmetrically*located around each precipitate (Figure 7). Following Trinkaus,[11,22] Larson and co-workers,[7–9] and Krivoglaz,[23] Eq. [13] for diffuse scattering can be written as follows:

*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,*

*I*

_{D}(

**Q**) ∼

*Q*|

*C*|, in contrast to the region of small

*q*, where

*I*

_{D}(

**Q**) ∼

*Q*

^{2}

*C*

^{2}. The last term in Eqs. [24] through [26] describes oscillations of the product

*I*

_{D}

*q*

^{4}. For positive

*q*value functions,

*B*(

*φ*) = 3.78, while for negative

*q*values,

*B*(

*φ*) = 0. That is why

*I*

_{D}

*q*

^{4}oscillations should be observed only for the positive

*q*branch, as observed experimentally (Figure 6). Experimental

*I*

_{D}

*q*

^{4}distributions are shown for (200) and (600) reflections at Figures 6(a) and (b). Pronounced oscillations of the

*I*

_{D}

*q*

^{4}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

*I*

_{D}

*q*

^{4}product for (200) reflections are shown together in Figure 6(c) for comparison. A theoretical analysis of the

*I*

_{D}(

**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

*I*

_{D}

*q*

^{4}distributions for (200) and (600) reflections in Figures 6(a) and (b)). There is a narrow interval of the 2

*W*~ 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.

Equations [24] through [26] were used further for the quantitative analysis of precipitate strength from *I*_{D}*q*^{4} oscillations. Theoretical *I*_{D}*q*^{4} 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 *I*_{D}*q*^{4} oscillations with the theoretically calculated oscillations according to Eq. [25], the estimated strength of defect *C* is equal to 510.

*F*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

*I*

_{D}=

*I*

_{H}∼ 1/

*q*

^{2}and RSM symmetry depends on the symmetry of the self-deformation tensor of precipitate,

*L*

_{ij}

^{0}. 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.

In the intermediate range of parameters, |*ΔF*| ∼ *FQR*_{0}|*L*^{0}|, 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 *L*_{ij}^{0}. 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**.

## 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 L_{1}2 type ordered Ni_{3} (Al, Si) precipitates after stress annealing reveals precipitation-induced anisotropic strain in the matrix. Oscillations of *I*_{D}*q*^{4} product are observed at large *q* near fundamental <*h*00> type reflections caused by nonvanishing contributions from scattering by the *symmetrically* located pairs of points **r**_{k} = **R**_{t} ± **R**_{st} around each precipitate. The strength of the precipitates is estimated by fitting the experimentally observed *I*_{D}*q*^{4} oscillations with the theoretically calculated ones.

## 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.