Caution in building a Burgers circuit for studying secondary dislocations

1. Introduction

Dislocations and steps (ledges) are common features in faceted interfaces between different phases [1], [2], [3], [4], [5], [6], [7]. Identification of the Burgers vectors associated with the dislocations and steps is essential for understanding the interfacial structure and properties. The Burgers circuit method is broadly used for determining the Burgers vectors of the dislocations within a crystal as firstly suggested by Frank [8]. In the so-called FS/RH convention [9], one firstly makes a closed Burgers circuit following a right-handed (RH) turn with the thumb pointing to the direction of a dislocation within a real crystal. A closure failure forms when a second circuit consisting of the same sequence of the lattice vectors is drawn in the ideal reference lattice. The Burgers vector b is defined by the closure failure as a displacement from the finish point to the start point (denoted by FS). The resultant b is strictly defined in the reference lattice, and it is independent of the size and actual path of the Burgers circuit to surround the dislocation. It is also valid and simpler to construct only one Burgers circuit surrounding a dislocation to define the closure failure, with the corresponding closed circuit in the reference crystal being implied and omitted. In this case, any difference of the determined Burgers vector in the simplified method from an ideal vector in the reference lattice is ignored.

It is reasonable to extend the application of this method for investigating the dislocations in a small angle grain boundary since the coherent region between the dislocations can be treated as a perfect crystal. This method has also been adopted for studying the Burgers vectors of the dislocations in interphase boundaries in high-resolution transmission electron microscope (HRTEM) images, such as the martensite-austenite interface [4] and the hcp-fcc (hcp: hexagonal close-packed; fcc: faced centered cubic) interface of titanium alloy [10]. In these applications, the structures between the dislocations are also coherent, called the primary preferred state according to Bollmann [11]. In this preferred state, there is a one-to-one lattice matching correspondence or an approximate atomic matching correspondence in three dimensions. In other words, there is continuity of lattices from different sides of the interface in the region between the dislocations. Therefore, the paths of the Burgers circuit can usually follow a sequence of the corresponding vectors in the two lattices. Either lattice of the real crystal can be used as the reference lattice for the Burgers circuit.

However, ambiguous paths of the Burgers circuit crossing an interface may occur to a system consisting of two lattices with significantly different lattice structures. Different Burgers circuits will lead to different closure failures, as shown in a recent study of an interface between Mg₃Sn and Mg matrix [12]. The interfaces in such a system usually contain secondary dislocations, whose Burgers vectors are not necessarily a lattice translation vector of either lattice. The structure between the secondary dislocations is in a secondary preferred state, also defined by Bollmann [11]. In contrast to the primary preferred state, the lattices from different sides of the interface do not hold a one-to-one lattice matching correspondence. To maintain a local low energy structure, the structure in a secondary preferred state tends to form a coincidence site (CS) coherent structure. The periodicity of the CS-coherent structure in a secondary preferred state can be represented by a coincidence site lattice (CSL). According to Bollmann, the Burgers vector of the secondary dislocations in the corresponding interface must be a lattice vector of the displacement shift completely lattice (DSCL) associated with the CSL [11]. Construction of the CSL/DSCL model for an interface is usually guided by the observations. Let the two lattices in a given system to interpenetrate according to the measured orientation relationship, with a point from each lattice being coincided at the origin. One can then find a certain pattern of good matching sites (GMSs). A periodic distribution of the GMSs near the origin defines an approximate CSL. Based on this approximate CSL, a constrained CSL can form by adding a small distortion to either lattice. Associated with this CSL, there is a DSCL with the following property [11]. The structure of the CSL is conserved after a translation between the two crystal lattices, only if the translation is defined by a lattice vector in the DSCL. Therefore, for areas between the secondary dislocations to have an equivalent structure of a secondary preferred state, the Burgers vector of the secondary dislocations must be a lattice vector of the DSCL. The DSCL can be calculated according to the reciprocal theorem between CSL and DSCL [13]. The lattice vectors in a DSCL is usually smaller than a lattice vector in a crystal. Therefore, the vector units for the Burgers circuit for analyzing a secondary dislocation are usually different from that for a dislocation within a crystal or in the primary preferred state. Olson and Cohen [14] have incorporated the CSL/DSCL concept with the Burgers circuit to identify the Burgers vector of twin dislocations. For a heterophase system, a CSL/DSCL can be built based on either lattice to represent the secondary preferred state. The two CSLs/DSCLs based on both lattices form the matching correspondence. The unique correspondence between the vectors of the DSCLs is crucial for guiding the paths of a Burgers circuit as it crosses the interface. The Burgers vector of a secondary dislocation can be specified by such a defined Burgers circuit, as applied to illustrate the secondary dislocations in the habit plane between cementite and austenite [15] and in a facet between ε'-Mg₅₄Ag₁₇ precipitates and Mg matrix [5].

It is rather common to apply the convention of a Burgers circuit for an interface in the primary preferred state for investigating an interface in a secondary preferred state. This paper aims to draw attention to the multiple results of Burgers vectors caused by arbitrary selections of the Burgers circuit and to show the use of the CSL/DSCL concept for guiding the construction of a proper Burgers circuit. As the example, different Burgers circuits will be constructed to analyze the dislocations in the habit plane (HP) between δ (Ni₃Nb) precipitate and matrix in an Inconel 718 alloy. This system has been investigated experimentally by various researchers [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26]. The existence of steps and dislocations have been confirmed experimentally by the contrast analysis and HRTEM study [16], [17], [18], [19], [20],23,25], but the agreement has not been reached about the Burgers vectors associated with these defects. One reason is that the Burgers vectors of the dislocations in the habit plane are not the lattice translation vectors in either lattice. A CSL/DSCL model joined with a Δg and O-lattice calculation was recently applied to this system [27]. The results of the joined CSL/DSCL model agree with the measured results in the orientation of the habit plane and the spacing of the observed dislocation array in the habit plane. However, the calculated Burgers vectors for the dislocations and the one-to-one association of the dislocations with steps do not agree with the results based on the HRTEM images provided by Liang and Reynolds [20]. In this study, the experimental HRTEM images will be carefully reexamined. The results of joined CSL/DSCL model will be adopted for guiding the paths of Burgers circuit in different sides of the interface, and the determined Burgers vector will be tested with the experimental results. An advantage of using this system as an example is that the expected Burgers vector lies in the plane of HRTEM image, allowing a direct comparison without a projection effect. For clarity, the classical method with two Burgers circuits will be used, i.e., one surrounding a defect in the interface, and the other in the reference lattice.

2. Methods

The results of the recent investigation with the joined CSL/DSCL model [27] are briefly reviewed below as a basis for further analysis. There are two calculated orientations of HPs between δ (Ni₃Nb) and the matrix (γ) phases in an Inconel 718 alloy, namely, HP_I (1.15 1.15 1)_γ and HP_II (1 1 1.31)_γ. Both HPs contain periodic steps with the same terrace plane parallel to (0 1 0)_δ/(1 1 1)_γ, but they are arranged in different inclinations. With a slight strain to the matrix lattice, the constrained 2D CSL/DSCL in the plane parallel to the terrace ((0 1 0)_δ/(1 1 1)_γ) is given in Fig. 1(a) and one in the plane normal to the step direction ([1- 10]_γ//[1 0 0]_δ) is given in Fig. 1(b). Here, the reference lattice is δ and the values of strain are shown in the figure. One can see the good matching in the terrace plane between each lattice point in δ and a corresponding point in γ, indicating the secondary preferred state in the terrace plane. The grids in these two figures are the DSCL in the corresponding plane. As seen from the figures, the horizontal DSCL vectors in Fig. 1(b) is much smaller than that in Fig. 1(a). This means that the Burgers vector of a secondary dislocation associated with the step is considerably smaller than the available Burgers vectors associated with coplanar dislocations. This explains why the interfacial steps are preferred in this system since the dislocations with smaller Burgers vector are usually favored energetically. Fig. 1(c) and (d) shows respectively a down step in HP_I and an up step in HP_II in the edge-on orientation along the step direction (defined by the intersection between the terrace plane and the step riser) [1 - 10]_γ(//[1 0 0]_δ). Note that the step height of a down step corresponds to one layer of (0 1 0)δ, but that of an up step corresponds to two layers of (0 1 0)δ. According to the misfit analysis based on the joined CSL/DSCL model [27], the steps of both types are associated with a secondary dislocation with identical Burgers vector of 1/6[11-2]_γ/1/3[0 0 -1]_δ. The two sets of parallel defects in the habit plane observed by Liang and Reynolds [20] possibly correspond to a mixture of these two types of steps with the majority being the down steps of the smaller in height. While the calculated dislocation direction and spacing are in a reasonable agreement with the observation, the step feature and the Burgers vector was not supported by the previous conclusion based on the HRTEM images [27].

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Fig. 1.   Constrained CSL/DSCL and edge-on view of two types of HP_I/HP_II calculated by a misfit analysis method [27]: (a) constrained CSL/DSCL on terrace plane; (b) constrained CSL/DSCL on plane normal to the step direction; (c) down step in HP_I surrounded by Burgers circuits; (d) up step in HP_II surrounded by Burgers circuits.

Similar to the 2D DSCL in Fig. 1(b), the fine grids of DSCLs were drawn in Fig. 1(c) and (d) as the reference for the construction of the Burgers circuits, since these two figures are in the same orientation as Fig. 1(b). In contrast to the DSCL in Fig. 1(b) which must conform to the lattice points of both lattices under a small constraint, the two DSCLs separated by the interface conform to the lattice points in different sides of the interface in Fig. 1(c) and (d). Therefore, the secondary misfit between the two DSCLs (and CSLs) in the terrace plane is clearly seen in Fig. 1(c) and (d). The secondary misfit can be canceled by the periodic secondary dislocations associated with the steps of either type with the same projected dislocation spacing on the terrace plane. The Burgers vector of the dislocation was determined with the RH/FS convention based on Burgers circuits in Fig. 1(c) and (d). The same Burgers vector, as enlarged in the insert, is obtained in the reference lattice selected as the γ phase, plotted in the up portion of the same figure.

3. Results and discussions

The determined Burgers vector of the secondary dislocations associated with both steps in Fig. 1(c) and (d) is 1/6[1 1 -2]_γ/1/3[0 0 -1]_δ. This result agrees fully with the result of the previous study [27]. To test the above result against the experimental result, the HRTEM images viewed from [1 - 10]_γ//[1 0 0]_δ were copied from Ref. [20] in Fig. 2. The projected atomic columns (assuming as the bright dots) in different phases were identified carefully according to the different stacking orders of planes in the two phases, i.e., ABCABCABC of (1 1 1)_γ planes in γ phase and ABABABAB of (0 2 0)_δ planes in δ phase. Corresponding rigid lattices are overlapped with the HRTEM images in Fig. 2 to illustrate the identification. Due to local distortion near the dislocations and certain degree of blur in the original image, some plotted points do not coincide with bright dot in the images. However, these local inconsistent does not affect the overall identification result of the interface position. The position of the interface, especially the existence or the location of a step in each HRTEM image, based on the present identification is different from the previous description of the same HRTEM images. Specifically, a step was identified in a location different from the one previously specified in Fig. 2(a) [20], leading to a different conclusion about the association of a dislocation with a step. The height of the newly identified step agrees with that of a down step in HP_I, with a single lattice layer of (0 1 0)δ (double atomic layers). The DSCLs in both lattices and Burgers circuit similar to these in Fig. 1(c) were added to the HRTEM in Fig. 2(a). The determined Burgers vector is 1/6[1 1 -2]_γ/1/3[0 0 -1]_δ, in agreement with result from Fig. 1(c) and the previously calculated results [27].

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Fig. 2.   Experimental HRTEM image Figs. 10(a) and 6 in Ref. [20] overlaid with identified atomic points: (a) based on DSCL to construct the Burgers circuits on Fig.10(a); (b) based on atomic points to construct the Burgers circuits on Fig. 10(a); (c) identified step on Fig. 6. Reprinted with permission from Springer (https://link.springer.com).

However, the above conclusion from the image in Fig. 2(a) is different from that given in the original work [20], in which no closure failure associated with a step was found. The present Burgers circuit differs from the previous one in two aspects: the location enclosed by the Burgers circuit (the original Burgers circuit is better seen at the left part of Fig. 2(b)) and the corresponding paths in the Burgers circuits. If the same small local interface as the one previously selected were used, the present method also could not yield a close failure, since no dislocation exists at the selected local interface. Let us see what would be the closure failure if the previous Burgers circuit were used to surround a dislocation. An equivalent Burgers circuit was drawn in Fig. 2(b) in a similar way as the one in Ref. [20]. The new circuit is now large enough to enclose the down step associated with a secondary dislocation. As seen from the enlarged inset in Fig. 2(b), the determined closure failure is 1/4[11 -2]_γ/1/2[00 -1]_δ, larger than the one measured with the Burgers circuit in Fig. 2(a). Such a Burgers vector was detected from another HRTEM image [20], which is copied in Fig. 2(c). In contrast to a flat interface marked in the original figure, a careful reexamining of this image also reveals a down step associated with a secondary dislocation, as marked in Fig. 2(c). Since the correspondences of Burgers circuits in Fig. 2(b) and (c) are equivalent, the same closure failure of 1/[11- 2]_γ/1/2[00 -1]_δ was obtained for the enclosed dislocation. It is different from the Burgers vector of 1/6[11 -2]_γ/1/3[00 -1]_δ suggested for the major M1 defects by Liang and Reynolds [20]. To explain the disagreement between the measured Burgers vector and the <112>/6 type dislocations expected in the interface, they regarded the closure failure as a sum of the projection of two different types <112>/6 (M1 and M2) dislocations, but the cores of two dislocations were not identified.

The Burgers circuits in Fig. 2(b) and (c) agree with the common practice of constructing a Burgers circuit based on an HRTEM image, i.e., the paths of circuit follow the traces of low index planes, shown as the rows of dense atomic columns. Coherency of an interface is usually examined according to the matching between edge-on planes, and the existence of a dislocation is manifested by an extra plane. While the plane matching analysis method is often appropriate for analyzing dislocations in an interface in the primary preferred state if the correct pair of corresponding planes related by the misfit strain is chosen, the result based on the plane matching or atomic columns in HRTEM images could be misleading for analyzing secondary dislocations in an interface. The step structures in HP_I and HP_II offer a good example to illustrate the possible ambiguity in characterizing the secondary dislocations. This system is special in that each of these HPs is normal to a group of parallel Δgs’ (each Δg = g_γ-g_δ) in reciprocal space [27], where g_γ and g_δ are reciprocal vectors of γ and δ lattices, respectively, if a slight rotation ($\widetilde{0}$.1°) in the ORs is neglected. According to the property of Moiré plane normal to a Δg, the planes corresponding to non-parallel g_γ and g_δ vectors must meet in an edge-to-edge manner in the interface parallel to their Moiré plane which is normal to the corresponding Δg [28]. Specifically, the average orientation of HP_I containing the down steps is normal to Δg_hkl (Δg=g_(002)γ-g₍₀₁₂)_δ) and that of the HP_II containing the up steps is normal to Δg_hkl (Δg=g_(002)γ-g_(022)δ) [27]. Therefore, the planes (0 0 2)_γ and (0 1 2)_δ used by Liang and Reynolds [20] should match each other in HP_II. Though there is no experimental result of an up step, we have plotted the edge-on planes of this pair to compare their matching status in both HPs in Fig. 3(a) and (b). An extra plane of (0 0 2)_γ is associated with the down step in HP_I in Fig. 3(a), in agreement with Fig. 2(b). However, Fig. 3(b) shows no extra plane associated with the up step in HP_II, as expected from the property of moiré plane. Similarly, the pair of edge-on planes (0 0 2)_γ and (0 2 2 )_δ were plotted in Fig. 3(c) and (d) to examine their matching status in both HPs. One sees that now the down step in Fig. 3(c) is not associated with any extra plane of (0 0 2)_γ, but the up step in HP_II in Fig. 3(d) is associated with two extra planes of (0 0 2)_γ. Apparently, it is improper to conclude that the up step or down step is not associated with a dislocation based solely on plane matching in either Fig. 3(b) or (c). Exact plane matching can be better demonstrated only if an HP is not decomposed into terrace and step, as seen in Fig. 3(b) or (c).

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Fig. 3.   Matching between planes (0 0 2)_γ/(0 1 2)_δ and (0 0 2)_γ/(0 2 2)_δ : (a) planes (0 0 2)_γ/(0 1 2)_δ on HP_I with down steps; (b) planes (0 0 2)_γ/(0 1 2)_δ on HP_II with up steps; (c) planes (0 0 2)_γ/(0 2 2)_δ on HP_I with down steps; (d) planes (0 0 2)_γ/(0 2 2)_δ on HP_II with up steps.

As expected, different results of the Burgers vectors were determined using the Burgers circuits following different edge-on planes, i.e., the Burgers vector of 1/4[1 1 -2]_γ/1/2[0 0- 1]_δ from Fig. 3(a), same as that in Fig. 2(b), and 1/2[1 1 -2]_γ/[0 0 -1]_δ as seen in Fig. 3(d). The different closure failures can be elucidated quantitatively by examining the shear displacement between the projected lattice points in correlated planes associated with a step. As shown by the arrow in Fig. 4(a), the shear displacement is approximately 1/12[1 1 -2]_γ/1/6[0 0 -1]_δ between the inclined corresponding vectors [1 1 0]_γ vs. 1/4[0 2 -1]_δ associated with the down step. After adding this part to the Burgers vector of 1/6[1 1 -2]_γ/1/3[0 0 -1]_δ of the secondary dislocation associated with a step, one obtains the total Burgers vector of 1/4[1 1 -2]_γ/1/2[0 0 -1]_δ, consistent with the closure failure in Figs. 2(b) and 3 (a). The shear displacement associated with the up step of two layers becomes -1/6[1 1 -2]_γ/-1/3[0 0 -1]_δ, which has the same value but different direction of the Burgers vector of the secondary dislocation associated with the up step. One would expect the misfit accumulated over the step spacing in the terrace plane to be canceled by this shear displacement, leaving exact matching in Fig. 3(b). Similarly, if the corresponding pair is selected between planes (0 0 2)_γ and (0 2 2 )_δ as shown in Fig. 4(b), a down step is associated with a shear displacement of approximately -1/6[1 1 -2]_γ/-1/3[0 0 - 1]_δ, which can exactly cancel the misfit accumulated over the step spacing in the terrace plane. Therefore, the corresponding plane fully matches in HP_I in Fig. 3(c). On the other hand, the displacement associated with a up step is 2/6[1 1 -2]_γ/2/3[0 0 -1]_δ, so the total closure failure after adding the Burgers vector of 1/6[1 1 -2]_γ/1/3[0 0 -1]_δ is 1/2[1 1 -2]_γ/[0 0 -1]_δ, which is the close failure in Fig. 3(d).

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Fig. 4.   Shear displacement vectors associated with down and up steps under different lattice correspondences: (a) Burgers circuit with paths of (0 0 2)_γ/(0 1 2)_δ; (b) Burgers circuit with paths of (0 0 2)_γ/(0 2 2)_δ.

While the multiple closure failures are generated due to different selections of the Burgers circuit paths, the physical existence of and the Burgers vector of a secondary dislocation should be unique and independent of the selection. Therefore, the construction of a CSL/DSC model to guide the selection of the Burgers vectors of secondary dislocations is strongly suggested. The Burgers circuit based on the correlated DSCLs yields the Burgers vector of 1/6[1 1 -2]_γ/1/3[0 0 -1]_δ for both type of steps, as shown in Fig. 1. It is consistent with the conclusion of the Burgers vector for the major M1 defects identified by Liang and Reynolds [20]. The Burgers vector of 1/6[1 1 -2]_γ also agrees with the experimental results in similar systems. A careful analysis based on the HRTEM image of a δ/γ HP concluded a displacement of <112>/6 associated with an interfacial step [19]. The dislocation with the Burgers vector of 1/6<112>_γ and its association with a step also agrees with the result of the HP between η and γ in ALLVAC 718Plus, where the structure of η is very similar to that of δ, except for difference in the composition and stacking sequence of the close-packed planes [23]. While it is often correct to regard the planes of similar spacing and inclination to be related by the matching correspondence in the primary preferred state for the construction of a Burgers circuit and the Burgers vector must connect two lattice points, it may not be so for an interface in a secondary preferred state. The Burgers vectors of a secondary dislocation, as a fraction of a lattice translation vector in either phase, may not always conform to a vector connecting nearby atomic columns within a phase viewed from an HRTEM image. The present case is such an example, as seen in the insert in Fig. 2(a). Thus, caution must be taken for selection of proper vector units in a Burgers circuit for analyzing the Burgers vectors of interfacial dislocations. A correct preferred state must be specified. In a secondary preferred system, a CSL/DSCL model should be constructed to provide the basis for defining the corresponding vector units in the Burgers circuit, as demonstrated in Fig. 1. The Burgers vector candidates based on the DSCL are also essential for a g⋅b contrast analysis for determination of Burgers vectors of the interfacial dislocations.

4. Conclusion

Caution must be taken in the selection of corresponding vectors, or edge-on planes, in constructing the Burgers circuits for the determination of the Burgers vector of a secondary dislocation in an interface. While the correspondence is often unambiguous in a system in the primary preferred state, the correct correspondence in a system in the secondary preferred state should be guided by a proper CSL/DCSL model for the interface. The HPs between γ/δ in a 718 alloy is given as an example to illustrate the different closure failures due to the Burgers circuits of different correspondences. The difference is explained in terms of plane matching geometry and the displacement between corresponding vectors associated with a step. The HRTEM images in a previous investigation were reexamined and the existence of down steps was confirmed. The Burgers vector of the dislocation associated with the step is 1/6[1 1 -2]_γ/1/3[0 0 -1]_δ, measured with the proper Burgers circuits. It is consistent with the calculated result based on the joined CSL/DSCL analysis and with the observed major defects in the habit plane.

Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (No. 51671111 and No. 51871131) and the National Key Research and Development Program of China (No. 2016YFB0701304). The authors wish to express thanks to Prof. Zhang-Zhi Shi for his helpful suggestions and discussions.

The authors have declared that no competing interests exist.