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Transmission electron microscopy characterization of precipitates
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Ultramicroscopy 30 (1989) 102-115 North-Holland, Amsterdam
TRANSMISSION ELECTRON MICROSCOPY CHARACTERIZATION OF PRECIPITATES U. D A H M E N National Center for Electron Microscopy,.Lawrence Berkeley Laboratory, Berkeley, California 94720, USA
Received at Editorial Office 6 December 1988; presented at Symposia August 1988
This paper reviews the methods and principles underlying the structural and morphological characterization of precipitates by TEM. Precipitate contrast mechanisms are reduced to the better-known contrast behavior of crystal defects such as dislocation loops, dipoles and stacking faults. Because of their importance, basic techniques of controlled specimen tilting and the analysis of double diffraction are treated in some detail. The remainder of this paper concentrates on more advanced concepts in precipitate characterization, not available in textbooks. Emphasis is placed on the use of crystal symmetry in practical problems of precipitate analysis, on the relationship between transformation strains of the real and the reciprocal lattice, and on the connection between transformation strain and morphology. These concepts are illustrated with selected examples.
1. Introduction Precipitates in solids are inclusions whose chemical and mechanical properties differ from those of the host matrix. Their characterization is important in understanding the microstructures and properties of materials spanning the range from semiconductors to metals to minerals. Precipitates are responsible for the stars in star sapphire, the strength of age-hardening alloys, and the toughening of zirconia ceramics. Procedures for precipitate analysis depend much on the context and the level of accuracy required. Sometimes it is sufficient to describe a precipitate by its size alone, but often it is necessary to determine the shape as well. Sometimes knowledge of the chemical composition is sufficient, but often the crystal structure is required, too. Electron microscopy allows very detailed characterization of m a n y aspects of precipitates, including composition, crystal structure, shape, orientation relationship with the matrix, distribution, degree of coherency and interface structure (e.g., refs. [1-9]). Compositional analysis is the realm of analytical microscopy and will not be covered here. The
remaining characteristics are structural in nature and are in fact interrelated.
2. Shape In the simplest scheme only three basic shapes (or morphologies) of precipitates exist: spheres, plates and needles. The intermediate morphology of a lath m a y be treated as a long plate or a flat needle. Further characteristics of precipitate shapes, i.e. whether they are facetted or rounded, regular or irregular, symmetrical or asymmetrical, are important but they do not change the mode of analysis which is different for spheres, plates and needles. Spherical precipitates appear identical in any crystal orientation and are characterized b y their size and the sign and magnitude of their strain field (interstitial or vacancy). Plates are the most c o m m o n precipitate morphology. They are characterized by their habit plane and aspect ratio, and by the sign, direction and magnitude of their strain. Needle-, rod- or lath-shaped precipitates in addition are characterized by the direction of their long axis.
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U. Dahmen / TEM characterization ofprecipitates
Precipitates can be made visible by two basic contrast mechanisms, the distortion they cause in the surrounding matrix (strain contrast) and the diffraction from the crystal structure of the precipitate (orientation and structure factor contrast). Precipitate analysis can thus be reduced to basic defect analysis conducted for the matrix and precipitate structure separately. With the first mechanism the precipitate is regarded as a defect. Precipitates may act like strain centers, dislocation loops, dislocation dipoles or displacement faults whose distortions can be probed in a g. b experiment. With the second mechanism, it is possible to form dark field images of the precipitate and, in fact, to conduct an entire g- b analysis of any defects that might be present inside the precipitate. These two mechanisms and other aspects such as symmetry relations between precipitates in a solid matrix will be examined below.
3. Controlled specimen tilting The basis of any precipitate analysis is choosing the right orientation and diffracting condition. Controlled tilting is therefore essential to the success of any systematic contrast experiment. In order to tilt the specimen into the desired Orientation, it is important to determine the location of the goniometer tilt axes relative to the diffraction pattern. In conjunction with Kikuchi lines in thick crystals or spot pattern intensity in thin crystals, this calibration can be used for tilt control. For example, if a pole is visible near the edge of the screen it can be brought to the center easily with the appropriate tilt drives. The position of a pole that is not visible on the screen can often be guessed from the directions of the Kikuchi lines that converge on it or from. the arcs of spot pattern intensity centered on it. Once this calibration is established, the area of interest can be maintained in the field of view during a controlled tilt sequence by tilting either in the image mode, or in the convergent-beam diffraction mode while observing the shadow image in the defocused diffraction disc. The choice of operating mode de-
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pends on how well the precipitate can be recognized in the shadow image.
4. Precipitates as defects 4.1. Small precipitates as strain centers
Small precipitates are most noticeably visible through their strain contrast, i.e. the distortion they cause in the surrounding matrix. The strain field can be mapped in a g - b experiment by bringing one set of crystal planes at a time into strong diffracting condition and thereby probing it for distortion. The contrast behavior under different diffracting conditions will depend on the character of the displacement field which, in turn, is related to the shape of the particle [10,11]. Spherical precipitates in an elastically isotropic matrix cause spherically symmetric strain fields and behave identically for all diffraction vectors [12]. Plate- and needle-shaped precipitates cause more complicated strain fields that give rise to very different contrast behavior for different diffraction conditions. Small (smaller than - ~ g / 3 ) strain centers exhibit oscillating contrast variations with depth in the foil. The best analysis can be performed on anomalous contrast that arises from strain centers located near the foil surfaces whose depth in the foil has been determined by stereomicroscopy. The calculated contrast oscillation of a small vacancy-type strain center (here a pure vacancy loop) with depth in the foil for bright field (BF) and dark field (DF) conditions is illustrated schematically in fig. 1 [13]. The sense of the black and white contrast reverses periodically through the thickness of the foil. The strong (anomalous) black and white contrast at the top and bottom surfaces is equal in DF and opposite in BF. The strongest image is observed at s = 0. For an interstitial strain center the sense of the contrast must be reversed. For particles causing larger strain the contrast oscillations are suppressed due to surface relaxations, and the sense of the black and white contrast will be that closest to the foil surfaces in fig. 1. As first shown by Ashby and Brown [10], the DF image exhibits a unique sense of this contrast.
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U. Dahmen / T E M characterization of precipitates
0.5_~Q
1-Z g
,~_
bottom
Fig. 1. Depth oscillationof black-white contrast from small vacancy-typestrain centers in BF and DF. Anomalous images appear for defects near foil surfaces. Only anomalous DF images have equal black-white direction at top and bottom surface. Interstitial-type strain centers show opposite contrast (after ref. [13]). Thus D F conditions at s - - 0 are required to determine the nature (interstitial or vacancy) of such a center of strain. 4.2. Precipitate plates as dislocation loops
Plate precipitates that are a few tens of nanometers in size (of the order of ~g) may show more extended ring-like loop contrast. This is because to first order the displacement field of a precipitate, like that of a dislocation loop, is caused by a plate of crystal that has been expanded, contracted or sheared. Here the fact that the two-beam BF image of a dislocation is located off to one side of the true position of the dislocation line can be used to determine the nature (interstitial or vacancy) of a dislocation loop or plate precipitate. An inclined plate will show inside/outside contrast for + g , interstitial/vacancy nature and posit i v e / negative habit plane inclination to the beam. To determine the nature of a plate from its inside/outside contrast, it is therefore essential to
know its sense of inclination relative to the g vector [14]. Fig. 2 illustrates a simphfied method [15] that can be used to avoid errors often arising due to the presence (or absence) of an inversion between image and diffraction pattern. Note that this analysis assumes the displacement vector is normal to the habit plane, usually a good approximation for small plate-shaped precipitates. First the plate is tilted edge-on so that the habit plane is parallel to the beam (fig. 2a). This crystal orientation (here [001]) is marked on a Kikuchi map. The sample is then tilted to an orientation where the precipitate is inclined. A two-beam BF condition with s > 0 is set up for a g vector pointing toward the pole (marked on the Kikuchi map) where the habit plane was edge-on. This is arbitrarily defined as + g (fig. 2c). If the plate shows outside contrast in + g , it is interstitial; if it shows inside contrast it is a vacancy-type precipitate. To distinguish between inside and outside contrast, images in both + g and - g should be recorded.
U. Dahmen / TEM characterization of precipitates
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d\
I-g |II
// Fig. 2. Simplified contrast analysis to determine the nature of plate precipitates and dislocation loops [15]. Precipitates marked A and B seen edge-on in a [001] zone (a) show outside contrast in + g (c) and are therefore interstitial in character. (Courtesy Acta Met.)
4.3. Precipitate needles as dislocation dipoles
4. 4. Precipitate plates as stacking faults
Long needle-shaped precipitates with small cross-section exhibit strain contrast very similar to that of a dislocation dipole. The true displacement field will, in fact, be different from that of a dipole, but its major component can often be described by a single displacement vector. A standard g - b analysis can provide the direction and approximate magnitude of this displacement vector and the nature (shear, edge) of the precipitate [16]. To determine the vacancy or interstitial character of a needle or dipole it may be regarded as an elongated loop and the same procedure as described above can be applied.
Under some conditions, uniformly thin plate precipitates give rise to displacement fringe contrast similar to that of a stacking fault. This is seen for a [100] 0' plate in an A1-Cu alloy in fig. 3b. The contrast arises if the matrix is in two-beam diffracting condition and if the precipitate is diffracting weakly and oriented inclined to the electron beam. Under these circumstances the primary contribution of the precipitate to the image c o n trast is that it causes a fractional displacement of the matrix lattice at the top and bottom interface. This displacement, and with it the fringe contrast, depends on the thickness of the plate [17]. Ledges
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U. Dahmen / T E M characterization of precipitates
Fig. 3. Displacement fringe contrast at an inclined plate-shaped 0' precipitate on a (100) plane in AI-Cu (b), When g is paratlel to the (100) habit plane the fringe contrast is absent (a) giving a clear view of interracial dislocations (numbered 1-5).
in the interface will thus lead to abrupt shifts in the fringe pattern, and the magnitude of the shift can be related to the height of the ledge [18]. To avoid fringe contrast it is necessary to set up a diffracting condition for which g . R is an integer. This is most easily accomplished by selecting a g-vector parallel to the plate surface (g . R = 0). As shown in fig. 3a it is then possible to see interface dislocations undisturbed by displacement fringes.
5. Precipitates as homogeneously distorted matrix In two-phase materials, it is often difficult to obtain separate diffraction patterns from the two phases, matrix and precipitate. Frequently, they will be seen as two interpenetrating patterns, and if the structures are similar, the precipitate pattern
may be interpreted as a distorted or transformed matrix pattern. The distortion, or transformation strain, can be read directly from the separation of corresponding diffraction spots (spot splitting) [19]. The transformation strain of the crystal lattices is inverse to the strain obtained from the spot splitting in the two interpenetrating diffraction patterns. This is shown more rigorously as follows [20]. For a crystal lattice described by lattice vectors r, diffraction spots occur only at positions g in reciprocal space for which g . r is an integer. This is the commonly known relationship between real and reciprocal space. It expresses the fact that the Fourier transform of a periodic object is itself periodic. Since g- r = integer, it is easy to see that the reciprocal lattice g must shrink if the crystal lattice r grows. More generally, if the crystal lattice undergoes a homogeneous transformation de-
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U. Dahmen / TEM characterization of precipitates
~
S
"°'....
Fig. 4. Illustration of typical plane strain precipitates with corresponding diffraction patterns and schematic transformation strains: (a) uniaxial strain, (b) simple shear and (c) general invariant plane strain. Diffraction spot splitting is normal to the habit plane n. The strain direction s lies normal to the plane of unsplit spots (marked by lines in diffraction patterns).
scribed by a tensor A, the reciprocal lattice will undergo the inverse transpose of that transformation: ,4-1T. This follows from the fact that ( g . A - 1 X A . r ) = i n t e g e r . Thus a cubic-to-tetragonal transformation in real space would manifest itself in a cubic-to-tetragonal transformation of the reciprocal lattice, but with inverse c / a ratio. This is illustrated with a few examples below. 5.1. Uniaxial strain
A tetragonal precipitate in a cubic matrix provides a simple example for a uniaxial transforma-
tion strain, assuming that the a-axes are identical in the two phases. One would expect a plate-shaped precipitate with a plane of contact (habit plane) on one of the cube faces of the matrix. The plate normal will be the direction of largest mismatch. An example is the formation of 0' precipitates in A1-Cu alloys, illustrated in fig. 4a, where the plates are seen edge-on. As shown in the schematic the transformation A can be described as a uniaxial expansion on the (001) plane in the [001] direction and the reciprocal lattice will undergo a corresponding contraction. Notice that the spot splitting in the diffraction pattern is normal to the
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U. Dahmen / T E M characterization of precipitates
habit plane. This transformation strain is a special invariant plane strain (it leaves the (001) plane invariant), but the result can be generaliTcd to other invariant plane strains.
5.2. Simple shear A twinning shear is a typical example of a simple shear transformation. As seen for a microtwin in fig. 4b, the plane of contact is the twin plane, and the corresponding diffraction pattern shows spot splitting normal to the habit plane. Notice the row of unsplit spots perpendicular to the habit plane. The accompanying schematic illustrates the transformation strain, a shear s on the twin plane n. The corresponding transformation strain in the reciprocal lattice is a simple shear n on the plane s.
5.3. General invariant plane strain In fig. 4c the precipitation of a bee phase in an fcc matrix (Cr in Cu) is treated in this manner. The orientation of the [111] bcc precipitate pattern relative to the [110] fee matrix pattern is such that the spot splitting lies normal to the (irrational) habit plane. Comparison of the diffraction pattern with the schematic of the transformation strain shows once more the inverse relationship of n and s in real and reciprocal space. As might be expected from these examples, all plate-shaped precipitates are related to their matrix by an invariant plane strain. The invariant plane is the habit plane since matrix and precipitate will be joined across the plane of minimum distortion. Assuming that no plastic deformation is involved in procuring this fit, i.e. the transformation is homogeneous, the reciprocal lattice will undergo the inverse transpose of this homogeneous invariant plane strain. It can be shown that this is again an invariant plane strain, but with the role of the invariant plane normal n and the strain vector s reversed. The invariant plane of the reciprocal lattice will be a plane with no spot splitting and its plane normal is s, the strain direction in real space. On the other hand all the distortions in reciprocal space (spot splitting) will be in the direction n, the habit plane normal. These results
are very useful in the analysis of plate-shaped precipitates and all transformations with planar (invariant plane) interfaces: for analysis by TEM it is important to obtain the special orientation containing both s and n in the plane of the diffraction pattern. Although the habit plane n is easy to find from images by trace analysis, the direction and magnitude of s can only be found from the diffraction pattern.
5.4. Invariant line strain Needle-shaped precipitates tend to be related to the matrix by an invariant line strain with the needle axis lying along the undistorted, or invariant, line [21,22]. (Note that the homogeneous component of the total transformation strain for many plato-shaped precipitates, including most martensite plates, also is an invariant line strain.) Again, it can be shown that an inverse relationship exists between real and reciprocal space transformation. The two characteristic vectors, the invariant line i and the invariant normal h, reverse roles in reciprocal space, i.e. the diffraction pattern contains a direction of no spot splitting h while the pattern seen when the beam is along the invariant line direction i contains all the spot splitting and thus allows the most accurate reading of the orientation relationship.
5.5. Double diffraction A homogeneous transformation does not lead to additional or forbidden reflections in the reciprocal lattice. If, apart from the homogeneous distortion, new reflections or absences are observed, the lattice has undergone ordering, shuffling or modulation. These lattice rearrangements are the subject of crystal structure analysis and are most elegantly solved by convergent-beam electron diffraction, a technique that lies outside of the scope of this presentation. However, a common source of extra reflections in second-phase analysis, unrelated to the crystal structure, is double diffraction. The simplest diffraction pattern is that from a single crystal, but often more than one crystal, for example matrix and precipitate, contribute to the
U. Dahmen / TEM characterization of precipitates
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Fig. 5. Diffraction patterns of fcc matrix with bcc precipitate: (a) without double diffraction and (b) with double diffraction due to overlapping of matrix and precipitate along the beam direction; (a) is the addition and (19) the convolution of the two patterns.
pattern, as shown for a bee precipitate in an fee matrix in fig. 5. If they originate in different regions the two patterns will simply add (fig. 5a), but if one is embedded in the other, i.e. they overlap along the beam path, extra spots may appear due to double diffraction. Each diffracted spot of the first phase may then act as an incident beam for the second phase, leading to the convolution of the two patterns (fig. 5b). In the absence of double diffraction the precipitate pattern is easily separated from the matrix pattern, and even with pronounced double diffraction it is usually easy to locate and identify the matrix pattern unambiguously. However, the problem of locating the primary precipitate pattern from the maze of non-matrix reflections and separating it from the doubly diffracted spots is sometimes difficult. Since the process of double diffraction is the convolution of the matrix and precipitate patterns it could, in principle, .be deconvoluted simply with a transparent overlay to separate the total pattern into a number of parallel-displaced precipitate (or matrix) patterns. Unfortunately, this deconvolution is not unique because it defines the precipitate diffraction vectors only modulo the matrix vectors. Thus from this procedure alone it is impossible to tell whether to
assign go as a primary precipitate diffraction vector or whether (g + gmatrix) is the correct assignment. The safest and by far the easiest way to make this distinction is by tilting off the zone axis to eliminate double diffraction paths. However, often the problem of double diffraction is not apparent until a spot pattern is examined on the negative. It is then necessary to use further information hidden in the intensity distribution and shape of the extra spots. The intensity of a doubly diffracted spot is proportional to the intensity of the matrix reflection that acts as its origin. Since the transmitted beam has the highest intensity, the primary precipitate pattern should be the strongest of the extra reflections. The first assignment of possible primary spots is therefore always to the most intense extra reflections in the pattern. This assit,nment must then be checked by parallel-translating these diffraction vectors, with the help of a transparent overlay, to see if in fact the precipitate reflections appear as if each matrix reflection acted as a transmitted beam. A further hint is sometimes given by the shape of a precipitate spot, such as the shape of an arc due to an angular spread of several similar surface precipitates contributing to the same diffraction
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U. Dahmen / TEM characterization of precipitates
pattern. Such an arc must always be centered on its origin, i.e. on the spot in the diffraction pattern that acts as its transmitted beam. This often allows a quick elimination o f ambiguities.
6. Precipitates as sin~e crystals The view of a precipitate as a single crystal separate from the matrix and its distortion focuses
Fig. 6. Example of a diffraction condition used in analysis of defects within a precipitate. As seen in the BF (a) the matrix is diffracting weakly while the particle is in a strong two-beam condition, used to form the DF image in (b).
U. Dahmen / TEAt characterization of precipitates
attention on the defects inside the particle. One way to analyze a precipitate in this manner is to separate it physically from its matrix by extraction. A contrast analysis can then be performed easily by setting up a series of two-beam diffracting conditions without interference from the matrix crystal. The same kind of experiment can be performed with somewhat greater difficulty if the precipitate is contained in the matrix (e.g., ref. [23]). In order to eliminate matrix contrast, all such images must be recorded in the DF mode. It is best to set up simultaneously conditions of strong diffraction in the particle and weak diffraction in the matrix, illustrated for a 0' precipitate in A1-Cu in fig. 6. This limits the number of orientations where clean interpretable images can be obtained. If precipitates are small and show only very weak diffraction spots it is possible to use Kikuchi bands of the matrix as a navigational aid for obtaining the desired diffracting conditions.
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precipitate are related by a point symmetry operation of the matrix [25-27]. This is readily apparent from the hypothetical sequence of operations shown in fig. 7: A precipitate is embedded in a matrix crystal (a). A spherical region including the precipitate is cut from the matrix (b), removed (c), rotated, inverted or mirrored by a matrix symmetry operation (d) (here a 90 o rotation), replaced in the matrix (e) and rewelded (f). The matrix crystal is undisturbed by this sequence because by definition it is invariant under a symmetry operation. However, a new variant of the precipitate has been generated. The number of possible variants is therefore equal to the number of symmetry elements in the point group of the matrix. If, however, the precipitate shares one of the symmetry elements of the matrix, for example a 180 ° rotation in fig. 7, then this element will not produce a new variant. Thus only those symmetry elements of the matrix that are not shared by the precipitate generate new variants. 7.1. Determination of needle axis
7. The use of synunetry
The shape, orientation and distribution of precipitates all bear inherent synunetry properties that can be exploited to facilitate and improve the accuracy of a morphological analysis b y TEM [24]. Ignoring translations, any two variants of a 8
Needle- or lath-shaped precipitates are characterized by their long axis [hkl], conventionally determined by trace analysis. A general axis direction
L \
",L \
)li f
/" I
Ill
L~\ /
/
/
in Fig. 7. Schematicillustrationof variant-generatingmatrix symmetryoperation[24].
U. Dahmen / TEM characterization of precipitates
112
hkl
110
hhl
111
hkO
100
Fig. 8. (001) Stereograms illustrating multiplicity of needle precipitates in a cubic crystal [24].
ments, and enumerated by the permutations of the three indices hkl and their negatives. This is most graphically illustrated in a stereogram, for example an f001) orientation as shown for the general direction in fig. 8. Twenty-four different variants of (hkl) needles are possible if the needle-morphology shares only the inversion with the matrix. If the needles lie on a {110} mirror plane, i.e. their indices are of the type fhhl), this mirror is a shared symmetry element and only 12 different variants exist. The remaining stereograms show other special directions in order of increasing symmetry. Each of the variant-generating symmetry operations may be thought of as a specimen tilt in a trace analysis. Needle axes can therefore be measured directly from a single micrograph in an f001) orientation: It is clear that simple (100) needles will be immediately apparent (stereogram f). However, even in the most complicated case of needles along a general f h k l ) direction (stereogram a), measuring three characteristic angles (a, fl, "t) enclosed between precipitate variants will yield their direction easily through the relations h 2 / l 2 -- (1 - c o s a ) / ( 1 + cos a),
h 2 / k 2 -- (1 - c o s / 5 ) / ( 1 + cos/5), k 2 / l 2 ~ (1 - cos V)/(1 + cos y).
Two angles are sufficient to determine the indices hkl but the third may be used to check the accuracy of the measurement. More details are available in the literature [24]. It should be pointed out that conventional trace analysis yields greater accuracy, while the analysis based on symmetry provides better statistics as well as additional information on the distribution of precipitate variants. 7.2 Increased accuracy of needle or plate habit If the crystallographic orientation of a needle or plate precipitate is known approximately, symmetry properties can be used to determine their orientation with greater precision. For example, it was found that needle precipitates of Cr in a Cu matrix lie on {111} planes [28]. To find their precise orientation in the {111} plane, the angle between two precipitate variants viewed along a (111) zone axis, shown in figs. 9a-9d, can be measured. This angle is twice the deviation of the needle direction from a {110} mirror plane. The same method can be used to improve the precision of a habit plane determination [29]. For example, large ~,' plates in F e - N alloys form in a characteristic butterfly-like arrangement. The two wings are two individual plates joined together at an acute angle. Each plate has two sets of stria-
U. Dahmen / TEM characterization ofprecipitates
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Fig. 9. (a)-(d) Two variants of lath-shaped Cr precipitates in a single grain of Cu matrix, illustrating horizontal mirror symmetry between particles and between corresponding diffraction patterns; (e) and (f) mirror symmetry between two variants of "t' plates in Fe-N alloy.
tions typical of two variants o f an invariant line strain. Since b o t h variants clearly have the same habit plane and since a n y two variants are related b y a s y m m e t r y operation of the matrix, it follows that the habit plane must have a s y m m e t r y ele-
m e n t in c o m m o n with the matrix, i.e. it m u s t be parallel or perpendicular to a mirror plane or r o t a t i o n axis. I n the present case it was f o u n d that the two variants were related b y an {001} mirror, a n d the habit plane m u s t be of the type { hkO}. In
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U. Dahmen / T E M characterization of precipitates
Fig. 10. High-resolutlonmierograph of peatagonally-twinaodC~ pr¢~'ipitate in A1 matrix with beam direction along (1i0) Ge and (I00) A1(CourtesyScience[301.) order to find the exact orientation, i.e. the ratio of h to k, the angle between two { hkO} plates that form the wings of the butterfly, when seen edge-on in an (001) foil orientation, can be measured accurately. As seen in figs. 9e and 9f, the angle of 42 ° is bisected by a (110) mirror, and the habit plane is (940} which may be compared with the (210] habit plane from less accurate measurements.
7.3. Symmetry of morphology and orientation relationship Symmetry is also apparent in the shape of precipitates. For example, the needle-shaped Ge
precipitates formed during aging of AI-Ge alloys exhibit a variety of shapes when viewed along their needle axis. High resolution micrographs show that the symmetry of the shape is usually that of the orientation relationship between the two crystals, i.e. the symmetry they have in common. Faulting inside the precipitates destroys this symmetry, a fact that is reflected in irregular shapes of heavily faulted particles. Sometimes, however, even the faults are arranged so as to be compatible with the symmetry of the orientation relationship. This leads to more regular morphologies. An example is seen in the fivefold twinning of a Ge particle (fig. 10) where the orientation relationship, the substructure and the shape all conform to mm2 orthorhombic symmetry [30].
U. Dahmen / T E M characterization of precipitates
8. Summary The major contrast mechanisms encountered in precipitate analysis have been reviewed and illustrated with selected examples. When precipitates are treated as strain centers, dislocation loops or dipoles, stacking faults, or isolated single crystals, the analysis of their shape, distribution and strain field becomes similar to defect analysis. The view of a precipitate as homogeneously distorted matrix illustrates the important relationship between transformation strains of the real and reciprocal lattice. In addition, crystal symmetry has been shown to be useful as a tool to improve the statistics and enhance the accuracy of precipitate characterization by TEM.
Acknowledgements I would like to thank Dr. K.H. Westmacott for the many enjoyable research discussions that form the basis for this review. This work is supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Materials Sciences Division of the US Department of Energy under Contract Number DE-AC03-76SF00098.
References [1] P.B. Hirsch, A. Howie, R.B. Nicholson, D.W. Pashley and M.J. Whelan, Electron Microscopy of Thin Crystals (Butterworths, London, 1965). [2] J.W. Edington, Practical Microscopy in Materials Science, Vols. 1-5 (Phllips, Eindhoven, 1974-77). [3] M.H. Loretto and R.E. Smallman, Defect Analysis in Electron Microscopy (Wiley, New York, 1975). [4] S. Amelinckx, R. Gevers and J. Van Landuyt, Eds., Diffraction and Imaging Techniques in Materials Science (North-Holland, Amsterdam, 1978). [5] G. Thomas and M.J. Goringe, Transmission Electron Microscopy of Materials (Wiley-Interscience, New York, 1979).
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[6] M. von Heimendahl, Electron Microscopy of Materials: An Introduction (Academic Press, New York, 1980). [7] D.B. Williams, Practical Analytical Electron Microscopy in Materials Science (Philips Electronic Instruments, Mahwah, N J, 1984). [8] L Reimer, Transmission Electron Microscopy (Springer, Berlin, 1984). [9] M.H. Loretto, Electron Beam Analysis of Materials (Chapman and Hall, London, 1984). [10] M.F. Ashby and L.M. Brown, Phil. Mag. 8 (1963) 1649. [11] K.H. Katerbau, Phys. Status Solidi (a) 59 (1980) 393. [12] M.F. Ashby and L.M. Brown, Phil. Mag. 8 (1963) 1083. [13] M. Wilkens, in: Modem Diffraction and Imaging Techniques in Materials Science, Eds. S. Amelinckx, R. Gevers, G. Remaut and J. Van Landuyt (North-Holland, Amsterdam, 1970). [14] H, FiSll and M. Wilkens, Phys. Status Solidi (a) 31 (1975) 519. [15] U. Dahmen, K.H. Westmacott and G. Thomas, Aeta Met. 29 (1981) 627. [16] G.C. Weatherly, P. Humble and D. Bodand, Acta Met. 27 (1979) 1815. [17] H. F~511,C.B. Carter and M. Wilkens, Phys. Status Solidi (a) 58 (1980) 393. [18] H. Gleiter, Acta Met. 17 (1969) 565. [19] U. Dahmen and K.H. Westmacott, in: Proc. 10th Intern. Congr. on Electron Microscopy, Hamburg, 1982, Vol. 2, p. 119. [20] A.G. Khachaturyan, Theory of Structural Transformations in Solids (Wiley, New York, 1983). [21] U. Dahmen, P. Ferguson and K.H. Westmacott, Acta Met. 32 (1984) 803. [22] C.P. Luo and G.C. Wcatherly, Acta Met. 35 (1987) 1963. [23] 1L Sankaran and C. Laird, Acta Met. 25 (1977) 51. [24] U. Dahmen and K.H. Westmacott, Mater. Res. Soc. Symp. Proc. 62 (1986) 217. [25] G. van Tendeloo and S. Amelinckx, Acta Cryst. A30 (1974) 431. [26] D. Gratias, R. Portier and M. Fayard, Acta Cryst. A35 (1979) 885. [27] J.W. Cahn and G. Kalonji, in: Proc. Intern. Conf. on Sofid-Solid Phase Transformations, Pittsburgh, 1981, Eds. H.I. Aaronson et al. [28] M.J. Witcomb, U. Dahmen and K.H. Westmacott, UItramicroscopy 30 (1989) 143. [29] U. Dahmen, P. Ferguson and K.H. Westmacott, Acta Met. 35 (1987) 1037. [30] U. Dahmen and K.H. Westmacott, Science 233 (1986) 875.
Ultramicroscopy 30 (1989) 102-115 North-Holland, Amsterdam
TRANSMISSION ELECTRON MICROSCOPY CHARACTERIZATION OF PRECIPITATES U. D A H M E N National Center for Electron Microscopy,.Lawrence Berkeley Laboratory, Berkeley, California 94720, USA
Received at Editorial Office 6 December 1988; presented at Symposia August 1988
This paper reviews the methods and principles underlying the structural and morphological characterization of precipitates by TEM. Precipitate contrast mechanisms are reduced to the better-known contrast behavior of crystal defects such as dislocation loops, dipoles and stacking faults. Because of their importance, basic techniques of controlled specimen tilting and the analysis of double diffraction are treated in some detail. The remainder of this paper concentrates on more advanced concepts in precipitate characterization, not available in textbooks. Emphasis is placed on the use of crystal symmetry in practical problems of precipitate analysis, on the relationship between transformation strains of the real and the reciprocal lattice, and on the connection between transformation strain and morphology. These concepts are illustrated with selected examples.
1. Introduction Precipitates in solids are inclusions whose chemical and mechanical properties differ from those of the host matrix. Their characterization is important in understanding the microstructures and properties of materials spanning the range from semiconductors to metals to minerals. Precipitates are responsible for the stars in star sapphire, the strength of age-hardening alloys, and the toughening of zirconia ceramics. Procedures for precipitate analysis depend much on the context and the level of accuracy required. Sometimes it is sufficient to describe a precipitate by its size alone, but often it is necessary to determine the shape as well. Sometimes knowledge of the chemical composition is sufficient, but often the crystal structure is required, too. Electron microscopy allows very detailed characterization of m a n y aspects of precipitates, including composition, crystal structure, shape, orientation relationship with the matrix, distribution, degree of coherency and interface structure (e.g., refs. [1-9]). Compositional analysis is the realm of analytical microscopy and will not be covered here. The
remaining characteristics are structural in nature and are in fact interrelated.
2. Shape In the simplest scheme only three basic shapes (or morphologies) of precipitates exist: spheres, plates and needles. The intermediate morphology of a lath m a y be treated as a long plate or a flat needle. Further characteristics of precipitate shapes, i.e. whether they are facetted or rounded, regular or irregular, symmetrical or asymmetrical, are important but they do not change the mode of analysis which is different for spheres, plates and needles. Spherical precipitates appear identical in any crystal orientation and are characterized b y their size and the sign and magnitude of their strain field (interstitial or vacancy). Plates are the most c o m m o n precipitate morphology. They are characterized by their habit plane and aspect ratio, and by the sign, direction and magnitude of their strain. Needle-, rod- or lath-shaped precipitates in addition are characterized by the direction of their long axis.
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U. Dahmen / TEM characterization ofprecipitates
Precipitates can be made visible by two basic contrast mechanisms, the distortion they cause in the surrounding matrix (strain contrast) and the diffraction from the crystal structure of the precipitate (orientation and structure factor contrast). Precipitate analysis can thus be reduced to basic defect analysis conducted for the matrix and precipitate structure separately. With the first mechanism the precipitate is regarded as a defect. Precipitates may act like strain centers, dislocation loops, dislocation dipoles or displacement faults whose distortions can be probed in a g. b experiment. With the second mechanism, it is possible to form dark field images of the precipitate and, in fact, to conduct an entire g- b analysis of any defects that might be present inside the precipitate. These two mechanisms and other aspects such as symmetry relations between precipitates in a solid matrix will be examined below.
3. Controlled specimen tilting The basis of any precipitate analysis is choosing the right orientation and diffracting condition. Controlled tilting is therefore essential to the success of any systematic contrast experiment. In order to tilt the specimen into the desired Orientation, it is important to determine the location of the goniometer tilt axes relative to the diffraction pattern. In conjunction with Kikuchi lines in thick crystals or spot pattern intensity in thin crystals, this calibration can be used for tilt control. For example, if a pole is visible near the edge of the screen it can be brought to the center easily with the appropriate tilt drives. The position of a pole that is not visible on the screen can often be guessed from the directions of the Kikuchi lines that converge on it or from. the arcs of spot pattern intensity centered on it. Once this calibration is established, the area of interest can be maintained in the field of view during a controlled tilt sequence by tilting either in the image mode, or in the convergent-beam diffraction mode while observing the shadow image in the defocused diffraction disc. The choice of operating mode de-
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pends on how well the precipitate can be recognized in the shadow image.
4. Precipitates as defects 4.1. Small precipitates as strain centers
Small precipitates are most noticeably visible through their strain contrast, i.e. the distortion they cause in the surrounding matrix. The strain field can be mapped in a g - b experiment by bringing one set of crystal planes at a time into strong diffracting condition and thereby probing it for distortion. The contrast behavior under different diffracting conditions will depend on the character of the displacement field which, in turn, is related to the shape of the particle [10,11]. Spherical precipitates in an elastically isotropic matrix cause spherically symmetric strain fields and behave identically for all diffraction vectors [12]. Plate- and needle-shaped precipitates cause more complicated strain fields that give rise to very different contrast behavior for different diffraction conditions. Small (smaller than - ~ g / 3 ) strain centers exhibit oscillating contrast variations with depth in the foil. The best analysis can be performed on anomalous contrast that arises from strain centers located near the foil surfaces whose depth in the foil has been determined by stereomicroscopy. The calculated contrast oscillation of a small vacancy-type strain center (here a pure vacancy loop) with depth in the foil for bright field (BF) and dark field (DF) conditions is illustrated schematically in fig. 1 [13]. The sense of the black and white contrast reverses periodically through the thickness of the foil. The strong (anomalous) black and white contrast at the top and bottom surfaces is equal in DF and opposite in BF. The strongest image is observed at s = 0. For an interstitial strain center the sense of the contrast must be reversed. For particles causing larger strain the contrast oscillations are suppressed due to surface relaxations, and the sense of the black and white contrast will be that closest to the foil surfaces in fig. 1. As first shown by Ashby and Brown [10], the DF image exhibits a unique sense of this contrast.
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U. Dahmen / T E M characterization of precipitates
0.5_~Q
1-Z g
,~_
bottom
Fig. 1. Depth oscillationof black-white contrast from small vacancy-typestrain centers in BF and DF. Anomalous images appear for defects near foil surfaces. Only anomalous DF images have equal black-white direction at top and bottom surface. Interstitial-type strain centers show opposite contrast (after ref. [13]). Thus D F conditions at s - - 0 are required to determine the nature (interstitial or vacancy) of such a center of strain. 4.2. Precipitate plates as dislocation loops
Plate precipitates that are a few tens of nanometers in size (of the order of ~g) may show more extended ring-like loop contrast. This is because to first order the displacement field of a precipitate, like that of a dislocation loop, is caused by a plate of crystal that has been expanded, contracted or sheared. Here the fact that the two-beam BF image of a dislocation is located off to one side of the true position of the dislocation line can be used to determine the nature (interstitial or vacancy) of a dislocation loop or plate precipitate. An inclined plate will show inside/outside contrast for + g , interstitial/vacancy nature and posit i v e / negative habit plane inclination to the beam. To determine the nature of a plate from its inside/outside contrast, it is therefore essential to
know its sense of inclination relative to the g vector [14]. Fig. 2 illustrates a simphfied method [15] that can be used to avoid errors often arising due to the presence (or absence) of an inversion between image and diffraction pattern. Note that this analysis assumes the displacement vector is normal to the habit plane, usually a good approximation for small plate-shaped precipitates. First the plate is tilted edge-on so that the habit plane is parallel to the beam (fig. 2a). This crystal orientation (here [001]) is marked on a Kikuchi map. The sample is then tilted to an orientation where the precipitate is inclined. A two-beam BF condition with s > 0 is set up for a g vector pointing toward the pole (marked on the Kikuchi map) where the habit plane was edge-on. This is arbitrarily defined as + g (fig. 2c). If the plate shows outside contrast in + g , it is interstitial; if it shows inside contrast it is a vacancy-type precipitate. To distinguish between inside and outside contrast, images in both + g and - g should be recorded.
U. Dahmen / TEM characterization of precipitates
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d\
I-g |II
// Fig. 2. Simplified contrast analysis to determine the nature of plate precipitates and dislocation loops [15]. Precipitates marked A and B seen edge-on in a [001] zone (a) show outside contrast in + g (c) and are therefore interstitial in character. (Courtesy Acta Met.)
4.3. Precipitate needles as dislocation dipoles
4. 4. Precipitate plates as stacking faults
Long needle-shaped precipitates with small cross-section exhibit strain contrast very similar to that of a dislocation dipole. The true displacement field will, in fact, be different from that of a dipole, but its major component can often be described by a single displacement vector. A standard g - b analysis can provide the direction and approximate magnitude of this displacement vector and the nature (shear, edge) of the precipitate [16]. To determine the vacancy or interstitial character of a needle or dipole it may be regarded as an elongated loop and the same procedure as described above can be applied.
Under some conditions, uniformly thin plate precipitates give rise to displacement fringe contrast similar to that of a stacking fault. This is seen for a [100] 0' plate in an A1-Cu alloy in fig. 3b. The contrast arises if the matrix is in two-beam diffracting condition and if the precipitate is diffracting weakly and oriented inclined to the electron beam. Under these circumstances the primary contribution of the precipitate to the image c o n trast is that it causes a fractional displacement of the matrix lattice at the top and bottom interface. This displacement, and with it the fringe contrast, depends on the thickness of the plate [17]. Ledges
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U. Dahmen / T E M characterization of precipitates
Fig. 3. Displacement fringe contrast at an inclined plate-shaped 0' precipitate on a (100) plane in AI-Cu (b), When g is paratlel to the (100) habit plane the fringe contrast is absent (a) giving a clear view of interracial dislocations (numbered 1-5).
in the interface will thus lead to abrupt shifts in the fringe pattern, and the magnitude of the shift can be related to the height of the ledge [18]. To avoid fringe contrast it is necessary to set up a diffracting condition for which g . R is an integer. This is most easily accomplished by selecting a g-vector parallel to the plate surface (g . R = 0). As shown in fig. 3a it is then possible to see interface dislocations undisturbed by displacement fringes.
5. Precipitates as homogeneously distorted matrix In two-phase materials, it is often difficult to obtain separate diffraction patterns from the two phases, matrix and precipitate. Frequently, they will be seen as two interpenetrating patterns, and if the structures are similar, the precipitate pattern
may be interpreted as a distorted or transformed matrix pattern. The distortion, or transformation strain, can be read directly from the separation of corresponding diffraction spots (spot splitting) [19]. The transformation strain of the crystal lattices is inverse to the strain obtained from the spot splitting in the two interpenetrating diffraction patterns. This is shown more rigorously as follows [20]. For a crystal lattice described by lattice vectors r, diffraction spots occur only at positions g in reciprocal space for which g . r is an integer. This is the commonly known relationship between real and reciprocal space. It expresses the fact that the Fourier transform of a periodic object is itself periodic. Since g- r = integer, it is easy to see that the reciprocal lattice g must shrink if the crystal lattice r grows. More generally, if the crystal lattice undergoes a homogeneous transformation de-
107
U. Dahmen / TEM characterization of precipitates
~
S
"°'....
Fig. 4. Illustration of typical plane strain precipitates with corresponding diffraction patterns and schematic transformation strains: (a) uniaxial strain, (b) simple shear and (c) general invariant plane strain. Diffraction spot splitting is normal to the habit plane n. The strain direction s lies normal to the plane of unsplit spots (marked by lines in diffraction patterns).
scribed by a tensor A, the reciprocal lattice will undergo the inverse transpose of that transformation: ,4-1T. This follows from the fact that ( g . A - 1 X A . r ) = i n t e g e r . Thus a cubic-to-tetragonal transformation in real space would manifest itself in a cubic-to-tetragonal transformation of the reciprocal lattice, but with inverse c / a ratio. This is illustrated with a few examples below. 5.1. Uniaxial strain
A tetragonal precipitate in a cubic matrix provides a simple example for a uniaxial transforma-
tion strain, assuming that the a-axes are identical in the two phases. One would expect a plate-shaped precipitate with a plane of contact (habit plane) on one of the cube faces of the matrix. The plate normal will be the direction of largest mismatch. An example is the formation of 0' precipitates in A1-Cu alloys, illustrated in fig. 4a, where the plates are seen edge-on. As shown in the schematic the transformation A can be described as a uniaxial expansion on the (001) plane in the [001] direction and the reciprocal lattice will undergo a corresponding contraction. Notice that the spot splitting in the diffraction pattern is normal to the
108
U. Dahmen / T E M characterization of precipitates
habit plane. This transformation strain is a special invariant plane strain (it leaves the (001) plane invariant), but the result can be generaliTcd to other invariant plane strains.
5.2. Simple shear A twinning shear is a typical example of a simple shear transformation. As seen for a microtwin in fig. 4b, the plane of contact is the twin plane, and the corresponding diffraction pattern shows spot splitting normal to the habit plane. Notice the row of unsplit spots perpendicular to the habit plane. The accompanying schematic illustrates the transformation strain, a shear s on the twin plane n. The corresponding transformation strain in the reciprocal lattice is a simple shear n on the plane s.
5.3. General invariant plane strain In fig. 4c the precipitation of a bee phase in an fcc matrix (Cr in Cu) is treated in this manner. The orientation of the [111] bcc precipitate pattern relative to the [110] fee matrix pattern is such that the spot splitting lies normal to the (irrational) habit plane. Comparison of the diffraction pattern with the schematic of the transformation strain shows once more the inverse relationship of n and s in real and reciprocal space. As might be expected from these examples, all plate-shaped precipitates are related to their matrix by an invariant plane strain. The invariant plane is the habit plane since matrix and precipitate will be joined across the plane of minimum distortion. Assuming that no plastic deformation is involved in procuring this fit, i.e. the transformation is homogeneous, the reciprocal lattice will undergo the inverse transpose of this homogeneous invariant plane strain. It can be shown that this is again an invariant plane strain, but with the role of the invariant plane normal n and the strain vector s reversed. The invariant plane of the reciprocal lattice will be a plane with no spot splitting and its plane normal is s, the strain direction in real space. On the other hand all the distortions in reciprocal space (spot splitting) will be in the direction n, the habit plane normal. These results
are very useful in the analysis of plate-shaped precipitates and all transformations with planar (invariant plane) interfaces: for analysis by TEM it is important to obtain the special orientation containing both s and n in the plane of the diffraction pattern. Although the habit plane n is easy to find from images by trace analysis, the direction and magnitude of s can only be found from the diffraction pattern.
5.4. Invariant line strain Needle-shaped precipitates tend to be related to the matrix by an invariant line strain with the needle axis lying along the undistorted, or invariant, line [21,22]. (Note that the homogeneous component of the total transformation strain for many plato-shaped precipitates, including most martensite plates, also is an invariant line strain.) Again, it can be shown that an inverse relationship exists between real and reciprocal space transformation. The two characteristic vectors, the invariant line i and the invariant normal h, reverse roles in reciprocal space, i.e. the diffraction pattern contains a direction of no spot splitting h while the pattern seen when the beam is along the invariant line direction i contains all the spot splitting and thus allows the most accurate reading of the orientation relationship.
5.5. Double diffraction A homogeneous transformation does not lead to additional or forbidden reflections in the reciprocal lattice. If, apart from the homogeneous distortion, new reflections or absences are observed, the lattice has undergone ordering, shuffling or modulation. These lattice rearrangements are the subject of crystal structure analysis and are most elegantly solved by convergent-beam electron diffraction, a technique that lies outside of the scope of this presentation. However, a common source of extra reflections in second-phase analysis, unrelated to the crystal structure, is double diffraction. The simplest diffraction pattern is that from a single crystal, but often more than one crystal, for example matrix and precipitate, contribute to the
U. Dahmen / TEM characterization of precipitates
109
Fig. 5. Diffraction patterns of fcc matrix with bcc precipitate: (a) without double diffraction and (b) with double diffraction due to overlapping of matrix and precipitate along the beam direction; (a) is the addition and (19) the convolution of the two patterns.
pattern, as shown for a bee precipitate in an fee matrix in fig. 5. If they originate in different regions the two patterns will simply add (fig. 5a), but if one is embedded in the other, i.e. they overlap along the beam path, extra spots may appear due to double diffraction. Each diffracted spot of the first phase may then act as an incident beam for the second phase, leading to the convolution of the two patterns (fig. 5b). In the absence of double diffraction the precipitate pattern is easily separated from the matrix pattern, and even with pronounced double diffraction it is usually easy to locate and identify the matrix pattern unambiguously. However, the problem of locating the primary precipitate pattern from the maze of non-matrix reflections and separating it from the doubly diffracted spots is sometimes difficult. Since the process of double diffraction is the convolution of the matrix and precipitate patterns it could, in principle, .be deconvoluted simply with a transparent overlay to separate the total pattern into a number of parallel-displaced precipitate (or matrix) patterns. Unfortunately, this deconvolution is not unique because it defines the precipitate diffraction vectors only modulo the matrix vectors. Thus from this procedure alone it is impossible to tell whether to
assign go as a primary precipitate diffraction vector or whether (g + gmatrix) is the correct assignment. The safest and by far the easiest way to make this distinction is by tilting off the zone axis to eliminate double diffraction paths. However, often the problem of double diffraction is not apparent until a spot pattern is examined on the negative. It is then necessary to use further information hidden in the intensity distribution and shape of the extra spots. The intensity of a doubly diffracted spot is proportional to the intensity of the matrix reflection that acts as its origin. Since the transmitted beam has the highest intensity, the primary precipitate pattern should be the strongest of the extra reflections. The first assignment of possible primary spots is therefore always to the most intense extra reflections in the pattern. This assit,nment must then be checked by parallel-translating these diffraction vectors, with the help of a transparent overlay, to see if in fact the precipitate reflections appear as if each matrix reflection acted as a transmitted beam. A further hint is sometimes given by the shape of a precipitate spot, such as the shape of an arc due to an angular spread of several similar surface precipitates contributing to the same diffraction
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U. Dahmen / TEM characterization of precipitates
pattern. Such an arc must always be centered on its origin, i.e. on the spot in the diffraction pattern that acts as its transmitted beam. This often allows a quick elimination o f ambiguities.
6. Precipitates as sin~e crystals The view of a precipitate as a single crystal separate from the matrix and its distortion focuses
Fig. 6. Example of a diffraction condition used in analysis of defects within a precipitate. As seen in the BF (a) the matrix is diffracting weakly while the particle is in a strong two-beam condition, used to form the DF image in (b).
U. Dahmen / TEAt characterization of precipitates
attention on the defects inside the particle. One way to analyze a precipitate in this manner is to separate it physically from its matrix by extraction. A contrast analysis can then be performed easily by setting up a series of two-beam diffracting conditions without interference from the matrix crystal. The same kind of experiment can be performed with somewhat greater difficulty if the precipitate is contained in the matrix (e.g., ref. [23]). In order to eliminate matrix contrast, all such images must be recorded in the DF mode. It is best to set up simultaneously conditions of strong diffraction in the particle and weak diffraction in the matrix, illustrated for a 0' precipitate in A1-Cu in fig. 6. This limits the number of orientations where clean interpretable images can be obtained. If precipitates are small and show only very weak diffraction spots it is possible to use Kikuchi bands of the matrix as a navigational aid for obtaining the desired diffracting conditions.
111
precipitate are related by a point symmetry operation of the matrix [25-27]. This is readily apparent from the hypothetical sequence of operations shown in fig. 7: A precipitate is embedded in a matrix crystal (a). A spherical region including the precipitate is cut from the matrix (b), removed (c), rotated, inverted or mirrored by a matrix symmetry operation (d) (here a 90 o rotation), replaced in the matrix (e) and rewelded (f). The matrix crystal is undisturbed by this sequence because by definition it is invariant under a symmetry operation. However, a new variant of the precipitate has been generated. The number of possible variants is therefore equal to the number of symmetry elements in the point group of the matrix. If, however, the precipitate shares one of the symmetry elements of the matrix, for example a 180 ° rotation in fig. 7, then this element will not produce a new variant. Thus only those symmetry elements of the matrix that are not shared by the precipitate generate new variants. 7.1. Determination of needle axis
7. The use of synunetry
The shape, orientation and distribution of precipitates all bear inherent synunetry properties that can be exploited to facilitate and improve the accuracy of a morphological analysis b y TEM [24]. Ignoring translations, any two variants of a 8
Needle- or lath-shaped precipitates are characterized by their long axis [hkl], conventionally determined by trace analysis. A general axis direction
L \
",L \
)li f
/" I
Ill
L~\ /
/
/
in Fig. 7. Schematicillustrationof variant-generatingmatrix symmetryoperation[24].
U. Dahmen / TEM characterization of precipitates
112
hkl
110
hhl
111
hkO
100
Fig. 8. (001) Stereograms illustrating multiplicity of needle precipitates in a cubic crystal [24].
ments, and enumerated by the permutations of the three indices hkl and their negatives. This is most graphically illustrated in a stereogram, for example an f001) orientation as shown for the general direction in fig. 8. Twenty-four different variants of (hkl) needles are possible if the needle-morphology shares only the inversion with the matrix. If the needles lie on a {110} mirror plane, i.e. their indices are of the type fhhl), this mirror is a shared symmetry element and only 12 different variants exist. The remaining stereograms show other special directions in order of increasing symmetry. Each of the variant-generating symmetry operations may be thought of as a specimen tilt in a trace analysis. Needle axes can therefore be measured directly from a single micrograph in an f001) orientation: It is clear that simple (100) needles will be immediately apparent (stereogram f). However, even in the most complicated case of needles along a general f h k l ) direction (stereogram a), measuring three characteristic angles (a, fl, "t) enclosed between precipitate variants will yield their direction easily through the relations h 2 / l 2 -- (1 - c o s a ) / ( 1 + cos a),
h 2 / k 2 -- (1 - c o s / 5 ) / ( 1 + cos/5), k 2 / l 2 ~ (1 - cos V)/(1 + cos y).
Two angles are sufficient to determine the indices hkl but the third may be used to check the accuracy of the measurement. More details are available in the literature [24]. It should be pointed out that conventional trace analysis yields greater accuracy, while the analysis based on symmetry provides better statistics as well as additional information on the distribution of precipitate variants. 7.2 Increased accuracy of needle or plate habit If the crystallographic orientation of a needle or plate precipitate is known approximately, symmetry properties can be used to determine their orientation with greater precision. For example, it was found that needle precipitates of Cr in a Cu matrix lie on {111} planes [28]. To find their precise orientation in the {111} plane, the angle between two precipitate variants viewed along a (111) zone axis, shown in figs. 9a-9d, can be measured. This angle is twice the deviation of the needle direction from a {110} mirror plane. The same method can be used to improve the precision of a habit plane determination [29]. For example, large ~,' plates in F e - N alloys form in a characteristic butterfly-like arrangement. The two wings are two individual plates joined together at an acute angle. Each plate has two sets of stria-
U. Dahmen / TEM characterization ofprecipitates
113
Fig. 9. (a)-(d) Two variants of lath-shaped Cr precipitates in a single grain of Cu matrix, illustrating horizontal mirror symmetry between particles and between corresponding diffraction patterns; (e) and (f) mirror symmetry between two variants of "t' plates in Fe-N alloy.
tions typical of two variants o f an invariant line strain. Since b o t h variants clearly have the same habit plane and since a n y two variants are related b y a s y m m e t r y operation of the matrix, it follows that the habit plane must have a s y m m e t r y ele-
m e n t in c o m m o n with the matrix, i.e. it m u s t be parallel or perpendicular to a mirror plane or r o t a t i o n axis. I n the present case it was f o u n d that the two variants were related b y an {001} mirror, a n d the habit plane m u s t be of the type { hkO}. In
114
U. Dahmen / T E M characterization of precipitates
Fig. 10. High-resolutlonmierograph of peatagonally-twinaodC~ pr¢~'ipitate in A1 matrix with beam direction along (1i0) Ge and (I00) A1(CourtesyScience[301.) order to find the exact orientation, i.e. the ratio of h to k, the angle between two { hkO} plates that form the wings of the butterfly, when seen edge-on in an (001) foil orientation, can be measured accurately. As seen in figs. 9e and 9f, the angle of 42 ° is bisected by a (110) mirror, and the habit plane is (940} which may be compared with the (210] habit plane from less accurate measurements.
7.3. Symmetry of morphology and orientation relationship Symmetry is also apparent in the shape of precipitates. For example, the needle-shaped Ge
precipitates formed during aging of AI-Ge alloys exhibit a variety of shapes when viewed along their needle axis. High resolution micrographs show that the symmetry of the shape is usually that of the orientation relationship between the two crystals, i.e. the symmetry they have in common. Faulting inside the precipitates destroys this symmetry, a fact that is reflected in irregular shapes of heavily faulted particles. Sometimes, however, even the faults are arranged so as to be compatible with the symmetry of the orientation relationship. This leads to more regular morphologies. An example is seen in the fivefold twinning of a Ge particle (fig. 10) where the orientation relationship, the substructure and the shape all conform to mm2 orthorhombic symmetry [30].
U. Dahmen / T E M characterization of precipitates
8. Summary The major contrast mechanisms encountered in precipitate analysis have been reviewed and illustrated with selected examples. When precipitates are treated as strain centers, dislocation loops or dipoles, stacking faults, or isolated single crystals, the analysis of their shape, distribution and strain field becomes similar to defect analysis. The view of a precipitate as homogeneously distorted matrix illustrates the important relationship between transformation strains of the real and reciprocal lattice. In addition, crystal symmetry has been shown to be useful as a tool to improve the statistics and enhance the accuracy of precipitate characterization by TEM.
Acknowledgements I would like to thank Dr. K.H. Westmacott for the many enjoyable research discussions that form the basis for this review. This work is supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Materials Sciences Division of the US Department of Energy under Contract Number DE-AC03-76SF00098.
References [1] P.B. Hirsch, A. Howie, R.B. Nicholson, D.W. Pashley and M.J. Whelan, Electron Microscopy of Thin Crystals (Butterworths, London, 1965). [2] J.W. Edington, Practical Microscopy in Materials Science, Vols. 1-5 (Phllips, Eindhoven, 1974-77). [3] M.H. Loretto and R.E. Smallman, Defect Analysis in Electron Microscopy (Wiley, New York, 1975). [4] S. Amelinckx, R. Gevers and J. Van Landuyt, Eds., Diffraction and Imaging Techniques in Materials Science (North-Holland, Amsterdam, 1978). [5] G. Thomas and M.J. Goringe, Transmission Electron Microscopy of Materials (Wiley-Interscience, New York, 1979).
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[6] M. von Heimendahl, Electron Microscopy of Materials: An Introduction (Academic Press, New York, 1980). [7] D.B. Williams, Practical Analytical Electron Microscopy in Materials Science (Philips Electronic Instruments, Mahwah, N J, 1984). [8] L Reimer, Transmission Electron Microscopy (Springer, Berlin, 1984). [9] M.H. Loretto, Electron Beam Analysis of Materials (Chapman and Hall, London, 1984). [10] M.F. Ashby and L.M. Brown, Phil. Mag. 8 (1963) 1649. [11] K.H. Katerbau, Phys. Status Solidi (a) 59 (1980) 393. [12] M.F. Ashby and L.M. Brown, Phil. Mag. 8 (1963) 1083. [13] M. Wilkens, in: Modem Diffraction and Imaging Techniques in Materials Science, Eds. S. Amelinckx, R. Gevers, G. Remaut and J. Van Landuyt (North-Holland, Amsterdam, 1970). [14] H, FiSll and M. Wilkens, Phys. Status Solidi (a) 31 (1975) 519. [15] U. Dahmen, K.H. Westmacott and G. Thomas, Aeta Met. 29 (1981) 627. [16] G.C. Weatherly, P. Humble and D. Bodand, Acta Met. 27 (1979) 1815. [17] H. F~511,C.B. Carter and M. Wilkens, Phys. Status Solidi (a) 58 (1980) 393. [18] H. Gleiter, Acta Met. 17 (1969) 565. [19] U. Dahmen and K.H. Westmacott, in: Proc. 10th Intern. Congr. on Electron Microscopy, Hamburg, 1982, Vol. 2, p. 119. [20] A.G. Khachaturyan, Theory of Structural Transformations in Solids (Wiley, New York, 1983). [21] U. Dahmen, P. Ferguson and K.H. Westmacott, Acta Met. 32 (1984) 803. [22] C.P. Luo and G.C. Wcatherly, Acta Met. 35 (1987) 1963. [23] 1L Sankaran and C. Laird, Acta Met. 25 (1977) 51. [24] U. Dahmen and K.H. Westmacott, Mater. Res. Soc. Symp. Proc. 62 (1986) 217. [25] G. van Tendeloo and S. Amelinckx, Acta Cryst. A30 (1974) 431. [26] D. Gratias, R. Portier and M. Fayard, Acta Cryst. A35 (1979) 885. [27] J.W. Cahn and G. Kalonji, in: Proc. Intern. Conf. on Sofid-Solid Phase Transformations, Pittsburgh, 1981, Eds. H.I. Aaronson et al. [28] M.J. Witcomb, U. Dahmen and K.H. Westmacott, UItramicroscopy 30 (1989) 143. [29] U. Dahmen, P. Ferguson and K.H. Westmacott, Acta Met. 35 (1987) 1037. [30] U. Dahmen and K.H. Westmacott, Science 233 (1986) 875.