Symmetric stacking fault nodes in anisotropic crystals

SYMMETRIC STACKING FAULT NODES ANISOTROPIC CRYSTALS*

IN

R. 0. SCATTERGOOD Materials Science Division, Argonne National Laboratory. Argonne. IL 60439, U.S.A. and

D. J. BACON Department of Metallurgy and Materials Science, The University. Post Office Box 147. Liverpool L69 3BX, England (Rrcriced 3 Norember 1975; in rerisedform

26 January 1976)

Abstract-The equilibrium configurations of symmetric, 3-fold stacking fault nodes in f.c.c. crystals were determined by a linear elastic self-stress method which accounts for the combined effects of dislocation interaction and elastic anisotropy. Based on the results, a set of empirical equations was developed which compactly summarizes the solution to the anisotropic node problem, and which can be used to analyze node measurements. In principle, the stacking fault energy Y and the dislocation core radius r0 can be obtained from node measurements; however, analysis of data for silver displayed too much scatter for both values to be reliably extracted and a best fit value of 7 = 23.5 erg/cm’ was obtained with the assumption of r,, = b. The appropriate choice of the Poisson ratio and shear modulus to obtain an isotropic approximation to the anisotropic node problem was given, and the deviations from isotropy in the node configurations were rationalized, qualitatively. in terms of the line tension of the partial dislocations. R&sum&-On a determini la configuration B I’equilibre de noeuds de difaut d’empilement symetriques dans les cristaux c.f.c. en Clasticite IinCaire par une mtthode d’auto-contrainte qui tient compte des etTets combints de I’interaction entre dislocations et de I’anisotropie ilastique. A partir de ces resultats. on dr%eloppe un systems d’tquations empiriques qui rtsument de maniire compacte la solution du probleme du noeud anisotrope. et que I’on peut utiliser pour analyser les risultats de mesures sur les noeuds. En principe. on peut d&Iuire I’energie de defaut d’empilement 7 et le rayon r0 du coeur de la dislocation a partir des mesures de noeuds; toutefois. les resultats dans le cas de I’argent etaient trop dispersis pour qu’on puisse obtenir des valeurs sCires pour 7 et r0 i la fois; on a obtenu le meilleur accord avec ;’ = 23,5 erg/cm’. en supposant que r o = b. On indique igalement le choix du coefficient de Poisson et du module de cisaillement qui permet d’obtenir une approximation isotrop-e atI probleme du noeud anisotrope; on a rationalis qualitativement les diviations par rapport & l’isotropie dans les configurations de noeuds, d partir de la tension de ligne des dislocations imparfaites. Zusammenfassung-Mittels einer linearen elastischen Methode der Selbstspannungen. die kombinierte EiTekte von Versetzungsu-echselwirkung und elastischer Anisotropie beriicksichtigt. wurden die Gleichgewichtskonfigurationen svmmetrischer Dreifachknoten von Stapelfehlem in kfz. KristaUen bestimmt. .Aufbauend auf den Ergebmssen wurde ein Satz empirischer Gleichungen entwickelt. welcher die LGsung des anisotropen Knoten-problems zusammenfa& und zur Analyse von Messungen an Knoten benutzt werden kann. Im Prinzip kann die Stapelfehlerenergie ‘J und der Radius des Versetzungskemes r,, aus Messungen an Knoten ermittelt werden; allerdings wiesen die Analyse von Silberdaten zu starke Streuung der beiden Werte auf, urn zuverllssig zu sein. Mit der Annahme r. = b wurde ein bester Wert fiir 7 von 23.5 erg,cm’ erhalten. Urn eine isotrope NBherung fiir das anisotrope Knotenproblem zu erhalten, wurden geeignete Werte von Poisson-Verhliltnis und Schermodul, amzegeben. Abweichungen von der Isotropie in den Knotenkonfigurationen werden mit Hilfe der LinLnspannung dcr Partialversetzungen qualitativ erklsrt.

INTRODUCTION The measurement of the width of symmetric 3-fold stacking fault nodes in f.c.c. crystals, as originally developed by Whelan [l]. has become an established technique for evaluating the stacking-fault energy. The increased widths afforded by the 3-fold node geometry mean that measurements may be possible even though the partial dislocations in an extended straight

beam methods will prove to be valuable in stackingfault energy measurements; nevertheless, the extended

* Work supported by the U.S. Energy Research and Development Administration. \\I 213 \

dislocation in the same crystal cannot be resolved. Furthermore, the symmetry of the 3-fold node provides a check on the possible existence of unwanted side effects such as internal stresses, image stresses, impurity pinning, etc. Weak beam electron microscopy techniques have now been developed to improve the resolution in transmission micrographs and it may be possible to resolve the partials in extended straight dislocations in cases where they heretofore could not be resolved [Z]. Undoubtedly the weak

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34old node geometry considered here will still prove useful. The primary intent of the work reported here was to obtain the static equilibrium configurations for the symmetric 3-fold node geometry found in f.c.c. crystals, taking into full account the combined effects of the dislocation interactions and the elastic anisotropy of the crystal. The calculations were based upon linear, anisotropic elasticity, and they represent a complete solution to the problem within the framew’ork of the linear theory. A typical, extended 3-fold stacking-fault node in an f.c.c. crystal is shown schematic~ly in Fig. 1. The width Iv of the node is defined to be the radius of the inscribed circle, and the Burgers vector orientation for the node is given by the angle 8 between the partial Burgers vector b (b will always denote the magnitude of the partial Burgers vector) and the direction of the straight sidearm asymptote; the nodes lie on jlll) planes and they are assumed to exhibit 3-fold symmetry. Note that as the perfect (undissociated) straight side-arm dislocation goes from screw to edge, 0, as defined in Fig. 1, goes from 0” to 90”. The angle 0 also gives directly the Burgers vector orientation of the partial dislocation at a point ap proximately in the center of the node branch (the branches need not be symmetric about their centers). With suitable procedures, both 0 and W can be obtained from transmission electron micro~aphs, and these two quantities should be considered the measured values from which the stacking-fault energy 7 is to be obtained. The basic problem in interpreting node measurements is the exact relationship between y and the /

4

,r+,;

D

\

\ Fig. 1. Schematic of the symmetric, 3-f&d stacking fault

node geometry. The three node branches consist of Shockley partial dislocations bounding a stacking fault, and the configuration is assumed to have 3.fold symmetry. The node size W is given by the radius of the inscribed circle. The width D of the sidearm asymptotes is equal to the

spacing of extended, infinite straight diifocations, and the angle 6, defined as shown, gives the orientation of the Burgers vector at the sidearms.

STACKING

FAULT NODES

node parameters 8 and M! Various analyses of this problem have appeared in the literature [S-9]. In calculations based on linear elasticity, there will also be a cutoff parameter r. to account for the divergence of the dislocation stress field at points on the dislocalion line, and r. is linked to the so-called ‘Core radius” for the dislocation self-energy. The core of the dislocation cannot be properly treated in linear elastic calculations, and this remains as the major drawback of the present results; as yet- there is no adequate means by which the core can be taken into account. If the form of the relation between q 8, y, and r,, is known, then in principle y and re can be obtained from measurements of W and 8. Since there are two unknowns viz. y and ro, at least two independent node measurements must be made. As an alternative procedure, one could use weak beam methods to obtain a value for y (re does not enter into the width of extended straight dislocations, at least in linear elastic calculations), and then use the 3-fold node geometry to obtain information on Q., The 3-fold node measurement appears to be one of the most direct means for estimating the parameter ra, which appears in numerous theories describing dislocation properties. The success of such measurements in extracting and 7 and r. values will of course depend upon the reliability and the scatter in the experimental data. Since the dependence on r. is not strong, in the sense that it can be expected to be logarithmic, one very often chooses r. 2 b, and then only y need be determined. We have varied r. in our calculations in order to display the specific dependence, and we emphasize that in principle, the 3-fold extended node geometry contains information on both y and ro. In order to obtain the equilibrium configurations for the node geometry shown in Fig. 1, the dislocation interactions and the elastic anisotropy of the crystal should be taken into account. It is obvious that a dislocation element on one of the three node branches will interact, by means of the dislocation stress field, with other elements of the same branch and with all elements on the remaining two branches. We cannot invoke the line tension approximation for this problem since this would ignore all but the local interactions at each element of dislocation (and would introduce another parameter-a logarithmic outer cutoff distance), and, by the very nature of the node geometry, it seems clear that all the dislocation interactions must be important. Brown [6] first used a selfstress method, which balances all forces on each dislocation element, to show the importance of the dislocation interactions in the node problem, but his analysis was confined to isotropic elasticity. The weakness of the isotropic assumption is difficult to assess, and, therefore, we have developed a self-stress method which allows the complete &isotropy to be included along with the dislocation interactions. Furthermore, in our method of solution, we make no a priori assumptions about the shape of the node branches,

SCATTERGOOD

,0-D BACON:

Fig. 2. Schematic showing the representation of a node branch by a finite set of points; the points define the set of finite dislocation segments used to calculate the self stress. If the stress is to be evaluated at one of the points P. the segment containing P is taken as a circular arc as shown, while the remaining segments on all branches are straight. The angle r between adjacent segments is defined as shown. The points S and Tdenote the endpoints

of the curved part of the branch where it joins the straight sidearm asymptotes. as was done in an earlier investigation [9]; we have found that a priori shape assumptions can lead to unreliable results in self-stress calculations [lo]. The calculations were carried out for silver and nickel because these crystals display noticeable anisotropy, and their “anisotropic” Poisson ratio and shear-modulus values, as defined in Section 2, span a range which covers most of the pure f.c.c. crystals. In addition, these crystals form the host materials for many alloys in which stacking-fault effects can be important. We should remark that the methods employed in this investigation demonstrate the practicability of solving complicated problems involving curved dislocations, completely taking into account the combined effects of dislocation interaction and elastic anisotropy. DESCRIP’l-IOl\i

OF THE CALCULATIONS

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is evaluated. Equation (1) must be satisfied at all points along one of the node branches shown in Fig. 1, and then by symmetry it will hold on the remaining branches. In the general case considered here, T, must be obtained as a line integral over the entire 3-fold node configuration, so that equation (1) becomes a very difficult equation to solve, and it is amenable only to numerical methods. The principle features of the calculations are the evaluation of TV, in particular for the anisotropic case, and the procedure for relaxing the node shape into the configuration satisfying equation (1). These features are outlined in the following. A branch of the 3-fold node is represented by a finite set of points, as is shown in Fig. 2. The boundary conditions imposed on the problem are: First, that the sidearm asymptotes of each branch be extended straight dislocations whose spacing D is equal to the spacing of the partials for the corresponding infinite straight extended dislocation in the given crystal. Second, the node must have 3-fold symmetry, and therefore the entire node configuration is obtained from a 3-fold rotation of the branch shown in Fig. 2, subject to maintaining the proper width D for the sidearms.* Other than this, the node geometry is arbitrary, and, for example, the branches need not be symmetric about their centers, although this symmetry does exist for 0 equal to 0’ or 90”. T, can be obtained for crystals with arbitrary elastic anisotropy by means of the theorems due to Brown [ 1l] and Indenbom and Orlov [12] ; in essence, these theorems reduce the problem to the determination of the fields of infinite straight dislocations, and these fields can be obtained in a straightforward manner by Green’s function methods [13,14]. If we consider a planar dislocation configuration r, the in-plane traction vector T due to r, at a point P in the plane, can be written as [ll], (2)

Consider the node geometry shown in Fig. 1. If we restrict all variations of the node configuration to the [ill) plane containing the node (the node plane), then the equation governing the static equilib rium configuration of the node can be written as follows, bs, + ‘i = 0.

STACKfXG

(1)

where br, gives the force on each point of the dislocation due to its own stress field. rI is generally called the self stress of the dislocation and it is the shearstress field in the node plane, due to the entire 3-fold configuration, resolved along the Burgers vector of the branch containing the point at which the field *Asymmetric nodes, not having 3-fold symmetry, are also possible and can be treated by the methods outlined here. In practice, symmetric 3-fold nodes can be observed and we shall consider only the symmetric geometry for this investigation.

where r is a vector joining P to the line element ds, w and $ are, respectively, the angles from a fixed (arbitrary) reference direction in the plane to r and the tangent to r at ds, and r is the angular part of the traction for an infinite straight dislocation lying along r and having the same Burgers vector as r. Equation (2) thus reduces the problem of determining the field due to r to evaluating an integral involving straight-line fields. At a point P on one of the 3-fold node branches, TV is obtained by integrating over each branch and resolving the tractions along the Burgers vector direction of the branch containing P. For the numerical evaluation, the integral in equation (2) is obtained as a sum over fmite dislocation segments whose endpoints are determined by the set of points shown in Fig. 2; the sum is to be taken over the entire 3-fold configuration. If T, is to be evaluated at point P in Fig. 2, the segment containing

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P is taken to be a circular arc through P and the two adjacent points on either side, while all the remaining segments are taken to be straight. Formulae for the contribution to rs from these two kinds of finite segments can be obtained from the integration of equation (2) and the integration procedure is analogous to that used by Scattergood and Bacon [lj] in an earlier investigation. The only difficulty which appears in this procedure is the singularity which arises for the circular arc segment since it contains the point P. This singularity is removed by carrying out the integration only up to a distance p0 from P; p. then becomes the “core cutoff parameter” which is always required in linear elastic calculations. In the case of isotropy, this procedure for calculating the self stress follows from calculating the dislocation self energy as half the interaction energy between two coincident dislocation configurations, assuming a zero interaction between two dislocation elements closer together than p,,. The parameter p,, can be related to the conventional core radius rO, which appears to be the one used in Brown’s method [6], by p. = ro/2, as was discussed by Scattergood and Bacon [ 151. In this context, one can view equation (1) as the minimization condition for the total elastic energy, and the procedure for computing the self stress will depend upon how a non-singular dislocation elastic self energy was evaluated; there is no a priori method for evaluating this energy within the framework of the conventional linear elastic theory. In a very recent investigation, Bamett and Gavazza [I63 have derived a procedure for calculating the self stress which is completely self consistent with minimizing the energy where the dislocation elastic self energy is calculated by excluding a tube of material of radius r. surrounding the dislocation. (Brown’s procedure, which appears at first sight to be based on the same energy evaluation, gives rise to certain inconsistencies as Brown himself pointed out) [6]. Their procedure gives a result rather similar to Brown’s [6] original formulation and, furthermore, the procedure can be applied to media of arbitrary anisotropy. The procedure used in this investigation for obtaining the self stress must be considered ad hoc in the case of anisotropy because there is no representation for the elastic self energy of the dislocation in terms of interaction energies. Our motivation for extending this procedure to the anisotropic case was its usefulness in developing a simple formula for the circular arc stress field; no such simple result could be obtained using Brown’s prescription. The various procedures for obtaining the self stress, when applied as in this investigation to dislocations of arbitrary shape in anisotropic media, will differ primarily in the non-logarithmic terms in the circular arc formula (the choice of p. = ro/2 makes the logarithmic terms identical), and these tend to be the least dominant terms in actual numerical applications. The “long-range” part of the self stress, as given by the summation over finite straight segments, stands essen-

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tially unchanged and it is the long-range part that is important insofar as it can modify the conventional line tension results, i.e. line tension is rather analogous to the contribution to the self stress from the circular arc since it depends strongly on the curvature of the line. As far as the present authors are concerned, the procedure developed by Bamett and Gavazza stands as the only completely correct one, and one which is developed from the most reasonable assumptions which can be made as far as calculating a non-singular dislocation self energy in linear elastic theory is concerned. We are now in the process of analyzing the differences between the various procedures, and this will be reported on later. As we have already noted, the basic differences will be manifested in the non-logarithmic terms in the circular arc contribution to the self stress, and we do not anticipate that this will have an appreciable effect on numerical results. As was discussed by Scattergood and Bacon [lj], the evaluation of equation (2) for finite segments can be made by expanding T: and its derivatives as Fourier series in terms of the angle between the Burgers vector and the infinite straight line, and then the basic input data for the calculations becomes the Fourier coefficients in these expansions; these coefficients will of course depend upon the elastic constants. In the case considered here. we require the coefficients for the partial dislocations. and these have been given in a recent tabulation [la]. For convenience, we have summarized the tinite segment stress field formulae in the Appendix. along with all additional information required for carrying out the present calculations. In order to obtain the static equilibrium configuration of the node, as determined by equation (1). we have used a relaxation technique which adjusts the lengths of the segments joining the points shown in Fig. 2. The included angle x between adjacent seg ments (Fig. 2) is held constant during the relaxation. and this has the desirable feature of distributing the points in proportion to curvature along the line so that the shape is represented every-where with good accuracy. In practice., we actually adjust the lengths of the chords joining the midpoints of the segments, and if the superscript n denotes values at the nth iteration, the relaxation procedure for the chord lengths h is h”’ ‘ =

h”

1 f

c

“: + ‘i, i’

(3)

,

where C is a constant of order one. and for each chord the corresponding value of r, is the value at the common endpoint of the two segments whose midpoints define the chord. The node branch shape is constructed from the chord lengths by geometry. Equation (3). which is to be applied at all points in Fig. 2, is in essence a procedure which scales the chord lengths (expanding or contracting the node) in proportion to the unbalanced force at each point.

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method is efficient in practice and converges without problems due to oscillatory shape changes. The + sign in equation (3) is chosen to be positive if the net force tends to expand the node. A solution for the equilibrium configuration of a node is considered to be achieved when the left-hand side of equation (1) is equal to zero to within 2”; accuracy of the current value of 7. At the point where the curved node branch meets the straight sidearm asymptote (points S and 7’ in Fig. 2), a somewhat larger deviation than the 2SLaccuracy was tolerated; this was usually of the order of jp/,. The reason for this is that the sidearm asymp tote represents a condition that is achieved only at quite large distances from the node center, and, in order to maintain fast convergence with a reasonable number of segments to represent the node branch. we constrain the junctions at the sidearms to lie at somewhat smaller distances from the center than are actually required. The deviations at the junctions could be reduced by adding more segments, and also by appropriately scaling the angles x (Fig. 2). but this had virtually no effect on the overall node geometry; therefore, the larger deviation at the junction was permitted in the interest of computing efficiency. We carried out the calculations with twenty segments on each branch. and with a sidearm asymptote segment that was always the order of 105W or more in length. With this numerical representation, it took about l-2 set of computing time to obtain an equilibrium configuration on a CDC Model 7600 machine. At fixed 7, and with r. = b, the results obtained here could be checked against the results obtained by Brown [6] for the case of isotropy; Brown’s selfstress method was different from the one used here. Values of CVfor 0 = 0’ and 90’ were in agreement, the differences being within the overall accuracy of the calculations. (Brown’s reported W values for other values of 0 may be somewhat in error due to an assumed false symmetry.) The largest discrepancy occurred for the case where the Poisson ratio v = l/2,

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709

This rela..ation

13’ 30’ 45’ 60. 75. 90’ 8-

Fig. 4. Node size It. as a function of orientation 0 for silver at various values of the stacking-fault energy ;’ and core radius rO.The upper to lower groups, containing four curves each. corresoond to values of IO’yiGb = 0.5, 1. 2. and 4, respkctive1y’t.G is defined by equation (1) in the text), and within each group, the value of r,,,b = 0.5, 1, 2, and 4, for the upper to lower curves, respectively. Note the logarithmic scale. and this probably reflects the fact that Brown’s selfstress evaluation for the circular arc type of segment differs from ours in the non-logarithmic terms, although these terms are not the dominant contribution and appear to have only a small influence on the final results.

RESULTS

/j A/ -----X 60° Fig.

:, / ,’

_> ‘ii’,, ,‘,

9o”

\

3. Equilibrium node shapes for silver with lO’y/Gb = Z and ro/b = 1 (G is defined by equation (4) in the text). The orientation 0 of each shape is indicated. The first-order change in node shape with 7 appears to be simply a scaling of the shape.

Calculations of the node configurations were carried out over a range of 7, r,, and 8 values for silver, nickel and isotropy at Poisson ratio values of 0 5 v I 4. To facilitate the tabulation and comparison of results, we shall define here the shear modulus G and the Poisson ratio v by the following relations, (4a)

(W

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where E,” and E: are, respectively, the prelogarithmic factors for the self energy of infinite straight screw and edge partial dislocations in the given crystal. Note that these definitions reduce to the usual values of G and v for the isotropic case. We adopt these definitions because they have proved useful in dealing with anisotropy in other instances [IO, 151, although for the node problem there is no simple heuristic argument to show that this is the appropriate choice. It does turn out that normalizations based on this choice of G and v are useful for the node problem The range of the parameters used for the calculations was as follows: r,/b = 0.5, 1, 2, and 4; lo3 y/Gb = 0.5, 1, 2,4; 8 = O”-90” in steps of 1.5’. Typical node shapes obtained by the methods described earlier are shown in Fig. 3 for the case of silver. As 8 changes from 0” to 90”, the node size W decreases whereas the spacing D of the infinite straight sidearms increases; this behavior is in agreement with Brown’s [6] earlier observations and appears to hold for all cases. For silver and nickel, W does not decrease monotonically as 6 goes from 0” to 90”, but instead displays a maximum at 8 z 30”, as is shown for silver in Fig. 4. A pronounced maximum of this kind did not appear for any of the isotropic cases over the entire range 0 4 v s 4. Increasing r,,, with all else held constant, leads to a decrease in w and the decrease is greatest for 0 = O”,as can be seen from Fig. 4. In the following Section, we shall develop a compact set of equations to describe the relationship between ct: 0, r. and y, and we shall attempt to rationalize the differences between anisotropic and isotropic behavior. DISCUSSION

OF RESULTS

We shall develop the form of the relationship between IY 8, r. and y on the basis of the following heuristic argument. At the equilibrium node configur-

420, 400

:

1”

I

320

“0

300

‘e

I-

/30’

so*

280 260

73.

240

SO*

220 200

mo---.‘~’ IO'

102

’

“““’

’

IO'

““‘I

IO'

Fig. 6. Node size values iV for silver plotted in accord with equation (6) of the text. The value of 0 appropriate to each straight line is given.

ation, the force 7 at any point P on the node branch must equal the dislocation self force br, We assume that br, consists of two additive contributions. First, there is the familiar line tension term which is in effect a result of the dislocation interaction in the immediate neighborhood of P, and this has the form R-L(Em + d2Ex/d~2)ln(X/ro) where R is the radius of curvature at P, EP + dZE”/db2 is the prelogarithmic factor in the line tension of an infinite straight dislocation lying along the tangent at P (E” is the prelogarithmic factor, or angular part of the self energy and 4 is the angle between the Burgers vector and the dislocation line), and X is an outer cutoff distance for the self energy. Second, there is a nonlocal, or long-range contribution to br, from the more distant elements of the branch containing P and from the remaining branches in the 3-fold configuration. We assume this long-range contribution to have the form Eo/d where E. is an angular term, dependent on the Burgers vector orientation 4, and d is some interaction distance. We then obtain from equation (1)

R, d and X will be functions of position along the node branch. To describe the situation, we shall take an average of equation (5) over the node branch, thereby replacing all of the various factors by their average values. We furthermore assume that the averages of R, d and X are proportional to w i.e. these values scale with the node size. On this basis, and after normalizing by Gb2, which should be a common factor in the right-hand side of equation (5), we get the result, The distances

Fig. S. Schematic representation of the approximation suggested in the text for the long-range interaction force at P (P is supposed to denote the average position along a node branch). Dislocations 1,2 and 3 are infinite straight partial dislocations in a symmetric 3-fold configuration centered on the actual node as shown. The spacing of the dislocations is not important here since only the angular dependence of the interaction force is considered.

YW Gb’=

.-l In 141 + B, I.0

(6)

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eFig. 7. Plot of the values of A’and B, used in equation (6) of the text, as a function of orientation for silver (solid) and isotropy (dashed). The values of G and Y for the isotropic case are determined by the partial dislocation in silver as defined by equation (4) in the text.

where A and L?are constants, and for a given crystal should depend only on the node orientation 0 because all size dependence is included in the terms which were assumed to be proportional to !t: We expect A to vary with 0 in a fashion similar to the dependence of the line tension. factor E” + d2Ex/d@ on the Burgers vector orientation angle $J. On the other hand, B should vary with 0 in a fashion which reflects the Burgers vector orientation dependence of the long-range interaction forces acting on a node branch. This dependence is difficult to predict, but we could argue that its form is given by the geometry shown in Fig. 5. The long-range interaction force at P is approximated by the infinite straight dislocations shown: dislocation 1 represents the interactions due to the branch containing P while dislocations 2 and 3 represent the interactions due to the remaining branches in the 3-fold configuration. For the symmetry of the arrangement shown in Fig. 5, the interaction force at P, due to the straight dislocations, has an angular dependence on 4 which is identical with the dependence of the energy factor E” on the Burgers vector orientation; therefore, we could expect B to display the same form of dependence on 8 as that of E” on 4. Figure 6 shows the W values for silver, obtained over the range of 0, :i and r,, values used here, plotted according to equation (6). This equation gives a surprisingly good fit, and the maximum deviation does not exceed the 2% accuracy maintained for the calculations. Equation (6) gave a similarly good fit for the nickel results and for all isotropic cases over the range 0 I v < i (some additional runs for copper also

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711

showed the same agreement). The values of B and A in equation (6) were obtained as functions of 0 from the intercepts and slopes, respectively. of the least squares lines drawn through the points shown in Fig. 6. The values of A and B as functions of 0 are shown in Fig. 7 for silver and for isotropy. Figure 8 shows the values of E” + d’E”jd4’ and E” as functions of C#J, the angle between the line and its Burgers vector for silver and the isotropic case. It is evident from these two figures that, qualitatively. the values of B and A as functions of 8 follow the same trend as the values of E” and E' + d’EX db’ as functions of 4, as was expected on the basis of the arguments given above. Note that. in contrast to E’, the B values for silver fall off somewhat as 0 approaches 90’, i.e. as the center part of the node branch becomes pure edge. This is probably a manifestation of the fact that the line tension for silver becomes negative in the edge orientation (4 = 90’ in Fig. 7). leading to a sharp curvature in the center of the node branch along with some concomitant perturbation in the long-range interaction force. Otherwise, the overall trends for B and A vs 8 are similar to those of E” and E” + d2E”/d#2 vs 4. In view of the trends exhibited by the curves shown in Figs. 7 and 8, it is reasonable to try and develop some empirical relationships for B and .A as functions of 0 in terms of the Burgers vector orientation dependence of Ef and E” + d’E”/d@ for the infinite straight line. Relative to solving the general node

Fig. 8. The prelogarithmic energy factor E” and line tension factor E” + d’E:“/dr#? of infinite straight Shockley partial dislocations as a function of Burgers vector orientation r#~for silver (solid) and isotropy (dashed). The values of G and Y for the isotropic case are determined by the partial dislocation in silver as defined by equation (4) in the text. All values are in units of Gb’,‘4n.

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problem the latter dependence is quite easy to establish, and thus such relationships would provide both a compact description and useful extension of the results obtained in this investigation. We have carried out such an analysis for the results obtained in this investigation and have developed the following relations, B(8) = 1.588 B,, + 0.910 B,

cos 20 -

8.94 Bz

required in equation (8) can be obtained in a straightforward manner by Green’s function methods described elsewhere [13]. A new tabulation of these coefficients for the common f.c.c. crystals is given in Table 1. In the case of isotropy, the leading coefficients are, 2-b

B”= Sic(I

cos

48 (7a)

B,

-

2 - Y

A0 = ___ Sn(1 - v)’

A(6) = 0.208 A, + 0.128 A, cos 26 + 0.232 A2 cos 48,

=

_

v)’

\’ %I( 1 - V)

‘A1 = j,. Srr(l - v)

(9a)

(W

(7b) and all other coefficients are identically zero. It is worthwhile to illustrate the use of equations (6 and 7) as working equations in analyzing node measurements. If we a priori assume a value for ro, a typical choice being r. = b. then a value of 7 is obtained from a set of K f_7values by the direct solution of equations (6 and 7). Alternatively. if we have at least two sets of IY 0 values, then in principle one should be able to determine both 7 and ro, as was mentioned earlier. For each 0 value, the appropriate constants A and B must first be determined from equation (7). Next, assume a range of r. values and plot 7 vs ln(r,) from equation (6), using the value of W appropriate to that 0. This results in straight line plots for each set of w 8 values, and the intersection point of the lines establishes the proper 7 and r0 values (due to scatter in the measured values, there could be up to n(n - 1)/2 intersection points for II sets of w 0 values and the best values of ‘J and r. would have to be obtained from, say. the centroid of the intersection points). We have attempted to apply this procedure to the node measurements made on silver by Ruff and Ives [ 171; however, there is a large amount of scatter in the data. and in fact

where the coefficients A,,, Al, etc. are to be obtained from the following expansions for the prelogarithmic energy and line tension factors for infinite straight Shockley partial dislocation in the given crystal E’/Gb’

= BO + B, cos2$ -t- B,cos44

(W

(Ei+$)/Gb’=AO+A,cos2q5+A,cos4& (Sb) where 4 denotes the angle between the line direction and the Burgers vector. The overall accuracy of equation (6). when used in conjunction with the A and B values given by equation (7), was about 6-7x for silver, nickel and all isotropic cases with 0 < v I ) (the fit for the isotropic cases was generally better than this). From these results, we would conclude that equations (6 and 7) will completely describe the anisotropic, symmetric 3-fold node problem to an accuracy of about 5-10% for all f.c.c. crystals. Before illustrating the use of these equations in analyzing node measurements, and also discussing the deviations of the anisotropic results from the isotropic approximation, we should mention that the Fourier coefficients

Table 1. Values of the Fourier coefficients to be used in Fourier expansions for the prelogarithmic line tension and energy factors of infinite straight Shockley partial dislocations in various f.c.c. crystals. The coefficients given are the ones to be used in the expansion lo3 (Er + d2E”/d@)/Gb2 = Aa + t,., d, cos(2n4). where 4 is the angle between the Burgers vector and the line. The nth row of Table 1 gives the value of A.-,, e.g. the first row gives Ao, the second A,, etc. and using coefficients up to the maximum given for each crystal will result in an expanston accurate to O.S’$gor better. The coefficients in the expansion 103E’/GbZ = B, + &_, B, cos(2n1$) are obtained from the A. coefficients by the relations B,, = A,, and B, = A./(1 - 4n2), and this expansion will be about an order of magnitude more accurate than the E” + d’E”/d+’ expansion. For completeness, the last two rows of Table 1 give. respectively, the values of G (units of 10” dyn/cm?) and c as defined by equation (4) in the text. The single crystal elastic constants used to obtain the results shown in Table 1 were the room temperature values obtained from the following sources: Ir-MacFarlane er al. [18], Pd-Rayne [19], Pt-MacFarlane et al. [ZO], all others from Hirth and Lothe [Zl]. Item

Ag

Al

Au

cu

Ir

Ni

Pb

Pd

Pt

A. n=O

108.552 1 92.2275 2 - 16.8004 3 -9.14599 4 -23.3158 1.60417 6 -2.14179 7

8 G Y

2.78 0.434

101.843 115.219 66.9201 112.369 -0.333969 - 18.5875 -0.309214 -5.03441 -0.645821 -27.6059 2.23536 -2.73656 0.625565 2.59 0.359

2.59 0.484

106.026 85.0794 - 17.6487 - 12.4630 -23.6078 1.59854 - 2.34463 4.42 0.413

94.5664 100.341 45.4398 65.3808 -1.10816 -8.62528 - 1.70980 -8.81289 -2.17634 -12.6152 0.434567 -0.667932 20.9 0.275

8.07 0.351

113.935 111.997 -30.2241 -11.3821 -40.3571 4.87831 -6.33546 0.906158 -0.522035 0.789 0.482

110.369 97.1276 - 15.3474 -1.46172 -22.1018 1.45229 - 1.83536 4.46 0.4-t7

108.695 88.2347 - 2.70384 - 1.34110 -4.72409

6.10 0.425

SCATTERGOOD

h?jD

BACON:

this scatter is so large that we could not obtain a reliable estimate of both 7 and ro. Since the dependence on r0 is not strong, to the extent that it enters logarithmically in equation (6), we have instead adopted the conventional procedure of assuming r,, = b and then obtained the best value of 7. Figure 9 shows the measured W values as a function of 8 along with the curve obtained from equations (6 and 7); the value y = 23.5 erg/m* was determined from equations (6 and 7) and was based on the best least squares fit to the data points. It compares with a value of 16.3 + 1.7 erg/cm’ obtained by Cockayne et al. [22] using the weak-beam technique of observing dissociated dislocations and a value for the force between infinite, straight partials given by anisotropic elasticity. Even after allowing for the uncertainty in our value, which from the data of Fig. 9 is approx. + 4 erg,cm’ near 0 = 0 and & 7 erg/cm’ near Q = 30’. it is difficult to reconcile these two values. One possibility is that the choice r0 = b is an underestimate, but even with a choice of r. = 46, the least squares fit to the data of Fig. 9 is only reduced to 18.3 erg/cm’. Another possibility is that dislocation core displacements may modify the value of y obtained from the relatively small partial separations observed for straight dislocations, as was discussed in detail for copper by Cockayne and Vitek [23]. The node method may suffer less from this core effect. However. the scatter in the points shown in Fig. 9 seems unreasonably large in view of the expected uncertainties in the measurements, and this large scatter is very likely evidence for some other varying factor,

60

I ZIP

30 0 20

-

RUFF

AND

IVES

CALCULATED

Fig. 9. Values of the node size W as a function of node orientation for silver. The points are values measured by Ruff and Ives [17]. The curve drawn through the data points was obtained by a least squares fit using equations (6) and (7) in the text. The least squares fit gave a value of 7 = 0.00507Gb = 23.5 erg/cm’ with G = 2.78 x 10” dyn/cm’ (see Table 1).

STACKING

713

FAULT NODES

0

1

I

I

300

90. 8

.-do*

Fig. 10. Node size W as a function of orientation 0 for silver (solid) and isotropy (dashed). The stacking-fault energy 7 has a value of 10’yiGb = 4; note that this is close to the value determined for silver (see Fig. 9). but is not equal to it. The values of G and v for the isotropic case are determined by the partial dislocation in silver as defined by equation (4) in the text. Note that the curve for silver differs slightly in detail from that given in Fig. 9 because the latter curve was obtained from the empirical equations developed in the text.

such as deviation from symmetry or another boundary condition; however, the precise cause is not known. In order to apply fully the results of the present investigation. measurements with less scatter will be needed, or the source of the scatter identified and taken into account in the calculations, e.g. an asymmetry in the node configurations. Finally, we shall consider the effects of including elastic anisotropy into the node calculations. Figure 10 shows the variation of W with 0 for silver and for the isotropic case with G and v chosen according to equation (4) for the partial dislocation in silver. As one can see from these figures, the isotropic curve is a good overall fit to the anisotropic curve, and the maximum deviation between the curves is the order of IO-15%. The same kind of agreement between the isotropic and anisotropic results was found for nickel. It therefore appears that a good overall isotropic representation of the anisotropic node problem is obtained with the Poisson ratio and shear modulus chosen according to the prescription given in equation (4). Since the choice of these values is not a priori selfevident, one result of the present investigation has been to establish that the definitions given in equation (4) are in fact appropriate, and also to show that the deviations from isotropy, using the definitions in equation (4), tend to be the order of 10-15”~ at the maximum. Table 1 gives a summary of G and v values.

71-1

SCATTERGOOD

&\LUD BACON:

The shape of the curves shown in Fig. 10 can be rationalized in terms of the line tension factors E” + d’E’“/d@ shown in Fig. 8. In the first approximation. the dislocation self force br, is the line tension, although this ignores the effect of all the longrange interaction force. When the line tension over the center portion of the node branch is large, the dislocation strongly resists inward bending and therefore the node size W is large. This effect is clearly evident from Figs. 8 and 10, noting that the value of the line tension for I$ = 0 gives the value appropriate for the center portion of the node branch (Fig. 2). The maximum in the W vs 6 curve at f3A 30 coincides rather well with the maximum in the E” + d’E”/d@ vs 4 curve at 4 A 30”, and there is also some indication at a slight “shoulder” in the W vs 0 curve at 13A 75’, which corresponds to the shoulder in the line tension at 4 + 70” (these effects in the LVvs 19 curve become more noticeable as 7 is decreased). The isotropic curves are also in qualitative agreement, except that PVstays relatively constant for increasing 0 up to about 60”, whereas the line tension value decreases monotonically through this range. In the case of nickel, the E” + d*E”/d# curve resembles that of silver, displaying a similar maximum at 4 + 30”, and the Wvs 0 curve shows a similar trend. As a final remark, it is interesting to note that for the perfect (undissociated) dislocations in silver and nickel, the line tension does not display a maximum value, greater than either the screw 0; edge value, as 4 goes from 0” to 90”.

SUMMARY The static equilibrium configurations of symmetric 3-fold stacking fault nodes in f.c.c. crystals were determined bv. a self-stress method. based on linear elasticity, which takes into account the combined effects of dislocation interaction and elastic anisotropy of the crystal. Specific calculations were made for silver, nickel and isotropy, and the analysis of the results should be generally applicable to f.c.c. crystals with arbitrary anisotropy. The results of the investigation support the following: (1) In principle, both the stacking-fault energy y and the dislocation core radius r. can be obtained from node measurements. Equations (6 and 7) provide a compact description of the general solution to the node problem for the anisotropic case. An analysis of node measurements for silver obtained by Ruff and Ives [ 173, which shows too much scatter for a reliable estimate of both r. and y, gave a value of y = 23.5 erg/cm’ for silver, assuming r. = b. This value is higher than that obtained from the observation of

a single dislocation by the weak-beam technique, and the origin of this discrepancy is unclear. (2) With an appropriate choice of the Poisson ratio and shear modulus, as defined in equation (4), an isotropic

approximation

to the anisotropic

node

solu-

STACKING

FAULT NODES

tions can be made which deviates by no more than about IO-15%. (3) The general trend of the node size vs node orientation relationship for the anisotropic cases can be reationalized in terms of the line tension for the partial dislocation in the given crystal. In particular, for silver and nickel, the node size goes through a maximum as a function of orientation and this reflects the fact that the line tension goes through a similar maximum as a function of the Burgers vector orientation. (4) The general methods developed for the solution of the anisotropic node problem can be applied to any dislocation problem in which the equilibrium configurations are determined by the in-plane tractions. These methods permit the combined effects of dislocation interaction and elastic anisotropy to be taken into account in computer-based calculations without excessive cost in terms of computer time. of us (R.O.S.) acknowledges the S.R.C. for the award of a Senior Visiting Fellowship held

Acknowledgemenrs-One

at the University of Liverpool, where most of the work was carried out, and continued support under the auspices of the U.S. Energy Research and Development Administration.

REFERENCES 1. Whelan M. J., Proc. R. Sot. (Land.) A, 249. 114 (1959). 2. Stobbs W. M. and Sworn C. H., Phil. Maq. 24, 1365 (1971). 3. Howie A. and Swann, P. R. Phi/. Mug. 6. 1215 (1961). 4. Siems R., DeLavignette P. and Amelinckx S., Z. Phys. 165, 502 (1961). 5. Thornton P. R., Mitchell T. E. and Hirsch P. B., Phil. Mug. 7, 1349 (1962). 6. Brown L. M., Phil. Mug. 10, 441 (1964). 7. Brown L. M. and ThaIen A. R., Discussions Faraday Sot. 38, 35 (1964). 8. Jessang T., Stowell M. J., Hirth J. P. and Lothe J., Acta Mer. 13, 279 (1965). 9. Petterson B., Phys. Status Solidi (b) 54. 253 (1972). IO. Scatteraood R. 0. and Bacon D. J.. Phvs. . Sratus Solidi (a) 25,1395 (1974). Il. Brown L. M.. Phil. Mug. 15, 363 (1967). 12. Indenbom V. L. and Orlov S. S., Sot. Phys. JETP Lett. 6, 274 (1968); Sou. Phys. Cryst. 12, 849 (1968). 13. Bacon D. J. and Scattergood R. 0.. J. Phxs. F: Metal Phys. 4, 2126 (1974). 14. Bacon D. J. and Scattergood R. O., J. Ph_vs.F : Metal Phys. 5, 193 (1975). 15. Scattergood R. 0. and Bacon D. J., Phil. &fag. 31, 179 (1975). 16. Barnett D. M. and Gavazza S. D., private communication, to be published. 17. Ruff A. W., Jr. and Ives L. K., Can. J. Phvs. 45, 787 (1967). The values of node size as a function of orientation for silver were kindly supplied by Dr. Ruff. 18. MacFarlane R. E.. Rayne J. A. and Jones C. K., Phys. Lert. 20, 234 (1966). 19. Rayne J. A., Phys. Reo. 118, 1545 (1960). 20. MacFarlane R. E., Rayne J. A. and Jones C. K., Phys. Lert. 18, 91 (1965). 21. Hirth J. P. and Lothe J., Theory of Dislocarions, p. 762. McGraw-Hill, New York (1968).

SCATTERGOOD

.ASD

BACON

STACKING

FAULT NODES

715

12. Cockayne D. J. H,, Jenkins St. L. and Ray I. L. F.. Phil. Msg. 24, 1353 (1971). 23. Cockayne D. J. H. and Vitek V.. Ph~s. Srurrrs Solidi (b) 65. 751 (1974).

APPENDIX 1. Sness field due to finite circular arc Xs explained in the text. a point P where the field is to be evaluated lies on the arc. and consequently the procedure outlined in the text must be used to avoid singular behavior. Since we require the shear stress resolved along the Burgers vector direction of the arc segment itself, the angular factor TF in equation (2) is v = X”. where E” is the prelogarithmic (angular) part of the self energy of an infinite straight dislocation: in other words, the in-plane component of traction along the Burgers vector direction of an infinite straight dislocation. due to that dislocation, has exactly the angular dependence of the self energy of the dislocation. The general form of the circular arc stress field has been given by Scattergood and Bacon [15]; however. for the sake of completeness. we shall give the appropriate form here for the (partial) dislocation in f.c.c. crystals. This is obtained by expanding the Tp + d’T:/do’ term in equation (2) as a Fourier series, and then integrating over the circular arc segment shown in Fig. 11. The resolved shear stress rars at P is obtained as

5

2

+

x

n=,

(-lI)“+‘A,cos2n~

f t,

Fig. 11. Schematic of the finite dislocation segment geometry. The various parameters are defined as shown with the positive sense for angles indicated by single-ended arrows.

1 2k - 1

X [ 2cos(,k - 1,1” - cos(2k - 1): 2

-

cos(2k -

1)

31-

2 ;

(-

2. Stress field due ro finite straight segment

I).+ ’

“=,

x A,sinZnfl

1

i sin(2k - 1) 2 t=, 2k - 1 2

+ sin(2k - 1) 9

11

(AlI

and we have. d’E’

E’ + 1

d+-

E

= A,, +

7

“7,

A,cos2n&

where we define. (-l)“A,cosZn~.

(A3)

The quantities appearing in equation (Al) are defined in Fig. 11. The coefficients AO, A, ,..., & are determined by the expansion in equation (AZ) where I#J is the angle between the Burgers vector and the line direction: a tabulation of these coefficients for partial dislocations in f.c.c. crystals is given in Table 1, they were obtained by methods described elsewhere [13]. The term in equation (A3) is the line tension factor of a straight dislocation lying along the tangent to the circular arc at P. In our expression for equation (Al), the B, coefficients given by Scattergood and Bacon [15] do not appear. because for f.c.c. symmetry these are identically zero. The angle x,, 3 pa/R = r,/ZR in equation (Al) is always positive, and the magnitudes of x, and x2 must be greater than x0, i.e. the arc must have a finite length.

We wish to obtain the resolved shear stress due to a finite straight segment at a point P not contained on the segment. In this case, we require the complete in-plane traction vector for the finite straight segment. because the Burgers vector direction of the dislocation element containing P may not be the same as that of the segment; in other words, the factors T; do not reduce to the energy factors E” when we consider the interaction force due to the node branches which do not contain the point P. (The subscript i = 1, 2 or 3 refers to a set of rectangular cartesian axes, with the one labelled i = 3 taken normal to the plane of the dislocation.) Once we have the in-plane traction vector at a point P, we can resolve this vector along the Burgers vector direction of the element containing P to obtain the resolved shear stress at P. Integration by parts of equation (2) over a finite straight segment immediately gives the traction vector Fs for the segment, -

T,“(t?)cos(B- x) +

dT;(P) 7

? sin(0 - xl

I

(A-t)

where the various quantities are as indicated in Fig. 11 for the straight segment. The angular part of the traction 7: of an infinite straight dislocation, lying along the 0 direction and having the same Burgers vector as the segment, and its derivative dTcr:‘dO with respect to Burgers vector orientation are all that is required for the complete evaluation of equation (A4). The determination of 7:: and its derivative can be obtained by methods described elsewhere [14], and a complete tabulation of the results for partial dislocations in f.c.c. crystals has been given by Bacon and Scattergood [14]. If bi is the Burgers vector of the dislocation element containing P, then the resolved

716

SCATTERGOOD

k\lD

BACON:

shear stress rrcp at P due to the finite straight segment is given by r .rl = $ ,i,

T1’*b,.

3. Coordinate systems and sign contentions We consider all directed angles shown in Fig. 11 for the circular arc to be signed, with counterclockwise being the positive sense. The coordinate system and sign convention for the traction vector components 78 follows that of Bacon and Scattergood [14]. Note that for consistency, the perpendicular distance d used by Bacon and Scatter-

STACKING

FAULT NODES

good [14] should be signed such that if the d direction (Fig. 11) is obtained from the segment direction by a counterclockwise rotation, ti is positive. otherwise it is negative; alternatively. in the coordinates used by these authors, the sign of d is positive in the direction of n x t and is negative otherwise. This ensures that the sign of the tractions for an infinite straight dislocation remain invariant when the directions b and t are both reversed. When all these conventions are followed. the siens of r,,, and r_ have the following interpretation: the-line direction of the element containing P, the slip plane normal (out of the paper) and the direction for positive force bs, = bs,,; + br,,# at P form a right-handed, orthogonal set.