Dislocations in Lithium Fluoride Crystals

Dislocations in Lithium Fluoride Crystals J. J. GILMAN Brown University, Providence, Rhode Island AND

W. G. JOHNSTON General Electric Research Laboratory, Schenectady, New York

I. Introduction.. . . 11. Methods for Obs

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Dislocations. ..

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3. X-Ray Microscopy.. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5. Source. . . . 6. Dislocation

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10. A Measurement of Dislocation Line Energy.. .

13. Stress Dependence of Dislocation Velocity

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.................................. 176 15. Pinning of Dislocations.. VI. Origins of Dislocations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 177 16. The First Glide Dislocations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17. Observations of the Multiplication of Glide Dislocations. . . . . . . . . . . . . . 180 18. The Multiplication Mechanism. . . . . . . . . . . . . . . . . . . . . . . . . 19. Growth of Dislocation Structure During Defor ........ VII. Theoretical Considerations of Dislocation Mobility. . . . . . . 20. Drag of Trails.. . ....................................... 194 21. Phonon Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 22. Nucleat.ion Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 23. The Temperatur endence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 24. Impurity Effects .....................

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28. Structural Changes During Irradiation. . . .

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33. Weakening Caused by Dislocations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34. Surface Energy Measurements. . . . .......... XIII. Dielectric Breakdown . . . . . . . . . . . . . . . ................... XIV. Mechanism of Etch-Pit Formation. . . . . 35. Effect of Dislocations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................... 36. Inhibition of Step Motion. . . 37. Effect of Segregated Impuriti ......... ..

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1. Introduction

Except for an accident, the present article could not have been written. The accident happened in a series of experiments and was observed by persons prepared to take advantage of it, but nevertheless if it had not occurred the story could not be told. It had unusual consequences in that before it occurred almost no knowledge of the plastic behavior of lithium fluoride existed, while a few years after the accident, more knowledge exists about the mechanical behavior of this crystal than almost any other, and the dislocation theory of crystal plasticity has been given a firm experimental basis. In addition, much has been learned about the role of dislocations in chemical dissolution, in fracture phenomena, and in dielectric breakdown. In 1955 it had been reported that dislocations can be observed by a photoelastic technique. We thought that this technique might be a p plicable to LiF and late in 1955 we began exploratory experiments on this substance. It seemed to us that in order for the photoelastic method to work the dislocation density should be quite low. Therefore we tried to obtain a preliminary estimate of the dislocation density in a LiF crystal by dipping it into the solution C P 4 * with the hope that dislocation etch pits would develop. The first time the etching was attempted, some rather illdefined pits were formed so the crystal was dipped in the CP-4 solution This solution, which had been developed in semiconductor technology, consists of 50 vol parts HNO, (conc.); 30 pts HF (conc.); 30 pts glacial acetic (HAc) acid; 1 pt Br.

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once more. This time, dislocation etch pits of beautiful uniformity and symmetry resulted, and it was realized immediately that a powerful tool for studying dislocations in LiF had been discovered. Later, the significance of the initial failure followed by success was learned. Iron ions of a critical concentration in the etching solution are needed for successful results with LiF. The iron tongs that were used to hold the crystal had provided the iron ions, but the concentration only became adequate during the second dip. After the dislocation etchant had been discovered, it was found that LiF crystals are remarkably well suited to studies of crystal plasticity. They’ have a simple crystal structure (the NaCl structure), are transparent, very resistant to atmospheric corrosion, and are easily cleaved into rectangular specimens along (100) planes. They are moderately plastic a t room temperature, yet creep resistant, and are relatively inexpensive since they are grown commercially for use as optical windows and prisms. Our studies were mainly devoted to three topics, namely: the role of dislocations in crystal plasticity; the fracture of crystals; and the theory of the etching process as a technique for observing dislocations. Some other topics, such as radiation damage and dielectric breakdown, also were investigated, but much less extensively. Several techniques have been developed in recent years for observing dislocations in crystals, but the etch-pit method has the unique advantage of being applicable to quantitative measurements of dislocation velocities. In crystals of low initial dislocation density, isolated dislocations can be etched before and after they undergo a motion. Then, if the time duration of the applied stress is known, the average velocities of the dislocations are determined. Measurements of dislocation velocities have shown that the motions are resisted by viscous forces and it is primarily these viscous forces that determine the yield stress of a crystal. Also, it has been shown that dislocation motions account fully and quantitatively for macroscopic crystal plasticity. Etch-pit studies have also been used to advantage to find out how dislocations originate in stressed crystals, and how they multiply. A finding of considerable significance is that multiple-cross-glide is often a more important mode of dislocation multiplication than the classical Frank-Read process. The studies have revealed important new information concerning the fracture process. Thus it was found that dislocations are often nucleated just in front of a propagating crack. Also, it has been established that there is a close connection between cleavage steps on fractured surfaces and screw dislocations. The remarkable geometric regularities of dislocation etch pits in lithium fluoride crystals stimulated us to study the etching process. It waa found

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that the production of pits depends very sensitively on the adsorption of highly specific ions (Fe3+and A13+are best) on the surfaces of the crystals. This adsorption inhibits the motion of kinks in surface steps without markedly influencing the nucleation of surface steps at places where dislocation lines intersect a surface. The highlights of our studies that are mentioned above indicate the scope of the work that has been done on LiF crystals. Now the methods that were used, and the results, will be discussed in more detail. II. Methods for Obserying Dislocations

Etch-pit formation,' internal colloid precipitation,2 and X-ray microscopy3 have been used for direct observations of dislocations in LiF crystals. The advantages and shortcomings of these techniques are such that they are complementary to one another rather than competitive. The etching method has been most widely used for detailed studies of dislocation behavior in LiF so it will be discussed more thoroughly than the others. Under favorable conditions, the different methods give identical results. This strengthens the conclusion that each may be used to reveal dislocation lines. 1. ETCHING

Dislocation etch pits. Etching is the most convenient method for revealing dislocations in many crystals. Shockley and Read4 first suggested that etch pits corresponding to individual dislocations might be seen microscopically. Successful, deliberate attempts to etch the surface terminal points of dislocations were made by Horn5 and Gevers and c o - w o r k e r ~ ~ ~ ~ who produced etch pits within growth spirals in silicon carbide. Soon thereafter, Vogel and co-workerss showed that in Ge the spacing of etch pits in grain boundaries corresponded to predictions based on the theoretical model of Burgersg and Bragg.'" This strongly supported the idea that etch pits can be formed preferentially at dislocation terminals. J. J. Gilman and W. G . Johnston, J . Appl. Phys. 27, 1018 (1956).

* J. Washburn and J. Nadeau, A& Met. 6.665

(1958).

J. B. Newkirk, Trans AZME 215 483 (1959). W. Shockley and W. T. Read, Phys. Rev. 75. 692 (1949). F. H. Horn, Phil. Mag. 43, 1210 (1952). R. Gevers, S. Arnelinckx, and W. Dekeyser, Naturwissenschaften 39, 448 (1952). R.. Gevers, Nature 171, 171 (1953). 8 F.L. Vogel, W. G. Pfann, H. E. Corey, and E. E. Thomas, Phys. Rev. 90,489 (1953). J. M. Burgers, Proc. Phys. SOC.London B52. 23 (1940). lo W. L. Bragg, Discussion to ref. 9, Proc. Phys. SOC. London B52, 54 (1940).

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A freshly cleaved (100) surface of a LiF crystal, when dipped into pure water, becomes highly polished and slowly dissolves at a rate of about 250 A/sec.” The exact dissolution rate depends on the temperature of the solution and its agitation, but the rate given is typical for room temperature and moderate agitation. If small amounts of FeF3 are then added to the water, small regularly shaped pits will appear on the surface. As the concentration of FeF3 is increased, at constant etching time, the pits become more sharply defined and decrease in size as shown in Fig. 1. The rate of dissolution at the centers of the pits remains at about 250 A/sec, but the rate of dissolution for the rest of the surface becomes too small to be measured.* Thus the FeF3 “inhibits” the dissolution process. At an optimum concentration of FeF3, the pits have the form of square pyramids with the straight edges of their bases parallel to (100) directions in the crystal surfaces. Contrary to the impression given by a photograph, the sides of the pyramids do not lie parallel to planes with rational indices. The angles that they make with the surface increase as the concentration of inhibiting ion increases. The,se angles have been measured by cutting sections through some pits, and by microinterferornetry.l2 They increase to a maximum of about 10 degrees in the series shown in Fig. 1. Similar results have been obtained by adding FeFa to concentrated mixtures of hydrofluoric and acetic acids; in fact, these were the essential constituents of the reagent that first produced dislocation etch pits in LiF.’ The behavior of the strongly acidic etchant is compared with that of the nearly neutral one in Fig. 2.t In order to make the figure,13fresh glide dislocations were put into a LiF crystal by means of localized deformation. The crystal had been slowly cooled from the solidification temperature so it contained many grown-in dislocations as well as the dislocations introduced by plastic deformation. The neutral etchant (Fig. 2a) distinguishes between fresh and grown-in dislocations quite clearly by forming sharp, pyramidal pits at the fresh ones; but only shallow, indistinct pits at the grown-in ones. Next, the crystal was polished chemically in order to erase the first set of etch pits. Then it was immersed in the

* For a { l o o } surface on a spherical crystal, the rate changes from

-900 A/sec to -90 A/sec upon the addition of FeF3.11 t The composition of the two etchants is as follows: neutral etch: Hs0 plus 2 ppm FeFa (time: 60 sec); acidic etch: 1 pt conc. HF, 1 pt glacial acetic acid, and 1 vol % conc. HF saturated with FeFI (time: 20 sec). The etchants are conveniently prepared by dropping water saturated with FeFa into distilled water, or HF saturated with FeF3 into the mixed acids. I ‘ J . J. Gilman, W. G. Johnston, and G. W. Sears, J . Appl. Phys. 29, 747 (1Y58). M. B. Ives and J. P. Hirth, J . Chem. Phys. 33, 517 (1960). J. J. Gilman and W. G . Johnston, “Dislocations and Mechanical Preperties of Crystals,” p. 116. Wiley, New York, 1957.

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FIG.1. Dissolution of LiF in water containing various amounts of FeFa. (a) Distilled mole frac. FeFs; (d) 8.0 X 10-a water; (b) 0.5 X 10-6 mole frac. FeF1; (c) 2.0 X mole frac. FeF,. Magnification: 1OOOX.

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FIG.2. Comparison of the action of the neutral and acidic etchants on the same field.

The crystal was chemically polished between etching treatments. The two horizontal rows of pits were produced by plastic deformation. 400X. (100}surface photographed.

(a) Neutral etchant-fresh dislocations more distinct than grown-in ones. Pit edges parallel to (100). (b) Acidic etchant-no distinction between fresh and grown-in dislocations. Pit edges parallel to (110).

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acidic etchant which reveals fresh and grown-in dislocations with equal distinction (Fig. 2b). The ability of the neutral etchant to discriminate between grown-in and fresh dislocations apparently is due to the effects of impurities that segregate a t dislocations in slowly cooled crystals. This ability has been very useful because it permits ready observation of the behavior of glide dislocations against a complex background of grown-in dislocations. The pits produced by the two reagents have different crystallographic orientations. Pit edges parallel to (100) directions in the (100) crystal surface are produced by the neutral etchant, whereas the pit edges are parallel to (110) directions in the case of the acidic etchant. The details of the chemical reactions that are responsible for this difference are not known. There is substantial evidence that pits like those shown in Figs. 1 and 2 mark the sites where dislocations emerge from a crystal surface, and that pits are produced at all places where dislocations intersect a (100) surface.13 It is evident that the formation of the pits is associated with defects of some kind in the crystals because: (a) their concentration ( #/cm2) is independent of the time and temperature of the etching; (b) they are arrayed in nonrandom patterns typical for dislocations; ( c ) when a surface is polished and then re-etched they reappear at the same positions, which indicates that the defects are extended along lines. The linearity of the defects is proved by the exact match, as shown in Fig. 3, of the patterns of pits on two surfaces that are formed by cleaving a crystal. This could only happen if linear defects were cut into pairs by the cleavage crack. The only linear defect in a crystal that should move easily under the action of a shear stress is a dislocation. Therefore, if a defect is shown to be both linear and readily moved by a shear stress it is almost certainly a dislocation. The first part of this test has been described above and evidence for the second part is provided in Fig. 4. A surface is shown that was initially etched to produce an etch pit a t the emergence site of a presumed dislocation. Then the crystal was subject to a shear stress, of about 600 gm/mm2, followed by re-etching. After the second etching treatment, it was found that the original etch pit had changed its shape (becoming flat-bottomed) and a new pointed etch pit appeared nearby it. On the basis of evidence that is discussed in detail elsewhere,13 it has been concluded that the second etch pit represents the final position of a dislocation that was initially located a t the position of the flat-bottomed pit. Evidence that all of the dislocations in LiF crystals are revealed by the etch-pit method has been provided by X-ray studies. A study of the over-all extinction of X-rays was made by the present authors,13and more detailed studies have been made by Newkirk13and by Yoshimatsu and

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FIG.3. Etch pita on matched cleavage faces of an “as-grown” crystal. Acidic etchant. (a) Normal printing. (b) Printing reversed to simplify comparison.

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FIG.4. Movement of an isolated dislocation in a lithium fluoride crystal. The crystal surface waa etched, then a stress was applied to move the dislocation, and the surface was re-etched. The initial dislocation position is shown by the flat pit; the final position by the pointed pit. 8OOX.

Kohra.’* These authors have shown that individual dislocations in the LiF lattice reduce X-ray extinction near themselves, causing intensity contrast in a diffracted X-ray beam. This allows the positions of dislocations to be seen in X-ray microphotographs. Furthermore, they have shown that an etch pit forms a t each and every place where an “X-ray dislocation” emerges from a crystal surface. Since the etching process not only reveals the surface positions of dislocations, but also is sensitive to motion of these dislocations, it is a powerful tool for dislocation studies. The virtue of this tool which is unique a t the present time is that it allows the motions of isolated individual d i e locations to bk studied in large specimen crystals. Because of their large size, accurately measured stresses and temperatures can be readily applied to these specimens. Similarly, the method can be used for studying annealing processes. For example, the pointed pits in Fig. 5 represent the final positions of dislocations that moved during an annealing treatment from the initial positions indicated by the flat-bottomed pits. The etch pits are always arrayed as would be expected for dislocations on the basis of theory and other observations. In undeformed crystals many pits are found in subgrain boundaries, and many others are distributed randomly within the subgrains. In deformed crystals pits are found on the traces of the known (110) glide planes. The changes of the pit positions during heat treatment and under an applied stress is in accordance with the expected behavior of dislocations; thus the inferred dislocation motion under stress is along the (110) glide M. Yoshimatsu and K. Kohra, J . Phys. Soc. Japan 15, 1760 (1960).

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Fig. 5. Dislocation movements during annealing of an “m-grown” LiF crystal. Crystal was etched 40 sec, annealed 16 hr at 400°C, re-etched 20 sec. W X .

planes. Also the apparent interaction of the defects is as expected for dislocations; for example, they pile up a t obstacles16and form polygonal walls. Although nearly all etch pits in LiF crystals correspond to dislocations, some correspond to other types of defects. Filamentary precipitates cause pyramidal etch pits to form,lBas do the tracks of fission fragments.” J. J. Gilman and W. G. Johnston, Discussion, “Dislocations and Mechanical Properties of Crystals,” p. 357. Wiley, New York, 1957. I e J . J. Gilman, J . Ap pl . Phys. 30, 1584 (1959). D. A. Young, Nature 182, 375 (1958). 16

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In addition to the two etchants discussed above other etchants for LiF have been reported18-z0but they have not been investigated as thoroughly. A very useful supplement to etching is chemical polishing, which consists of dissolving away surface layers in such a way as to produce a fresh, smooth surface. It is convenient for removing accidently damaged surface layers, or to erase etch-pit patterns. The three-dimensional dislocation structure of a crystal can be deduced, by alternate polishing and etching. A chemical polish for LiF consists of 1.5% conc. ammonium hydroxide in distilled water.13 Ammonium hydroxide apparently removes, by complex formation, residual impurities from,the distilled water that tend to cause etch-pit formation. The solution should be kept below 25"C, and with agitation it will dissolve about 1-1.5 p/min from the crystal surface. Fluoboric acid polishes much more rapidly (20-50 p/min), but it leaves a rougher surface.z1 2. DECORATION

Dislocations are sometimes revealed in transparent crystals by precipitates which form on them preferentially; each dislocation is then delineated by a thread-like precipitate, or a chain of precipitate particles, which can be seen with an optical microscope. Hedges and Mitchellz2first decorated dislocations in AgBr by causing photolytic silver to deposit on them. Amelinckx, Dekeyser, and co-workers at the University of Ghent have used the decoration technique extensively to study dislocation configurations in ionic crystals. This work has been reviewed by A m e l i n c k ~ . ~ ~ Washburn and h;adeau2 decorated dislocations in LiF crystals grown from melts which contained either Ca or Mg. The dislocations that were present at high temperatures were decorated with CaFzor MgFz precipitate particles during cooling (Fig. 6). The decoration method provides a view of the three-dimensional dislocation structure in a large crystal. However, the procedure usually requires that crystals be heated to quite high temperatures so that only the dislocation structure of an annealed crystal can be observed. 19

L. S. Birks and It. T. Seal, J . A p p l . Phvs. 28, 541 (1957). A. A. Urusovskaia, Kristallograjiyu 3, 726 (1958); English translation Soviet Phys.

20

A. P. Kapust,in,Kristallograjiya 4,265 (1959); English translation Soviet Phys. Cryst. 4,

Cryst. 3 , 731 (1958).

247 (1959).

J. Nadeau, The influence of point defects on the mechanical properties of lithium fluoride, Thesis, Univ. of California, Berkeley, California, 1960. zz J. M. Hedges and J. W. Mitchell, Phil. Mag. 44, 223 (1953). 23 S. Amelinckx, Nuouo cimento 7 (Suppl.), 569 (1958).

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FIG.6. Decorated dislocations in a LiF single crystal observed by ultramicroscopy. The crystal was grown from a melt containing 0.5% Mg. 4OOX. (Courtesy of Washburn and Nadeau.*’)

3. X-RAY MICROSCOPY

Crystals of LiF that have been carefully grown are relatively perfect, containing only about lo5 dislocations/cm2. Diffracted X-ray beams are highly extinguished by such crystals, but the diffracted X-ray intensity increases about thirteen-fold if the crystals are deformed.13 Newkirk13 using the Berg-Barrett reflection technique, has shown that the reduction in extinction is especially marked near dislocation lines. Figure 7 shows one of his photographs of a LiF crystal, in which the X-ray images of dislocations can be correlated with etch pits. The reflection method is restricted to regions relatively near the surface of the crystal, although the low X-ray absorption coefficient of LiF allows dislocations to be seen to a depth of over 50 p . Lang24 has developed a transmission technique which is not restricted to the surface layers, but it has not yet given as good resolution as the reflection technique. “ A . R. Lang, J . A p p l . Phys. 30, 1748 (1959).

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FIG.7. Dislocations revealed in LiF by the Berg-Barrett X-ray reflection technique. Enlarged X-ray photograph at left, and photomicrograph of etched surface at right. 120X. (Courtesy of Newkirk13.)

A useful feature of the X-ray technique is that it can be used to determine the Burgers vector of a dislocation line. The change of X-ray extinction is quite anisotropic about a dislocation line, being greatest when the beam'is reflected from planes that lie perpendicular to the Burgers vector (the distorted planes). There is little change for diffracting planes lying parallel to the Burgers vector. Therefore, by examining the contrasts within X-ray beams that have been diffracted from various planes at a dislocation, one can identify the Burgers ~ e c t o r . ~ J ' The reflection technique has recently been used by Schultz and Washburn2s to study effects of heat treatment on the dislocation distribution near the surface of LiF crystals. J. M. Schultz and J. Waahburn, J . Appl. Phys. 31, 1800 (1960).

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4. COMPARISON OF METHODS

Of the three methods, etching offers highest resolution. Etch pits can be resolved at separations of -1 p with an optical microscope, and at less than 0.05 p on a surface replica in an electron microscope.26It appears from published photographs that the limit of resolution of the decoration method is about 0.5 p. The X-ray transmission method can resolve dislocations about 5 p apart,Z6and the reflection method is useful down to about 2 p.27 For studies of the movements of dislocations, the etching method is the faste,st and most convenient by far, but the X-ray method could be adapted to the purpose. The decoration method is not useful for studying motions because the decoration process completely immobilizes the dislocations. All three methods can be used to study the three-dimensional structures of crystals, but the decoration method is somewhat more convenient and effective than the others. 111. Some Properties of Lithium Fluoride Crystals

5. SOURCE Lithium fluoride crystals are grown commercially for use in optical equipment, and all of our experiments have been done with crystals grown by the Harshaw Chemical Company. They were grown in the form of 6-in. diameter ingots that were cooled very slowly from the melting temperature. Our crystals, as supplied, had the form of blocks measuring 13 in. X 13 in. X 3 in. Each block was cleaved into a number of specimens. Cleavage occurs on the (100) planes, so rectangular rods can be obtained conveniently. Each large crystal that was received yielded numerous specimens having the same mechanical properties. However, there were significant differences between various large crystals that were obtained over a period of five years. In 1956 the as-received crystals were relatively hard, having a yield stress in compression of over 1OOO gm/mmZ. In succeeding years, the as-received crystals became softer and softer, and some of the crystals now have a yield stress of less than 120 gm/mm2. While the hard crystals obtained earlier cleaved well, it is unfortunately very difficult to obtain good cleavage in the soft crystals obtained recently. The change in the as-received crystals has been due to the elimination of divalent metal impurities; particularly magnesium. The harder crystals 26

A. M. Turkalo and U'. G. Johnston, Unpublished observations. J. M. Lommel, Private communication, 1959.

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contained over 80 ppm of magnesium, while the recent soft crystals contain no detectable impurities. The impure crystals responded markedly to heat treatment"; they were hard after being slowly cooled from above 30O0C, and soft after being air-cooled from the same temperature. Thus they seem to have undergone precipitation hardening during the slow cooling. 6. DISLOCATIONS IN AS-GROWNCRYSTALS

The dislocation contents of the as-received LiF crystals ranged from about 3 X 104/cm2in better crystals, to about 106/cm2 in poorer ones. About half the dislocations were in subgrain boundaries, and half in the interior of the subgrains. The average diameter of subgrains ranged from about 1 to 5 mm, individual dislocations in crystals with large subgrains and low dislocation content could be observed readily. The dislocation structures of the as-grown crystals did not depend on the impurity content. This observation is consistent with the findings of Washburn and Nadeau2 who studied the formation of substructure during growth of LiF crystals. As a crystal was being withdrawn from a molten bath, they added 0.5% CaF2 to the melt and continued withdrawing the crystals. The dislocation structure did not change upon addition of the impurity. They concluded that the dislocation content was insensitive to growth rate, impurities, and temperature gradient in the solid. Most of the dislocations in melt-grown crystals result from accidental, thermal, and mechanical stresses. The dislocations in annealed and slowly cooled crystals are immobile; apparently because they are strongly pinned by i r n p ~ r i t i e s . Observa'~~~~ tions on hundreds of specimens indicate that there is little, if any, stressinduced motion of annealed dislocations. Therefore, such dislocations play only a passive role in the plastic behavior of the crystals.

7. GLIDESYSTEMS Lithium fluoride crystals prefer t o glide plastically on { 110) planes and in (110) directions, as do most alkali halide crystals that have the rocksalt structure. The (110) directions in the rocksalt structure are the only low-index directions that lie parallel to rows of ions of the same charge sign, and glide can occur in these directions without juxtaposing ions of the same sign. Also, the smallest crystallographic repeat distance lies in the (110) direction. Since the energy of a dislocation line is proportional to the square of the repeat distance, the energy of a (110) dislocation is lower than that of other dislocations in this structure. W. G. Johnston and J. J. Gilman, J . A p p l . Phys. 30, 129 (1959).

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For a given glide direction, metal crystals prefer to glide on whatever crystal planes have the widest atomic spacing. This is not the case for LiF which prefers to glide on { 110) planes even though its (100) planes are more widely spaced. At temperatures above about 300°C glide does occur but the preference for { 110) planes is strong a t readily on { 100) lower temperatures (Fig. 8 ) . It is thought that the (110) planes are pre6

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J. J. Gilman, A& Met. 7, 608 (1959).

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ferred over (100) planes because the core energy of a dislocation is considerably smaller for the { l l O ) planesao; and particularly because the variation of the core energy with position of a dislocation on its glide plane is smaller. 8. CROSS-GLIDE OF SCREWDISLOCATIONS Although dislocations normally move on the { 110) planes of LiF, under some conditions it is observed that screw dislocations move on { 100) planes, or along curved trajectories. This indicates that screw dislocations can move without great difficulty on planes other than { 110), and therefore tend to move on the planes of maximum applied shear tress.'^.^^ This cross-glide of screw dislocations often occurs in the vicinity of glide bands where there is a sharp gradient in plastic strain. The driving force for the cross-glide seems to be residual stresses (Fig. 9). Screw dislocations normally move along a ( 110) plane like ABC in the direction BC. Cross-glide means that the ends of the dislocations move onto (001) planes in the direction GH. Figure 9b shows a photomicrograph of a crystal that was etched immediately after it was deformed, and it may be seen that some of the dislocations cross-glided out of the primary glide plane as the etching proceeded. The observation that screw dislocations with a [ l i O ] Burgers vector move relatively easily on { 001 ) planes, whereas large scale glide occurs only with difficulty on (001) planes, implies that it is edge dislocations which behave differently on the two planes and thereby cause anisotropic plastic behavior. IV. Behavior of Individual Dislocations

The general purpose of our work was to gain a better understanding of the complex way in which crystals deform plastically. The first phase was to define and study the elementary parts of the process in some detail. Then came the task of combining the elements uniquely, in order to obtain a satisfactory description of the complete process. In this section, the present state of knowledge of the motions, origins, energies, etc., of individual dislocations in LiF crystals will be introduced. The work on the origins and mobilities of dislocations will then be discussed in detail; and, in Part IX, our present thoughts about the way in which the elementary properties interact to yield a stress-strain curve will be described. *I

H. B. Huntington, J. E. Dickey, and R. Thomson, Phys. Rev. 100. 1117 (1955). W. G. Johnston and J. J. Gilman, J . A p p l . Phys. 31, 632 (1960).

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a

FIG.9. Cross-glide of (110) ( 1 l O ) dislocations. (a) Sketch of two dislocations,the ends of which have moved by cross-glide out of the normal glide plane (ABC) along the (001) planes (DEF) and (GHI). (b) Photograph of LiF crystal showing motion towards the right of the ends of screw dislocations out of their (110) glide plane. 500X.

9. MOBILITYOF FRESHLY INTRODUCED DISLOCATIONS

Fresh dislocations must be employed for studying the behavior of glide dislocations during plastic deformation, because the grown-in or aged dislocations are usually securely pinned by impurities and take no part in the deformation. Isolated, fresh dislocations can be put into a crystal in a very simple manner.la A steel ball is rolled across the crystal, and wherever

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FIG.10. Method for putting single dislocation half-loops into crystals. (a) Clusters of dislocations produced by rolling a steel ball across a crystal. (b) Single dislocation halfloops that remain when -15 p have been dissolved from the surface of a crystal like that shown at (a).

the stress is concentrated at a piece of dust or an asperity, a cluster or “rosette” of dislocations forms (Fig. 10a). Next the surface of the crystal is dissolved away until only the deepest lying half-loop in each rosette remains. Each pair of pits in Fig. 10b corresponds to a single dislocation half-loop that was produced in this manner. Dislocation loops of two kinds can be produced by the method above (Fig. 11). For one type of loop the dislocations that intersect the surface are parallel to the Burgers vector and are therefore screw dislocations. For the other type the dislocations that intersect the surface lie perpendicular to the Burgers vector, i.e., they are edge dislocations.

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

167

FIG. 11. Schematic drawing of two dislocation loops that intersect the surface of a cleaved crystal. Screw components emerge at points S, and edge components emerge at points E. Cleavage plane, ( 1 0 0 ) ; glide plane, (110);glide direction, (1iO).

Dislocation loops such as those shown in Figs. 10 and 11 can be enlarged by applying a sufficiently large tensile stress to the specimen, or collapsed by applying compressive stress. When the loops are enlarged to more than 100 p diameter, the dislocations that intersect the surface behave as essentially independent glide dislocations. There is a minimum resolved shear stress T P , which can be determined* experimentally to an accuracy of &lo%, below which neither the screw or edge dislocations can be moved. For stresses greater than TP, the dislocation velocity is very sensitive t o the applied stress. Details of this stress dependence will be discussed in Part V. Here we merely emphasize that there appears to be a plastic resistance to dislocation motion, 7p, that must be overcome in order to move a dislocation. The plastic resistance changes, for example, with purity of the crystal, heat treatment,, radiation damage, temperature, etc. 10. A MEASUREMENT OF DISLOCATION LINE ENERGY

The line tension of small dislocation half-loops, such as those shown in Fig. lob, tends to collapse them so they move out of the crystal. This tendency toward collapse is opposed by the resistance to dislocation motion. The resistance TP can be measured by determining the smallest applied stress a t which a relatively straight dislocation will move. Then if D, is the smallest diameter loop that is stable in the crystal, the line tension (energy per unit length) is: (10.1) where b is the Burgers vector. In one instance the observed values were'*: TP = 4 X lo7 dynes/cm2, and D, = 4, so with Z, = 2.85 X 10-8cm, the

168

J. J. GILMAN AND W. G . JOHNSTON

line tension was:

T

=

2.3 X

erg/cm

which is in fair agreement with a theoretical estimates2: Ttheor =

3.02 X lod4erg/cm.

This technique for measuring dislocation line tensions should be quite useful, but it has not been fully explored. The theoretical expression used for calculating the line energy is:

Ttheor

=

Rb2In

(E,

(10.2)

+

where the first term is the elastic strain energy, and C is the core energy; R is an average elastic constant calculated from anisotropic elasticity theory and equals 4.4 X 1O1O dynes/cm2, for LiF32.The core radius T O was taken to be 2b. An estimate of the average core energy for the dislocation half-loop was based on the calculation of Huntington and associates30 for NaCl, and yielded a value of C = 1.9 X erg/cm. The relaxations at the surface of the crystal were not taken fully into account. FROM 11. GLIDEBANDFORMATION

A

DISLOCATION HALF-LOOP

The single dislocation half-loops shown in Fig. 10b produce a surprising effect when a crystal is stressed enough to cause plastic deformation. Figure 12a shows an etched crystal containing several fresh dislocation loops that are identified by pairs of etch pits. This crystal was bent, then examined with oblique illumination (Fig. 12b). It may be seen that a glide band passes through the position of each half-loop. A light etching (Fig. 12c) produced many thousands or tens of thousands of dislocations within each glide band. In other words, the original half-loops have multiplied profusely into swarms of new loops. The mechanism of this process will be discussed shortly. 12. IMPLICATIONS REGARDING CRYSTAL PLASTICITY The observations described above have certain clear implications regarding the mechanism of plastic deformation. The observation that grown-in dislocations play little or no part in the deformation means that theories which interpret the flow stress in terms of the stress required to activate Frank-Read sources of the grown-in network are not valid. The formation of glide bands from single dislocation loops implies that dis8

J. J. Gilman, Trans. A I M E 209, 449 (1957).

DISLOCATIONS IN LITHIUM FLUORIDE CRYSTALS

169

FIG.12. Growth of glide bands from dislocation half-loops. Strain rate =Z X 10-6/sec. (a) Single dislocation half-loops at the surface of a LiF single crystal. (b) Same crystal after bending shows glide bands paasing through five of the half-loops. (c) A light etch reveals many dislocations in each glide band.

170

J. J. GILMAN AND W. G. JOHNSTON I

N

o

I 3.01 E

A

Y

- AS-GROWN (SLOWLY COOLED) - COOLED FROY 500% IN 10 YIN.

z 25

a

1.5

i O . 5 t / " ,

'0

I

I

,

I

0.5 LO 1.5 2.0 2.5 3.0 3.5 4.0 CRITICAL BENDING YOYENT Kp ,mm 2 2xsocTloNyowws

4.5

~

FIG.13. Stress required to move a fresh dislocation vs. yield stress of various crystals.

location multiplication is not the limiting process in determining the flow stress. The observation that there is a resistance to motion of individual glide dislocations suggests that it is this plastic resistance which determines flow stress. This view is strengthened by the close correspondence between the plastic resistance encountered by individual dislocations and the flow stress that is measured in bending (Fig. 13). The plastic resistance is small in a soft crystal, and larger in harder crystals. Some of the questions that these implications raise are: (a) What determines the mobility of a dislocation? (b) Where do the very first glide dislocations come from when a crystal is stressed? (c) What is the microscopic mechanism whereby a single mobile dislocation forms a glide band? (d) How can we understand the macroscopic deformation of large crystals in terms of these microscopic dislocation properties? These questions will be considered, in turn, in the next few sections. V. Dislocation Mobilities

13. STRESSDEPENDENCE OF DISLOCATION VELOCITY

The essence of a method that was used to measure dislocation velocities was shown in Fig. 4.Shear stresses were applied to specimens for a known length of time and etching was used to find the "before and after" positions of dislocation lines. The distance between the before and after positions was measured with a microscope, and the average velocity was calculated. The time required for acceleration of the dislocations is short

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

171

compared with the time of stress application, so steady-state velocities are obtained by this method. This has been demonstrated experimentally by showing that the effect of one long stress pulse is the same as that of 10 pulses each one-tenth as long. We applied stress pulses of various lengths (from lop6to lo5 sec) and various magnitudes (from 0.2 to 20.0 kg/mm2) to LiF crystals and found that dislocations moved at velocities ranging from about 10 to about 10l2 atom distances per sec.28Data showing the behavior for this enormous range of velocities are presented in Fig. 14. Several features of the behavior are of interest. First, a certain critical stress (yield stress) must be applied to a.crystal before dislocations will move through it at an observable rate.

APPUED SHEAR STRESS (KGIMM~)

Fig. 14. Dislocation velocity vs. resolved shear stress in a LiF crystal. The yield stress and critical resolved shear stress (C.R.S.S.) indicated on the abscissa were measured in compression.

172

J. J. GILMAN AND W. G. JOHNSTON

Second, the velocity of the motion increases very rapidly with small increases of applied stress. Third, as the dislocation velocities increase toward the velocity of sound, increasingly larger stresses must be applied. The velocity of sound appears to be an upper limiting value of dislocation velocity. This is predicted by theory33because dislocations are elastic disturbances and hence should not be able to move faster than elastic waves under normal conditions. The two separate curves show the difference in velocities of the screw and edge components of dislocation loops. At the lower stresses the edge components move 50 times as fast as the screw components. At higher stresses the curves converge, consistent with the theoretical expectation that the two velocities have almost the same upper limit.a3 There is a threshold stress TP, below which dislocations will not move, and we have called this the “plastic limit” of the crystal. The existence of such a threshold stress is not obvious from Fig. 14. Each point in the figure is the average of the velocities of 40 to 100 dislocations that actually moved during a test. At stresses above 800 gm/mm2 it was found that all fresh dislocations moved during each test, at 700 gm/mm2 about 20% of the fresh dislocations moved, a t 650 gm/mm2 only about 2 or 3% moved, and a t 600 gm/mm2 there was no movement as great as 1 p in los sec. Thus as the applied stress is decreased, the average velocity of the moving dislocations decreases, and in addition some of the dislocations become stuck. Below TP (which can be measured to &lo% accuracy) none of the dislocations move. As was mentioned previously, the velocities shown in Fig. 14 are steadystate velocities. Therefore, all of the work that is being done by the driving forces on the dislocations is being dissipated as the dislocations move. The force on a dislocation due to an applied shear stress, T , is T b , where b is the Burgers vector, so the work that is done when the dislocation moves forward by one Burgers vector is ~ b 2(ergs/cm of dislocation). Thus, except for a constant factor, the curves of Fig. 14 may be considered to be plots of the energies dissipated by moving dislocations at various velocities. The data in Fig. 14 are for a moderately hard crystal of LiF that was slowly cooled from the solidification temperature. This typical behavior can be modified by heat treatment, a change in the test temperature, or radiation damage (Fig. 15). The primary effect of changed conditions is to shift the dislocation velocity-stress curves uniformly along the stress coordinate. Thus, the various treatments do not affect the behavior a t just one velocity, or for a narrow range of velocities, but they affect the motion over a wide range of velocities. It is through increasing the rate of Ia

J. D. Eshelby, Proc. Roy. Soc. A62, 307 (1949).

DISLOCATIONS IN LITHIUM FLUORIDE CRYSTALS

173

1,

E

a Y

,j IRRADIATED NEUTRON 300.K

Id'

'Oat

SOFTENED

' 100

17.K

AS-GROWN

(SCREW CONPONENT)

1000

10.000

APPLIED SHEAR STRESS (G/mm2)

Fig. 15. Velocities of the screw components of dislocation loops versus applied shear stress for crystals of various hardnesses.

energy dissipation by dislocations that are moving at a given velocity, that radiation damage, for example, increases the hardness of a LiF crystal. This rules out many possible models of the behavior, especially static pinning models. Although individual dislocations in LiF crystals can be made to move at very high velocities, in normal plastic flow the velocities which occur are not much greater than 1 cm/sec. This is indicated by the position of the critical resolved shear stress (C.R.S.S. on the abscissa of Fig. 14) as measured in a compression test at a strain-rate of about 5 X lod/sec for a specimen of the crystal that was used in the velocity measurements.

174

J. J. GILMAN A N D W. G. JOHNSTON

Extreme strain rates during macroscopic plastic flow would be required to produce high dislocation velocities. The only analytic equation that has been found to satisfactorily describe the experimental data of Figs. 14 and 15 has the form34: (13.1)

= Vg-AIr

where V is the dislocation velocity, 7 is the applied shear stress, and V , and A are constants. This equation describes the stress dependence of the velocity very well indeed as may be seen from Fig. 16.

0

2

4

I

6

(&

8

I 108)

10

1.2

FIG.16. Screw dislocation velocities in a LiF crystal. a4

J. J. Gilman, Australian J . Phys. 13, 327 (1960).

1.4

16

DISLOCATIONS IN LITHIUM FLUORIDE CRYSTALS

175

14. TEMPERATURE DEPENDENCE OF DISLOCATION VELOCITY The temperature dependence of dislocation velocity at constant stress is shown in Fig. 17 for a soft crystal and a hard crystal. To within the ac-

. ' 1 ' 6 1I

3.2

3.4

36

,

3.0

91 x

4.0

4.2

4.4

4.6

lo3 (DEG-')

FIG.17. Temperature dependence of dislocation velocity in an as-grown crystal and in a crystal softened by heat treatment.

curacy of the data the velocity vanes exponentially with 1/T. The slopes of the two curves are the same, and correspond to an activation energy of 0.7 ev, if the velocity is assumed to be thermally activated. However, it is not known to what process such an activation energy corresponds. It should be noted that the low-temperature behavior of LiF shows that the temperature dependence of the dislocation velocities cannot be described by an Arrhenius equation over a wide temperature range, 80 any

176

J. J. OILMAN AND W. G. JOHNSTON

interpretation of Fig. 17 must be made with caution. The temperature dependence of the yield stress of LiF crystals is best described by an equation of the formla: (14.1)

T y = T&-KT

where T is the absolute temperature and 7 0 and K are constants. This equation fits the data of Fig. 17 quite well and may describe the behavior at low temperatures as well. 15.

PINNING OF

DISLOCATIONS

Although static pinning of dislocations does not determine the over-all hardness of an impure or radiationdamaged LiF crystal, such pinning does occur. Pinning has been demonstrated directlyZ8by putting isolated dislocation loops into crystals and then subjecting the crystals to a lowtemperature aging treatment (Fig. 18). Two specimens, the first of which contained fresh dislocations, were aged a t 100°C and cooled to room temperature. Fresh dislocations were then introduced into the second specimen, and the stress required to move the fresh dislocations, and the lowest stress a t which any of the aged dislocations moved were measured. The aging treatment did not change the plastic resistance to motion of the dis-

:t P

w

FRESH DISLOCATIONS

I

0.2

k

0

0

2

4

6

8

AGING TIME AT 100.C

D

1

2

1

4

1

6

18

(HRS.1

FIG.18. Dislocation pinning by low-temperature aging.

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

177

locations that were introduced after aging. However, the aged dislocations were strongly pinned after short aging periods. After an aged dislocation became unpinned by application of a sufficiently high stress, it continued to move at the stress at which the fresh dislocations moved. At the same time that aged dislocations become mechanically pinned, there is a change in the appearance of their etch pits. This indicates quite clearly that something diffuses to the dislocations during the aging period and pins them. It is suspected that the particular impurity which does this is Mg. VI. Origins of Dislocations

Since the main theme of this work was the mechanism of plastic flow in crystals, we were especially concerned with how dislocations get into stressed crystals. The processes that allow nucleation of dislocations in crystals and multiplication of existing dislocations are very important parts of a description of crystal plasticity. One reason for this is that very large numbers of dislocations are needed to obtain large macroscopic strains. These large numbers are often not initially present in crystals, so they must be formed as plastic flow progresses. Another reason is that various processes should lead to different distributions of dislocations in strained crystals. These different distributions can lead to important differences in such phenomena as strain hardening and creep. 16. THE FIRSTGLIDE DISLOCATIONS

In general, dislocations can be formed in crystals in one of three ways: (1) Homogeneous nucleation

(2) Heterogeneous nucleation (3) Regenerative multiplication (a) Frank-Read sources, designated FR (b) Multiple-cross-glide, designated MCG. Prior to the work on LiF, it was generally assumed that plastic flow always begins through the operation of Frank-Read sources. However, the d i e covery that a single glide dislocation can form an entire glide band obviated the necessity for postulating the existence of Frank-had sources in undeformed crystals. Such sources are sometimes present, but they are not necessary for plastic flow. To account for the flow as it normally occurs we must understand origin of the first glide dislocations and the mechanism whereby they form glide bands. The origins of the first glide dislocations have been studied by subjecting LiF crystals to very short (-1 psec) stress pulses.16The glide bands that formed could not lengthen

178

J. J. OILMAN AND W. Q. JOHNSTON

FIG.19. Glide bands that formed at cleavage steps when a stress pulse waa applied to as-cleaved LiF. The lines that run from upper left to lower right are the cleavage steps. 8OX.

appreciably during the brief time that the stress was applied, so the point of origin of each glide band could be established. It was observed that glide bands form most easily at dislocations that have been introduced by surface damage, such as a t scratches, or points where the crystal has been touched accidentally with a hard object. Glide bands form a t these surface dislocations in the manner shown in Fig. 12. Large cleavage steps are also sources for glide dislocations, presumably because concentrated stresses form a t the bases of such steps. Figure 19 shows an as-cleaved crystal that was given a brief stress pulse, and it may be seen that numerous short glide bands are localized at cleavage steps. If surface damage and cleavage steps are removed from a crystal by dissolving away the surface, glide bands tend to form a t inclusions or precipitates (Fig. 20). It has been shown that particles a t least as small as lo00 A in size can cause glide band formation. Such precipitates a p parently can concentrate local stresses sufficiently to cause heterogeneous dislocation nucleation. Also, small particles might create small prismatic loops in their vicinity, because of thermal stresses, as observed in AgBr by Jones and Mitchell." The prismatic loops could then be a source for the first glide dislocations, but the existence of such prismatic bops in LiF has not been established. D. A. Jones and J. W. Mitchell, Phil. Mag. 3, 1 (1958).

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

179

FIQ.20. Short glide band that formed a “black” pit during a stress pulse: (a) etched 1 minute, stressed, re-etched 1 minute-acid etch; (b) after 3 minutes additional in water etch. The rectangular white marking is the ribbon-like precipitate. 1OOOX.

After special heat treatments, it is found that dislocations may move out of the grown-in network to form glide bands.16 Grown-in dislocations are usually strongly pinned, but the pinning impurities desegregate from the dislocations at high temperatures. Upon rapid cooling, the impurities remain desegregated and then the dislocations can be moved by moderately high stresses. It is not necessary for a general motion of the network dislocations to occur, because each small piece of dislocation line that breaks away can form a glide band. In the absence of surface damage, grown-in dislocations, or heterogeneities from a crystal, new dislocations can form only through homogeneous nucleation. That is, nucleation must occur in an essentially perfect crystal due to the action of a stress alone. It is expected that this process can occur only at very high stresses, of the order of G/4a, and it has been shown that stresses at least as high as about G/85 can be applied to perfect regions of LiF crystals without causing dislocation nucleation in them.I6 Furthermore, Sears36has shown that small “whiskers” of LiF can be subjected to very high stresses without occurrence of plastic flow. G . w. Sears, Phys. and Chern. Solids 6. 300 (1958).

180

J. J. OILMAN AND W.

a.

JOHNSTON

The possible ways, then, in which the first glide dislocation can arise in LiF, in order of increasing difficulty, are: (a) a t surface damage, including large cleavage steps, (b) a t inclusions or precipitates, (c) from the grown-in network, and (d) by homogeneous nucleation at very high stresses. 17. OBSERVATIONS OF

THE

MULTIPLICATION OF GLIDEDISLOCATIONS

Although heterogeneous dislocation nucleation may often start the process of plastic deformation, it does not contribute the vast bulk of the dislocations that finally participate in a large plastic deformation. All but a few of the dislocations are formed through regenerative multiplication. This process has been studied in some detail by etch pit techniques,lava1so the conditions that must be satisfied by any proposed mechanism are well known. A mechanism that satisfies these conditions will be described in the next section. The present section is devoted mainly to the observations. When single dislocation loops such as those shown in Fig. 10b are expanded slowly no new dislocations are found intersecting the surface (Fig. 21a). If the loop is expanded rapidly, many new dislocations appear (Fig. 21b). This behavior is described more quantitatively by Fig. 22, where the number of dislocations that .appear per centimeter within an

FIG.21. Multiplication of dislocationloops. (a) Loop expanded slowly in two stagesno multiplication. (b) Loop expanded rapidly with 30% greater stress-profuse multiplication. 320 X .

DISLOCATIONS IN LITHIUM FLUORIDE CRYSTALS

181

-B 10‘ \

; 3

I02

u-l

a

i

Id

18

urn

1

1.h

urn

’

~~~,!

22 ,a5 I*:o a . a 0 APPLIED SHEAR STRESS (Kq/mrn2)

2b

FIG. 22. Effect of stress on dislocation multiplication. The linear density of dislocation loops is shown as a function of stress, for glide bands that were formed by expanding single dislocation loops to a diameter of 4 . 5 mm under various applied stresses. The dislocations observed were the screw components of the loops. The yield stress of the hard crystal was 1350 gm/mm2, and that of the soft crystal was 300 gm/rnm*. The arrows on the abscissa shown the lowest stress at which dislocationswould move in each crystal.

expanding loop is plotted as a function of the stress at which the loop was expanded. Loops expand slowly when the stress exceeds the plastic resistance TP, but no multiplication is observed unless the applied stress exceeds a threshold stress (somewhat greater than TP) which causes rapid motion. Beyond the threshold stress, the rate of multiplication increases rapidly with increasing stress. The new dislocations that arise within an expanding loop do not all arise from one source, but form at many different points within the first loop. This was proved as follows: a loop was expanded slowly so that no new dislocations appeared. A pulse of higher stress was then applied so the original loop expanded slightly more, and some new dislocations appeared (Figure 23a). The resulting dislocation structure of the crystal was determined by alternate polishing and etching, and plotting a map of the small glide band (Figure 23b). From the map, it is clear that all the new loops that formed in the wake of the first one did not arise at the same point. Even if a large dislocation loop is completely collapsed and etching of the surface indicates that no dislocation is present, the crystal retains a “memory” of the dislocation that was there. Subsequent application of a stress, either tensile or compressive, will cause some new dislocation loops to appear within the area originally encompassed by the large loop that

182

J. J. GILMAN AND W. G . JOHNSTON

FIG.23. Structure of a dislocation loop after some multiplication has occurred within it. (a) Loop was expanded slowly a t a low stress and then given a short pulse of higher stress. Large pita show final position of slowly expanded loop. Small pita show dislocations after stress pulse. (b) Map of glide band shown in (a).

FIG.24. The “memory effect” in LiF. The large loop was pushed out of the crystal, and then a pulse of tensile stress waa applied. New dislocations (small pyramidal pits) formed within the former loop (marked by large flabbottomed pita). 4OOX.

DISLOCATIONS IN LITHIUM FLUORIDE CRYSTALS

183

was collapsed (Fig. 24). This memory effect implies that something is left in the wake of a dislocation as it moves through the crystal, and that this “something” can give rise to new dislocation loops. Detailed etching studies have shown that defects are indeed left in the wake of a moving screw dislocation. The defects have the form of lines that lie perpendicular to the screw dislocation (Fig. 25). The defects, which we have called “trails,” produce etch pits that are quite shallow on type I surfaces (Fig. 25), but look like dislocation etch pits where they intersect type I1 surfaces.* The trails are not simple edge dislocations, however, because they are not moved by stresses that move normal dislocations, and because they are stepped as shown in Fig. 25 rather than continuous. Similar trails have been observed in silicon by Dash.a7 The new dislocation loops that form in the wake of a moving dislocation always have the same Burgers vector as the parent dislocation (differing possibly only in sign, depending on the sign of the stress that caused the new loops to appear). The fact that glide bands become progressively wider as multiplication proceeds (Fig. 26) shows that the new dislocation loops do not, in general, form on the same plane as the parent dislocation.

--I

‘,*

SURFACE

STEPPED TRAILS

II

1 b, DISLOCATION LINE

FIG.25. Schematic representation of stepped trails within an expanded dislocation loop.

* A type I cleavage surface, as defined in Fig. 25, is parallel to the Burgers vector, while a type I1 surface is not. Edge dislocations intersect type I, and screws intersect type I1 surfaces. W. C. Dash, J . Appl. Phys. 29,705 (1958).

184

J. J. GILMAN AND W. G . JOHNSTON

FIQ.26. Lateral growth of glide bands in LiF. (a) Glide band formed at a single loop. Large pits show the position of the single loop. Small pits show that new dislocations lie on both sides of the glide plane of the original loop. (b) Widening of a glide band. Crystal waa deformed twice and etched after each deformation. WI and W I show the glide band widths after the two deformations. 300X.

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

185

The main features of the multiplication that a suitable mechanism must explain are: (a) new dislocations arise at many points in the wake of a moving dislocation; (b) the new dislocations have the same Burgers vector as the parent but are not necessarily on the same atomic plane, so that wide glide bands are formed; (c) the “memory effect,’’ which is probably associated with the defect trails left by moving screw dislocations; and (d) the rate of multiplication is very sensitive to the applied stress.

18. THE MULTIPLICATION MECHANISM

An important problem in the theory of crystal plasticity has been to understand how the large numbers of dislocations needed for large-scale plastic flow are created. The most widely accepted solution to the problem has been offered by Frank and Readas who proposed the well-known regenerative sources that bear their name. A Frank-Read source can create an unlimited number of dislocations merely through glide of the dislocations that comprise the source. Both types of sources that were originally proposed by Frank and Read have been observed in silicon,39so there is little doubt that such sources operate in crystals whenever the proper geometry and dislocation mobility exists. However, in the case of LiF and many other crystals, most of the glide dislocations are not created by conventional Frank-Read sources. One reason is that possible sources that may exist in the as-grown distribution network are securely pinned by impurities. A more important reason is that another regenerative process dominates the Frank-Read one. The process that competes with Frank-Read sources was originally proposed by K ~ e h l e r , and ~ ” by O r o ~ a n , and ‘ ~ is called the “multiple crossglide process.” In this process (Fig. 27a) a screw dislocation XY moves from left to right on plane I. In Fig. 27b, a portion of the screw dislocation OP has moved by cross-glide onto the parallel glide plane labeled 11. The jogs OM and PN are considered to be relatively immobile, so that although the horizontal segments of dislocation OP, XM, YN, continue to move on their respective planes, the ends connected to these vertical segments do not move. It is apparent from Fig. 27c that as the dislocations continue to move, there will be a Frank-Read source MN on plane I, and another Frank-Read source OP on plane 11. However, if cross-glide occurs very commonly, the Frank-Read sources may never complete an entire cycle, F. C . Frank and W. T. Read, Phys. Rev. 79, 722 (1950). W. C . Dash, J . Appl. Phys. 27, 1193 (1956). 40 J. S. Koehler, Phys. Rev. 86, 52 (1952). E. Orowan, Dislocations and mechanical properties, in “Dislocations in Metals” (M. Cohen, ed.), p. 69. AIME, New York, 1954.

186

J. J. GILMAN AND W. G. JOHNSTON

C

FIG.27. Cross-glide multiplication mechanism. ( a ) A screw dislocation XY gliding on plane I. (b) Part of the screw dislocation OP moves by cross-glide onto plane 11. (c) The dislocation segments OP, XM, and NY sweep across their respective planes.

and there will be one continuous dislocation line with sections lying on each of many parallel glide planes, and with jogs connecting the various sections. This process will always dominate the conventional Frank-Read process because the rate of multiplication is proportional to the length of screw dislocation that is present at a given time, instead of being constant. Thus, whereas the number of dislocations that comes from a Frank-Read (FR) source increases in proportion to time at constant velocity, the number of dislocations increases exponentially with time for the multiple cross-glide process. It may be seen that multiple-cross-glide (MCG) not only causes an increase in the number of dislocations on the original glide plane, but also causes glide to spread to other nearby planes. Therefore, the distributions

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

187

of dislocations in strained crystals that result from a collection of FR sources on the one hand and from the MCG process on the other are quite different. A FR source forms a set of concentric and coplanar loops, 60 its operation should result in atomically sharp glide lines due to gliding on only one plane for each source. In the limiting case of profuse cross-glide, the MCG process results in only one dislocation loop being formed each time a critical amount of cross-glide occurs. Then there are no concentric dislocation loops, and there would be monatomic glide lines clustered together to form broad diffuse glide bands. Of course, if many loops are formed each time that a dislocation cross-glides, then the MCG process will result in sets of concentric loops, and the final structure will be almost indistinguishable from the result of the operation of a collection of FR sources. Most of our observations for LiF can be accounted for by the multiple cross-glide mechanism. It would lead, for example, to the formation of a wide glide band from a single glide dislocation, and the secondary dislocations would all have the same Burgers vector as the primary dislocation. Another consequence is that trails should be observed if many small jogs form by cross-glide. A very small jog (-1-2b) might move along with the screw dislocation and create a string of vacancies or interstitials in its wake (Fig. 28a). If the jog is very large, the two dislocations move independently since their glide planes are very far apart (Fig. 28b). If the jog length is in the range of 3-300b, the jog will be too large to move along with the screw dislocation, and since the dislocations on the two planes will have strong elastic interactions they cannot pass over one another unless the applied stress is large (Fig. 28c). As a result, the jog MN will remain joined to the screw dislocation by an edge dislocation dipole MNOP. The two edge dislocations MO and NP will not be forced past one another unless the applied stress exceeds G b / 8 ~ ( 1 - v ) d , where d is the distance between planes I and 11, and v is Poisson’s ratio. The dislocation dipoles formed by intermediate size jogs have properties that are completely consistent with our observations of the trails left by screw dislocations. The smaller dipoles should not move at low to moderate stresses, but the more widely spaced dipoles should decompose into two single glide dislocations at moderate stresses. The existence of the dipoles would account for the “memory effect” in LiF. Cross-glide is very common in LiF, so a dislocation should not be able to retrace its movements exactly as it is forced out of the crystal, and the trails will not be eliminated by collapsing a surface half-loop. New trails might also be formed as the dislocation moves out of the crystal. A nonlinear distribution in the separation distance d of the dipoles might account for the observed stress sensitivity of the multiplication.

188

J. J. OILMAN AND W. O. JOHNSTON

a

/

:

.,-'L!!

C

FIG.28. Behavior of jogs with different heights on a screw dislocation. (a) Small jog is dragged along, creating point defects as i t moves. (b) Very large jog-the dislocations N Y and XM move independently. (c) Intermediate jog-the dislocations NP and MO interact and cannot pass by one another except a t a high stress.

Although cross-glide cannot occur on primary { 110) glide planes in LiF (unlike the case of fcc metals, each Burgers vector has only one primary { 110) plane associated with it), cross-glide of screw dislocations does occur fairly easily on { 001 ) planes. Further evidence of formation of large jogs is provided by recent electron microscopy studies of MgO. Since MgO and LiF have the same crystal structure, dislocation behavior in MgO is quite similar to that in LiF.Q Washburn et al." recently observed trails formed behind screw dislocations in MgO crystals and concluded that some of them consist of 43

J. Washburn and A. E. Gorum, Rev. met. 57.67 (1960). J. Washburn, G. W. Groves, A. Kelly, and G. K. Williamson, Phil.Mag.5,991 (1960).

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

189

edge dislocation dipoles. Amelinckx and Dekeyseru suggested that dislocation dipoles could break up and form a series of prismatic loops, and Washburn et al. observed that this does indeed happen in MgO. The breaking up into loops would cause the discontinuous etching effects that have been observed in LiF. Multiplication through multiple cross-glide seems to occur not only in LiF and MgO, but, according to Low and Guard,& also in Fe-3% Si crystals. The mechanism probably operates in all crystals for which glide bands form during deformation. A screw dislocation that intersects a free surface a t an acute angle can shorten its length by cross-gliding (Fig. 9). This is the cause of some of the cross-glide that leads to dislocation multiplication and may be responsible for surface environment effects.4sHowever, glide bands widen very nearly as rapidly in the interior of a crystal &s they do a t the free surface. A component of the applied stress must provide the initial driving force to make interior multiplication occur. After a glide band has been established, the internal stresses at its edges should be sufficient to cause cross-glide to widen it, so this discussion concerns only the initial multiplication process. Since only one { 110) plane is associated with a (170) Burgers vector in LiF, the stress components on other (110) planes do not play a role. The secondary glide planes (100) have no shear stresses on them when crystals are loaded parallel to a (100) direction, so these planes cannot play a role either. It is possible that the stress components on the tertiary (111) planes are the important ones. Although direct observations of glide on (111) planes in LiF have not been made, it is reasonable to assume that it occurs since it has been observed in NaCl crystals by Stepanov and B~brikov.~' Each (170) Burgers vector lies on two ( 111) planes so a dislocation could cross-glide to either side of its glide plane; e.g., a [ O l l ] dislocation could cross-glide on either the (111) or the (111) plane. For a crystal that is loaded parallel to a (100) direction, the shear stresses on the ( 111) cross-glide planes are equal, and 18% smaller than the shear stress on the primary glide plane. If it is assumed that cross-glide obeys the same stress dependence law @3q. (13.1)] as primary glide (albeit with different constants), then the forms of the multiplication rate versus stress curves of Fig. 22 can be understood. Although multiplication is not expected to occur every time that cross-glide occurs, it seems reasonable to suppose that the rate at which multiplication occurs will be proportional to the rate at which (I

S. Amelinckx and W. Dekeyser, J . Appl. Phys. 29, loo0 (1958). J. R. Low and R. W. Guard, Ada. Met. 7 , 171 (1959).

J. J. Gilman, Phil. Mag. 6 , 159 (1961). A. V. Stepanov and V. P. Bobrikov, Soviet Phys. Tech. Phys. 1, 77i (1957).

190

J. J. QILMAN AND W. Q. JOHNSTON

I

I

I

1

I

I

FIG.29. Stress dependence of dislocation multiplication in LiF crystals.

cross-glide occurs. Hence if m is the rate of multiplication, its stress dependence should be : m = moe-ljlr (18.1) by analogy with Eq. (13.1). With the exception of the high stress point, this stress dependence does seem to fit the observations as shown in Fig. 29. It is not surprising that the high stress point does not follow the rest because the dislocations become too densely packed along a glide plane at this stress level for free operation of sources. Also, many of the loops that might be formed are so small that they would collapse when the high stress was removed from the crystal. Before the evidence had been accumulated in support of multiple cross-glide as the dominant multiplication mechanism, Fisher48proposed ‘8

J. C. Fisher “Dislocations and Mechanical Properties of Crystals,” p. 513. Wiley, New York, 1957.

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

191

that condensed vacancy clusters form small prismatic dislocation loops that subsequently act as FR sources. It was further proposed that these loops would not be observable by means of etch pits. This idea was elaborated by Weertman4@ and Kuhlmann-Wilsdorf.soNow it seems clear that the prismatic loop hypothesis is unnecessary. Such dislocation sources may exist in special circumstances, but they Seem to play no major role in the plasticity of LiF.16 19. GROWTHOF DISLOCATION STRUCTURE DURING DEFORMATION

As plastic deformation proceeds in a LiF crystal the glide bands widen, and occasionally new bands appear. The glide bands eventually cover the entire crystal as shown in Fig. 30. As a glide band widens the dislocation density in its interior remains relatively constant, so it can be inferred that either no new dislocations are being formed in the interior, or dislocations are leaving the crystal as fast as they form. Washburn and Goruma have shown that in MgO the strain in the interior of a glide band remains nearly constant as the band widens. We conclude that in LiF dislocations just are not being created in the interior of glide bands during the widening. The dislocation density in the interior of glide bands reaches a saturation density because of strain-hardening there, which stops dislocation motions and hence stops multiplication. When a dislocation near the edge of the band cross-glides towards the interior, it runs into a strain hardened region. If the dislocation cross-glides away from the band, it moves into a region of low dislocation density where it can more easily move and multiply. The saturation dislocation density within a widening glide band varies linearly with the flow stress of the crystal. If a crystal is hard, either because it is impure, or has been damaged by radiation, or because it is tested at a low temperature, the saturation density will be higher than in a soft crystal. Figure 31 shows this difference in two crystals one of which was deformed at 25°C and the other a t -196"C, a t which the flow stress is five times greater than a t 25°C. Since glide bands of constant saturation density widen in proportion to the strain, the total dislocation density also increases linearly with plastic strain as shown in Fig. 32. In the early stages of deformation the dislocation density is very inhomogeneous, and the densities shown in Fig. 32 have been averaged over the entire crystal. To within the accuracy of the data, we see that for this particular crystal the dislocation density J. Weertman, J . Appl. Pjys. 28, 1068 (1957). D. Kuhlmann-Wilsdorf, Phil. Mag. 3, 125 (1958).

192

FIQ.30.Lateral growth of glide bands in a LiF crystal. The crystal waa photographed after various amounts of compression. Screw bands are shown, and they have a dislocation density of -3 X 107/cm*. 375X. (a) 0.03% compression. (b) 0.9% compression. (c) 8.0%compression.

DISLOCATIONS IN LITHIUM FLUORIDE CRYSTALS

193

FIG.31. Comparison of glide bands formed at -196" and 25°C in the same crystal. 0.36%strain in both caaes. (a) Temperature = 196°C;7, = 1100 gm/mm*, n, 3.6 X 107/cmz. (b) Temperature = 25°C; s, = 220 gm/mm2;n, 8.5 X 106/cm*.375X.

-

can be written as

n = 10%

(19.1)

where E is the plastic strain. VII. Theoretical Considerations of Dislocation Mobility

This discussion was deferred until Part VI had been presented because

it is important, in a discussion of dislocation mobilities, to appreciate that the structure of a screw dislocation cKanges as it moves. This probably ac-

194

J. J. OILMAN AND W. G. JOHNSTON

'I

5

108

FIG.32. Average etch-pit density versus plastic strain in a LiF crystal.

counts for the difference in behavior between edge and screw dislocations (Fig. 14), but since the effect does not exist for edge dislocations, it cannot be the only contribution to the viscosity. 20. DRAGOF TRAILS

The dragging stress TT produced by trails on a screw dislocation can be estimated as follows. The work done by TT for a forward motion b of a dislocation is: r T b 2 . This must equal the energy stored in the trails which is bnE, where n is the linear density of trails along a screw dislocation and E is the energy per unit length of trail. Then TT = nE/b. A lower limit for E is the energy of a single row of vacancies, say about 1.5 ev per atom spacing. In a particular crystal, n was estimated to be =3 X W/cm from a count of etch pits.31 These values give TT 7 75 gm/mm2. Thus this effect is sufficiently large to account for the difference in mobility between edge and screw dislocations.

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

195

21. PHONON VISCOSITY

The most critical aspect of the dislocation velocity curves is the unusual stress dependence. One attempt to interpret this has been made by Mason,s1 who divides the curve into three stress domains: low, medium, and high. The drag in the low stress domain (5-11 X lo7 dynes/cm*) he attributes to “cutting through a forest of cross dislocations or other obstructions,” spaced a distance 1 apart. No “forest” was present in our experiments so this means “other obstructions.” The analysis leads to a velocity that is proportional to exp - [U - ~ b ~ 1 / 2 ] / kinT the low stress domain. Mason attributes the drag in the medium stress domain (1570 X lo7 dynes/cm2) to breakaway from impurity atoms, and obtains an expression for the velocity that is similar to that for the low stress domain, except that 1 is replaced by IA, the distance between impurity atoms. The above stress dependence of velocity is not consistent with the empirical relation: V a exp ( - A / r ) shown in Fig. 16, and as we indicated in Section 14, the above temperature dependence is not valid over a wide temperature range. Finally, both the low stress and medium stress behaviors are affected by impurity concentration, so there is no reason to believe we are dealing with different obstacles in the two cases. It, therefore, appears that Mason’s treatment of the low stress and medium stress behavior is not satisfactory. Mason ascribes the drag at high stresses ( >lo9 dynes/cm2) to phonon viscosity and calculates a drag coefficient for this effect. Although the numerical results agree with our experimental curves (Fig. 14) at high stresses, the following objection should be raised. Mason calculated the core radius of a dislocation to be b/6, and integrated the elastic strain in to this limit. The core radius of b/6 was obtained by using a value of loo0 ergs/cm2 for the surface energy of LiF. If the correct surface energy of 340 ergs/cm2 62 is used, the core radius will be b/2 so that Mason’s numerical value for the phonon drag is high by at least one order of magnitude. If a core radius of 2b-3b is used, his calculated drag will be high by more than two orders of magnitude. It would appear that with a better choice of core radius, Mason’s treatment would yield a value for the phonon drag that is no greater than that obtained in the previous treatments by Eshelby and Leibfried (see reference 57, p. 66).

THEORY 22. NUCLEATION An interpretation of the stress dependence of dislocation velocities that is attractive both because of its simplicity, and because it can ac51

U

W. P. Mason, J . Acausl. SOC.Am. 32, 458 (1960). J. J. Gilman, J . A p p l . Phys. (1960).

196

J. J. GILMAN AND W. G. JOHNSTON

count for many observations is based on nucleation theory.a4It is assumed that dislocation motion occurs in a series of jerks; this being consistent with observations.Z*Between rapid motions, the dislocations rest in places of low potential energy. These might be Peierls-Nabarro potential wells, or places favorably disposed with respect to impurity atoms. Following the method of Fisher," the critical nucleation energy required to make a dislocation glide away from its potential well is calculated. If this is done with the realistic condition that the curvature of the dislocation line always must be in equilibrium with the applied stress (i.e., T = radius of curvature = S/br, where S is the line tension, b is the Burgers vector, and r is the applied shear stress), then the critical nucleation energy is:

s2 w*= -+) br

so

where Sois the reduced line tension in the potential well and f (So/&') = 2 for loose binding. The probability of glide nucleation is proportional then to exp ( - W*/kT) which is the observed dependence given in Eq. (13.1). The fact that this simple theory is consistent with: (a) the velocity versus stress behavior over an enormous range of velocities (Fig. 16), (b) the temperature dependence near room t e m p e r a t ~ r eand , ~ ~ (c) the multiplication behavior (Fig. 29), suggests that it may be a helpful model, although it needs further quantitative development. 23. THETEMPERATURE DEPENDENCE

According to Fig. 17, dislocation velocities at constant stress are proportional to exp ( - A / T ) . Combining this with Eq. (13.1), it appears that over a limited temperature range the following empirical relation holds; in agreement with the glide nucleation theory:

v

= V&-ClrT*

(23.1)

However, this is not a complete description of the behavior because it does not agree with the low-temperature behavior of LiF. In order to produce agreement it would be necessary to postulate a decrease in the binding energies of dislocations to their potential wells with increasing temperature. Such an effect has recently been proposed by KuhlmannWilsdorf,64who points out that, because of atomic vibrations, the position of a dislocation core is not sharply defined, but is distributed, probably in a Gaussian fashion. This tends to raise the average energy of the di5 53 64

J. C. Fisher, Trans.Am. SOC.Metals 47, 451 (1955). D. Kuhlman-Wilsdorf, Phys. Rev. 120, 773 (1960).

DISLOCATIONS IN LITHIUM FLUORIDE CRYSTALS

location core, thereby decreasing the binding stress lation of the form:

TB,

197

according to a re-

rB = r B p - G T .

Such a temperature variation could readily account for the observed behavior in LiF, but the data available a t present are not sufficient for a meaningful discrimination among the possible explanations of the combined stress and temperature dependences. 24. IMPURITY EFFECTS

The yield strengths of alkali halide crystals can be greatly increased by impurities, particularly divalent metallic ions, such as Mg+ +. This has been known since the early work of Edner, Metag, and Sch~enfeld.~~ Only now do we fully realize that the effect must be understood in terms of the interactions of mowing dislocations with the impurities. The results shown in Fig. 15 demonstrated this. On the basis of the effects of heat treatments we conclude that the impurities are more effective in raising the room temperature strength of LiF when they are clustered or precipitated, than when they are in solution.28Defects created by radiation damage also provide resistance to dislocation and again the defects are more effective at room temperature when the dose has been sufficiently high to form clusters. It may be that the reason that divalent impurities are more effective than monovalent ones is because their extra charge makes strong ionic bonds in the lattice or hard clusters of ions, but it is not clear why clusters interact so much more strongly with moving dislocations than do dispersed impurities. VIII. Individual Dislocations in Strain-Hardened Crystals

Several mechanisms have been proposed to account for strain-hardening in crystal^.^' All of the proposed mechanisms probably make some contribution to the hardening, but the one that is most important has not been clearly identified. This situation is largely due to the fact that it is difficult to investigate strain-hardening experimentally because it involves the motion of dislocations through strained material that has a complex structure. This makes it difficult to observe individual dislocations and the events that lead to strain-hardening. 66

6’

A. Edner, 2. Physik 73,623 (1932). J. J. Gilman and W. G. Johnston, J . Appl. Phys. 29,877 (1958). A. H. Cottreil “Dislocations and Plastic Flow in Crystals,” p. 151. Oxford Univ. Press, London and New York, 1953.

198

J. J. GILMAN AND W. G. JOHNSTON

A technique was devised for observing individual dislocation motions in strained LiF crystals.s8Thus the process could be studied in more detail than was previously possible so that the mechanism could be defined more clearly. The technique takes advantage of the difference between the etching behavior of “aged” and “fresh” dislocations.” One can strain a crystal, age the dislocations in it, and then introduce a few fresh ones by means of localized stresses (Fig. 33). Since etching allows the fresh dislocations to be distinguished from the aged ones, the velocities of the fresh dislocations can be measured as they move through a strained crystal containing many other dislocations. These velocities can then be compared with the velocities in unstrained crystals that have undergone the same heat treatment. The stress dependence of dislocation velocity in unstrained and strain hardened crystals is shown in Fig. 34. The figure shows that the velocities are sensitive to applied stress to about the same extent in the two crystals,

FIG.33. Screw dislocations moving in LiF crystals. Initial positions are indicated by flabbottomedpits; positions after stress was applied by pyramidal pits. 480X. (a) Undeformed crystal. (b) Strained crystal containing -5 X 10‘ dislocations/cm*. 68

J. J. Gilman and W. G. Johnston, J . Appl. Phys. 31,687 (1960).

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

199

5-

. 0

w m

tn

Po

z

*

8

>

z

-

-

1.0 7

P

-

4 0

-

I-

2 0.5 a

L w

a 0

m

-

I

d

FIG. 34. Effect of strain hardening on mobility of dislocations in lithium fluoride crystals.

but it requires a higher stress to produce the same velocity in the strainhardened crystal. That individual fresh dislocations in a strain-hardened crystal can move long distances compared to the spacing of background dislocations is apparent from Fig. 33. Individual dislocation loops can form glide bands that extend completely across the strain-hardened crystal as shown in Fig. 35. The only apparent differences between the behavior of dislocations in unstrained and strain-hardened crystal, is that a higher stress is required to produce the same phenomena in the latter case. Stress-strain curves show that the strain-hardening of LiF increases linearly with plastic strain. Dislocation density also increases linearly with strain (Fig. 32) so the strain-hardening is proportional to the dislocation density. This linear dependence is also evident from the way in which glide bands grow. As they grow the dislocation density within the

200

J. J. GILMAN AND W. G. JOHNSTON

FIG.35. Glide band that was formed by expanding a single dislocation loop in a strain-hardened and aged crystal. The pair of large pits near the center of the band show the initial position of the loop. -560X.

bands remains constant a t the “saturation value.” This saturation density is proportional to the applied stress16*and the proportionality constant is the same as for the hardening experienced by an individual dislocation (Fig. 24) ; namely, about 4 dynes/dislocation. This linear dependence is not predicted by either of the two more commonly employed theories of strain-hardening. Taylor hardening involves the interaction of a moving dislocation with the stress fields of other parallel dislocations, and the “forest” theory involves the interaction of a moving dislocation with a forest of perpendicular dislocations. In the Taylor theory, the flow stress increases as nt, where n is the dislocation density. In the forest theory the stress may vary in a more complicated manner with dislocation density, but in the simplest treatment it varies as nt. It therefore appears that neither of these treatments will predict the linear variation of flow stress with dislocation density that is observed in LiF. The nature of the strain-hardening in LiF is not yet understood, but we have suggested that strain-hardening probably results from the interaction of a dislocation with the trails of defects that have been left in the wakes of other dislocations (see Section 17). Since these trails are produced in proportion to the plastic strain, it seems reasonable that they should lead to proportionality between hardening and dislocation density. Also, since these defects have extended lengths, they should cause more impediment to a dislocation moving perpendicular to their lengths than to a parallel dislocation. This could be the reason that duplex glide causes much more rapid hardening than single glide.

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

201

IX. Macro-Plasticity

25.

PREDICTION O F

STRESS-STRAIN BEHAVIOR

The plastic behavior of a large crystal can be predicted from the microscopic dislocation properties that have been described above.2* The treatment is necessarily oversimplified, but it gives a considerable insight into macroscopic plastic behavior. We assume that the empirical relation between dislocation density and plastic strain (Fig. 32) is valid at all strains [Eq. (19.1)]. We also assume that the relation between dislocation velocity and applied stress as shown in Fig. 14 applies to all mobile glide dislocations in the crystal. Finally, we neglect strain-hardening. The weakest of these assumptions is that all mobile dislocations in the crystal move with the same velocity, because the dislocations in the interior of a widening glide band will not be able to move as readily as those at the edge of the band. The plastic strain rate of a crystal may be written as B = bnV

(25.2)

where b is the Burgers vector, n is the number of dislocations per unit area, and V is the average dislocation velocity. If only pure edge and screw dislocations are present, the strain rate depends on two velocities, B = b(n,V,

+ neV,)

(25.3)

where the subscripts refer to screw and edge dislocations. Edge components of dislocation IOOPS move about 50 times as fast as screw components, so the relative densities of screw and edge dislocations would be n8/ne= Ve/V, N 50.

(25.4)

The total number of dislocations is n

=

n.

+ n, = n, + n,/50 = n.

(25.5)

so that Eq. (25.3) may be written as B

= 26nV, (

T)

.

(25.6)

The experimental relation between dislocation density and strain, n = loQc, is employed to give (21i.7) B N 2 X log & V . ( T ) . Measurement of a stress-strain curve involves an interaction between a specimen and some kind of machine. Therefore the machine characteristics must be taken into account in interpreting the observations. The machine that we used for compressing crystals is depicted schematically

202

J. J. OILMAN AND W. O. JOHNSTON

i CROSS HEAD MOTON: CROSSN k

At

s

Sc=CONSIANT

FIG.36. Schematic drawing of compression apparatus. The spring represents the elastic deformation in the apparatus and the crystal.

in Fig. 36. The crosshead of the machine moves a t a constant speed, S,, so as to compress the crystal. Part of the crosshead motion goes into elastic strain of the deforming fixture and the crystal, and this elastic strain is depicted schematically by an imaginary spring for which the elastic displacement is Aye1 = K F ; F being the applied force and K the spring constant. If ALP is the amount of plastic deformation in the crystal, the crosshead displacement may be written as

Ay

=

S,t

=

+ ALP

(25.8)

- KF)/Lo.

(25.9)

Ayei

and the plastic strain of the crystal is Q =

ALP =

Lo

(Set

The plastic strain rate can be expressed by taking the time derivative of Eq. (25.9) i = (5.

- K -d -F) / L ~ . dt

(25.10)

During a test the machine records the applied force F as a function of

t, which is essentially a record of applied stress versus crossliead displace-

ment y . Equations (25.9) and (25.10) can be substituted into Eq. (25.7) to get a differential equation relating only the resolved shear stress r and y: (25.11)

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

203

y =UIOSsHEAD DISPLACEYENVLENGTH a 100%

FIG.37. Family of deformation curves calculated from data on dislocation velocities and dislocation densities.

where 7 = F/2A; A = cross-sectional area of crystal, B = (2KA)-', C = 2 X lo0 bB/S,, and V a ( r )is the velocity of screw dislocations. This equation has not been solved to give ~ ( y explicitly, ) but has been used to express the observed relation between dislocation velocity and stress. Arbitrarily selected curves from the family of deformation curves defined by Eq. (25.11) are shown in Fig. 37 for a particular set of experimental values of the constants, A, K, and S,. The curves were constructed by calculating the slope dr/dy at a number of points ( T , y) to form a linealelement diagram. The curves do not pass through the origin, but numbers 1-5 lie so close that they cannot be distinguished from the line OE. Under the assumption that n a el if a curve starts at the origin with e = 0, the crystal must initially contain no glide dislocations, and the curve should follow an extrapolation of OE up to the stress required for dislocation nucleation or unpinning. If the initial number of unpinned dislocations initially present is not zero, the curve starts with E # 0, and does not pass through the origin. The displacement from the origin is a measure of the initial glide dislocation density no and for the curves of Fig. 31 the initial displacements correspond to the initial dislocation densities listed in Table I. The density of mobile dislocations initially present in a test crystal should be less than 104/cm2,so we would expect the experimental curve to look like curves No. 3, 4, or 5. 26. OBSERVED STRESS-STRAIN BEHAVIOR The deformation curve of a crystal, that was compressed uniaxially in the [loo] direction is shown in Fig. 38. The dimensions of the crystal were

204

J. J. GILMAN AND W. G . JOHNSTON

TABLEI. INITIAL DENSITY OF GLIDE DISLOCATIONS no, dislocations/cm* ~~

~~~

Curve

1-5

6

7

8

9

10

< lW

1 X lo6

3 X 106

1.5 X 106

6 X lon

1.3 X 10'

Y I , , 2 , , , 3 , , , 4 , , 5

OO

CROSSHEAD DISPLAEMENTILENGM

I

100%

FIG.38. Deformation curve in compression (solid line). The dotted line is a calculated deformation curve based on measured dislocation velocities and densities.

0.209 in. X 0.261 in. x 1.390 in., and the crosshead velocity was 0.005 in./min. In the early stages of the deformation the experimental curve resembles curve No. 7 in Fig. 37, rather than No. 3, 4, or 5 as we had expected. The yield drop is smaller than expected because it is difficult to obtain perfectly flat and parallel ends on the specimen, and to align the specimen perfectly before the deformation begins. Thus the stress is not applied uniformly, and some regions of the crystal deform before others. We shall see in the next section how large yield drops can 6 produced. The experimental curve of Fig. 38 has an initial linear portion whose slope defines the spring constant K that is used in Eqs. (25.9) and (25.10). Deviation from linearity first occurs at the elastic limit, r y = 650 gm/mm2, which is slightly higher than the minimum stress a t which dislocations would move in the crystal. The curve levels off at rm, a t which point

dF/dt

=

0 so that

i,,, =

&/Lo

=

6.0 X

sec-'

according to Eq. (25.10). The plastic strain a t that point is 1.2 X 1O-a SO that from Eq. (25.7) we find that the dislocation velocity is V , = 1.1 X

DISLOCATIONS IN LITHIUM FLUORIDE CRYSTALS

205

1W8cm/sec. From the dislocation velocity versus stress curve of Fig. 14 we see that this velocity corresponds to a stress of raalc = 930 gm/mma. This calculated flow stress is close to the observed flow stress, r, = 920 gm/mm2. We can conclude that for small strains, once the specimen is aligned, our calculated curve is in reasonably good agreement with the observed flow curve. The calculated and experimental deformation curves diverge at larger strains, but this is to be expected because we have neglected strain-hardening in our treatment. If the entire stress difference is ascribed to strainhardening, the hardening increases linearly with plastic strain as shown in Fig. 39. The deformation curve we have constructed from the observed properties of dislocations may, therefore, be brought into agreement with the experimental curve by assuming a linear strain-hardening law. 27. PLASTIC INSTABILITY AT THE YIELDPOINT Large drops in stress at the yield points of crystals are often observed in testing machines that enforce a fixed strain-rate and measure the stress required to maintain this strain-rate. For such a machine (25.6) gives the density of dislocations that is required to maintain a particular strain-rate when the applied stress is causing the dislocations to move at a velocity given by V,(r). If the dislocation density is smaller than given by Eq. (25.6), the stress rises; and the stress drops if the density is greater. A rising stress level causes faster motion and faster multiplication so that eventually there will be enough dislocations present to make the strain700r

PLASTlC STRAIN, AL/Lo alOO%

FIG.39. Stress increment attributed to work hardening versus plastic strain. The strem increment is the difference between the solid and dotted curve^ of Fig. 38.

206

J. J. OILMAN AND W. G . JOHNSTON

rate of the crystal equal the applied strain-rate. Conversely, a dropping stress level means slower dislocation motion and less multiplication so that an equilibrium is eventually established between the strain-rate of the crystal and the imposed rate. If the mobile dislocations in our tests are ieitially few in number, they cannot move sufficiently fast to produce the required strain-rate, and the stress rises. As the stress rises, dislocations multiply exponentially at fist and then in proportion to the plastic strain; and faster motion occurs. Eventually, the stress stops rising when strain-rate of the crystal equals the applied strain-rate, but, since multiplication continues in proportion to the strain, more than enough dislocations are soon present. The stress will then drop until dislocation motion becomes so slow that the strainrate of the crystal equals the applied strain-rate. In accordance with the analysis given above, a crystal with only a small number of mobile dislocations ( <104/cm2) should exhibit a large stress drop upon yielding when tested properly in the type of machine we have employed. The yield drop should be at least as large as that shown in curve No. 5 in Fig. 37. Such large yield-stress drops are indeed obtained for LiF when a few mobile dislocations are initially present, provided that the stress is applied uniformly. The number of mobile dislocations can be kept small either by starting with a low dislocation content, or by immobilizing the existing dislocations. If a specimen is compressed about 0.01% it becomes well aligned, and the ends are in good contact with the deforming machine. Upon reloading, the applied stress is uniform, but then there will be more than lo6dislocations/ cm2present because of the prestrain. However, if the prestrained crystal is heated to about 110°C for 2 hours and then cooled to room temperature, the dislocations will become strongly pinned by impurities. If the specimen is heat treated in situ it remains properly aligned, and uniform stresses can be applied. A large drop in stress then occurs upon yielding as shown in Fig. 40. The magnitude of the drop in stress depends on the rate a t which dislocation velocity varies with stress. If the log V - log T curve of Fig. 14 is very steep, a large change in velocity will correspond to a relatively small stress change, and the drop in stress at yielding will be small. On the other hand, if the velocity change requires a larger stress change, the yield drop will be large. Strain-hardening tends to obscure the yield-stress drop in a crystal. When a linear strain-hardening term is added to the curve of Fig. 37 the drops in stress appear smaller. This effect increases as the rate of strainhardening increases.

“O01

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTAL8

I I

207

2 3 CROSS HEAD DISPLACEMENT I Lo, IN 46

FIG.40.Deformation curve in cornpreasionshowing a large drop in stress up011plastic yielding. The crystal waa prestrained 0.02y0,and aged in situ for 2 hours at 110°C.

It has been proposed in the past that a stress drop at yielding occurs because dislocations that are pinned by impurity atoms suddenly break free when the applied stress becomes sufficiently high.67A yield drop can, of course, occur in this manner. However, general unpinning is not necessary, but only the unpinning of a few segments of dislocation, each of which will rapidly produce a wide glide band as described in Section 11. X. Effects of Radiation Damage

Extensive damage of LiF crystals can be brought about by quite short bombardments with thermal neutrons. The neutral Li metal in the crystals contains 7.5% of the Li’ isotope. The capture cross section for thermal neutrons of this isotope is large (950 barns) and the following fission reaction results: Lis

+ n1

--f

H3 (2.7 MeV)

+ He4 (2.1 Mev).

Three species of defects are produced directly by this process (tritium atoms, He atoms, and Li+ vacancies) ; but more important is the 4.8 Mev of kinetic energy which the fission products dissipate through displace-

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J. J. GILMAN AND W.

G.

JOHNSTON

ments and ionization before they come to rest. About 1500 defect pairs (vacancy interstitial) are created per fission event, so that a neutron flux of about 9 X 1017nvt displaces 1% of all the atoms, and 1 ppm of defects is created by a flux of 9 X 1013nvLs6 LiF is also damaged by irradiation with X-rays or y-rays. The extent of the damage (as indicated by property changes) is about the same as for neutron irradiation when the colors of damaged crystals are the same. Thus the dominant part of the damage in all cases is caused by ionization.

+

28. STRUCTURAL CHANGES DURINGIRRADIATION*

Coloration is the most striking effect of irradiation. As little irradiation as 1Olo nvt causes a detectable F-band absorption in the ultraviolet at 2500 A wavelength. The tail of the F band causes very pale yellow coloration to appear at lo1*nvt which becomes canary yellow a t 10ls nvt, deep yellow a t 1014nvt, reddish brown at 1OlS nvt, and completely black at 10l6nvt. Substantial hardening occurs as the dose increases from 10l2 to lo1"nvt. Lattice expansion becomes detectable a t 10l6nvt and increases to about 0.4% after an irradiation of 5 X 1017 nvt. Finally, cavities and thin platelets of Li metal along { 100) planes appear when the dose reaches about 1OI8 nvt.ss No evidence of the direct production of dislocation lines at displacement spikes was found in as irradiated crystals. Thus no local collections of small dislocation loops appeared, and no isolated loops larger than about 25 A in size are formed. This is ' d i k e the case of damage by fission fragments which produces heavy and easily detectable tracks.17 Extended irradiation produces clustering of the isolated defects that appear for small doses. This causes a change in the mode of etching of the surface of a damaged crystal. After an annealing treatment at about 450°C (1 hr), etching reveals tiny (lo00 A) dislocation loops lying on { 110) planes (Fig. 41). By extrapolation, it seems likely that the "clusters" in as-irradiated crystals are associated with very small ( <50 A) dislocation loops. The loops anneal out completely a t about 600°C. Annealing treatments of heavily irradiated crystals ( > 5 X 1OlS nvt) at temperatures above 600°C result in the formation of small rectangular cavities (Fig. 42). It is believed that these cavities are not a result of the gaseous fission products, but result from the formation of molecular fluorine during irradiation which subsequently escapes from the crystal.

* For the sake of brevity, the many contributors to this subject will not be named. The 69

literature is reviewed in reference 56. M. Lambert and A. Guiner, Compt. rend. acad. sci. 244,2791 (1957); 245,526 (1957).

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

209

FIG. 41. Pairs of etch pits indicating (110) dislocation loops in an irradiated and annealed LiF crystal. The neutron dose w m 6 X 1010 nvt; the crystal wm heated at 500°C (1 hr); then etched. 2000X.

Whapham and Makineo have studied hardening in LiF caused by electron bombardment. They were particularly concerned with the de pendence of the hardness on radiation dosage, and they found that the increment in hardness TH is related to the radiation dose t$ by: 7H

= J[1

- e-Z+]1/2

where J and 2 are constants. This form of dose dependence is qualitatively the same as for neutron radiation.

29. CHANGESIN DISLOCATION MOBILITY

It has been shown that the hardening which irradiation produces is not caused by “pinning” of dislocations by radiation defects.6sThis was done by cooling irradiated crystals to -196°C and then rubbing their surfaces to introduce fresh dislocations. These fresh dislocations could not be “pinned” prior to testing because diffusion would be too slow at the low temperature. It was found that the fresh dislocations moved no more easily than did dislocations introduced a t room temperature and cooled to -196°C. A. D. Whapham and M. J. Makin, Phil. Mag. 5, 237 (1960); 3, 103 (1958).

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J. J. GILMAN AND W. G . JOHNSTON

FIG.42. Rectangular cavities in irradiated and annealed LiF. 500 X. (a) 3 X 1016nvt plus 1 hr at 840°C. (b) lo1' nvt plus 27 hr at 800°C; shown in view perpendicular to a subgrain boundary.

Hardening results from a dynamic interaction between moving dislocations and radiation defects. This is evident from the fact that it requires just as much extra stress to move a dislocatidn at a velocity of lo+' cm/sec through a damaged crystal as it does at cm/sec.28Thus static effects are not important.

21 1

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

That no hardening occurs during the development of intense F-band absorption, but rapid hardening occurs when the 4500 A optical band appears, suggests that clusters of defects are required for hardening. A similar conclusion has been reached by Aerts et aLglfor the case of X-irradiated NaCl crystals. A phenomenological understanding of the hardening process follows from the form of the V ( T )curve (Fig. 14), according to which any local internal stresses will result in a net decrease in the average velocities of dislocations. Suppose that internal stresses with amplitude AT and wavelength L are introduced by radiation damage. For simplicity, we assume a square wave form so the internal stress is uniformly -AT for a distance L/2, then jumps to +AT for a distance L/2, then back to -AT, etc. The argument can easily be extended to any waveform remembering that the average internal stress must always equal zero. The time required for a dislocation to move a distance L in the absence of the internal stresses is t o = L / V ( r ) . With the internal stresses it is: 1

V ( T- A T ) but! [V ( T - A T )]-I

+ V ( T+ A T )

is very large compared with [V ( T t=

+ AT)

]-I

so :

L

2V(r - A T )

and the average velocity over the distance L is 2 V ( r - A T ) instead of V ( T ) This . average velocity will be considerably smaller than V ( T )for a reasonable AT &cause V ( T ) is so nonlinear. Since the net average velocity is independent of wavelength, the stress fluctuations can be considered to be quite localized.

30. NEWDISLOCATION SOURCES Radiation damage can sometimes put dislocation sources into a crystal by inducing the formation of precipitate particles.lB I t has been found that this only occurs in impure crystals, but one must be alert to this possibility in studies of the macroscopic plasticity of radiation damaged crystals. XI. Cyclic Strains

The phenomenon of mechanical failure through fatigue shows in a general way that even very small plastic strains are irreversible. We can E. Aerts, S. Amelinckx, and W. Dekeyser, A&. Met. 7, 29 (1959).

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J. J. OILMAN AND W. 0. JOHNSTON

now see that the fundamental basis of this irreversibility is that screw dislocations undergo multiple cross-glide when they move. Since they do this randomly the process is irreversible and leads to a gradually increasing concentration of edge-dislocation dipoles in a crystal. Eventually this leads to fatigue failure. Keith studied the behavior of LiF crystals under cyclic loading conditions.s2 He found that the densities of dislocations within individual glide bands increases as the plastic strain is cycled, apparently reaching some limiting value that is characteristic of the testing conditions. The dislocation density does not increase with the total strain as rapidly as for monotonic loading, but it does not become reduced during strain reversals, rather it either increases or remains constant. This last is a consequence of the irreversibility of the individual dislocation motions. The average strains within glide bands often become reduced during macroscopic cyclic straining, but they do not decrease to zero, which is another consequence of the fundamental irreversibility that causes strainhardening within a glide band so that further straining tends to occur elsewhere in the crystal. It has been concluded that the trails left behind moving screw dislocations are the primary cause of fatigue damage. Surface contour changes (intrusions and extrusions) are secondary effects. XII. Fracture Studies

Having a means for finding the positions of dislocations in crystals has led to a clarification of several phenomena associated with the propagation of cracks. Since the ease with which a crack can propagate determines whether a crystal is brittle or ductile, the problem of brittleness in ionic crystals has also been clarified. In a perfect crystal, a fast-moving crack is only impeded in its motion by the energy required to make the new surfaces of the crack, and by the inertial mass of the crystal. In imperfect crystals, other effects act to impede the motion of cracks. The most important of these effects is the plastic deformation that can occur at the tip of a slowly moving crack. In a crystal that contains screw dislocations, a crack front must split into two parts each time it intersects a screw dislocation, and this provides another impediment to the motion of the crack. Also, edge dislocations can have an effect because the cohesion of a crystal is reduced a t their centers. R. E. Keith and J. J. Gilman, Progress report on dislocation behavior in lithium

fluoride crystals during cyclic loading, Symposium on Basic Mechanism of Fatigue, Am. SOC.Testing Maderials Spec. Tech. Pull. No. 237 (1959).

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

213

31. DISLOCATION NUCLEATION AT CRACKTIPS

Cracks can move as fast as 2 X lo6 cm/sec along { 100) planes in LiF crystals.6aThis maximum, or terminal, velocity is limited by the inertia of the crystal as it opens up to form the crack. In agreement with Cragg’s theoryle4it is nearly equal to the Rayleigh surface wave velocity of 2.95 X 106 cm/sec. At crack velocities near the terminal value, there is no time for dislocation nucleation; but at a much lower velocity (-6 X 1W cm/sec) disloce tion loops begin to form in front of a crack tip.” This is only an approximate value of the critical velocity because it varies with the flow stress of each’particular crystal. The higher the flow stress, the lower the critical nucleation velocity. If the velocity of a crack is allowed to drop below the critical value, the dislocations that are nucleated move increasingly large distances during crack propagation, and cause increasing amounts of plastic flow. At velocities less than about 50% of the critical velocity enough plastic flow occurs to make crack propagation unstable. Periodically such a crack slows down and forms a group of dislocations which tend to slow it down still further. During this time, the driving force on the crack increases and eventually becomes great enough to cause the crack to jump forward under a large driving force, and the cycle repeats.6a The dislocations that form in front of cracks are often small complete loops and they form at places where no dislocations existed previou~ly.~~ Thus a crystal need not contain dislocations initially in order for plastic flow to occur in it and impede crack propagation. By blunting the sharp edge of a crack, plastic flow reduces its stress-concentrating effect and thereby makes it difficult for the crack to move. However, according to . effect ~ ~ is not large in LiF notched-bar impact tests by Johnston et ~ 1 this until temperatures above about 400°C are reached. 32. CLEAVAGE STEP CREATION BY DISLOCATIONS

A screw dislocation converts the crystallographic planes that lie perpendicular to it into a helical ramp. Therefore, when a crack runs along one of the planes and intersects the screw dislocation, it splits into two parts. The two parts move along planes that are separated from each other by a distance equal to the component of the dislocation Burgers vector that lies perpendicular to the planes. Eventually the two halves of the crack join together along a “step.” Such steps can easily be seen on OrJ. J. Gilman, C. Knudsen, and W. Welsh, J. Appl. Phy8. 29, 601 (1958). J. w. Craggs, J . Mech. and Phy8. Solids 8 , 66 (1960). T. L. Johnston, R. J. Stokes, and C. H. Li, J . M e l d s 11, 66 (1959).

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J. J. GILMAN AND W. G. JOHNSTON

the surfaces of cleaved crystals (Fig. 43) and they correlate closely with etch pits at screw dislocations in the crystals.aaThe screw dislocations may be present in several forms: as twist-type sub-boundaries; within glidebands that result from plastic deformation; or as parts of the small dislocation loops that nucleate ahead of moving cracks. The energy of a surface that is roughened by the presence of many cleavage steps is higher than the energy of a smooth surface. Also, considerable tearing may accompany the formation of cleavage steps and this requires energy. If a step forms by cleavage perpendicular to the plane of the main crack, the energy absorption is simply proportional to the step height and is generally rather small. If; however, a step forms by plastic tearing or shearing, then the energy absorption is proportional to the square of the heighta7and considerable energy may be absorbed for steps that are greater than -100 A in height. This can slow down crack motion.aa 33. WEAKENING CAUSEDBY DISLOCATIONS

Several persons have observed that edge dislocations in the bubble model of a crystal can be fractured readily along their glide planes. Evidence that this effect or a similar one exists in LiF crystals was obtained by cleaving prestrained crystals under controlled conditions.as It was found that (100) cleavage cracks tend to branch and move along { l l O ) planes at places where gliding has occurred during the prestrain treatment. No such branching occurs in crystals that have not been prestrained. It has been concludedes that dislocations put into a crystal by plastic flow may reduce its cohesive strength. 34. SURFACE ENERGYMEASUREMENTS

At - 196°C where dislocation motion is sluggish, cracks propagate through LiF crystals in an almost completely elastic manner. The energy that is needed to cause crack propagation is then equal to the surface energy. By measuring the work expended in cleaving crystals at - 196"C, the surface energy of the {loo) cleavage plane has been determined.aQ The best value was taken to the 340 ergs/cm2; a value that is in good agreement with the liquid surface tension70 (350 ergs/cm*), and ionic lattice theory" (360 ergs/cm2). J. J. Gilman, Trans. AZME 212, 310 (1958). J. J. Gilman, J . Appl. Phys. 21, 1262 (1956). J. J. Gilman, J . A p p l . Phys. 32, 738 (1961). O9 J. J. Gilman, J . Appl. Phys. 31, 2208 (1960). 70 F. M. Jaeger, Z . unorg. u. dlgen Chem. 101, 1 (1917). 71 A. E. Glauberman, Zhur. Fiz. Khim. 23, 124 (1949). 66

67

DISLOCATIONS I N LITHIUM FLUORIDE CRYSTALS

215

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J. J. GILMAN AND W. G . JOHNSTON

XIII. Dielectric Breakdown

During attempts to nucleate dislocations by means of inhomogeneous It was found that electric fields, a new phenomenon was disco~ered.’~ prior to dielectric breakdown through the interior of LiF crystals, a complex pattern of surface breakdown occurs. This surface breakdown is not normally invisible, being only revealed by etching the surface (Fig. 44). It consists of dendritic branches that lie along (100) directions on (001} surfaces. The branches terminate by splitting into two short segments that lie parallel to (110) directions, forming a “Y)) configuration. At these termination points, it is often possible to find a colloidal thread that passes down into the bulk of a crystal along a zig-zag path. Etching studies have shown the dendritic branches to consist of multitudes of discoloration half-loops arranged as shown in Fig. 45. It might be argued that the surface breakdown figures are produced by secondary mechanical effects of what is primarily an electrical phenomenon. However, dislocation lines can be expected to exhibit unusually high ionic conduction in LiF crystals because of the relatively small size of the Li+ ions. Also, dislocation lines carry a charge7sand so their movements can relieve concentrated inhomogeneous fields. For these reasons it is reasonable to suppose that dislocations may play a very intimate role in some types of dielectric breakdown, and further investigation of this role should be fruitful. XIV. Mechanism of Etch-Pit Formation

When it had been established that preferential etching occurs at dislocation sites (Section 11-1)) it became possible to deduce many details of the etching process.” Crystal dissolution is thought to occur by the movement of monomolecular steps across a crystal surface (Fig. 46). The active sites on the steps are the kinks where an atom is less tighlty bound than elsewhere along a step.’* At temperatures above absolute zero, a high concentration of kinks will form along existing steps because of thermal fluctuation^.^^ However, the steps themselves do not form with similar ease,7sand the availability of steps may limit the rate of dissolution. At a perfect region of the surface, far removed from the edges, new steps may form by nucleation of pits that are of monomolecular depth (P in Fig. 46). Kinks form along the edges of such a “unit pit” and as atoms leave the kinks to go into solution, the steps sweep across the surface. J. J. Gilman and D. W. Stauff, J . Appl. Phya. 29, 120 (1958). R. L. Sproull, Phil. Mug. 5, 815 (1960). ’I4 W. Kossel, Nachr. Ges. Wiss. Gottingen Math. physik. K1. Fachgruppen p. 135 (1927). 76 W. K. Burton, N. Cabrera, and F. C. Frank, Phil. Trans. Roy. SOC. London A243, ?*

299 (1951).

DISLOCATIONS IN LITHIUM FLUORIDE CRYSTALS

217

FIG.44. Surface breakdown pattern on LiF crystal revealed by dislocation etchant. crystal ww etched, broken down, re-etched. Dendritic branches lie parallel to (100) directions. (a) entire pattern 9.4X. (b) center of pattern 56X.

218

J. J. O I L W AND W. G. JOHNSTON

FIQ.45. Arrangement of dislocation loops that form surface breakdown patterns in LiF crystals-the region near a “Y”junction is shown.

FIG.46. Schematic drawing of features on the surface of a dissolving crystal. AB is a monomolecular step on a close-packed plane, and K a kink from which atoms go into solution. A unit pit that has formed by two-dimensional nucleation is depicted a t P.

DISLOCATIONS IN LITHIUM FLUORIDE CRYSTALS

219

35. EFFECTOF DISLOCATIONS The increased internal energy near the center of a dislocation line may cause rapid nucleation of unit pits at the point where it intersects the surface. Repeated nucleation of unit pits may lead to the formation of a dislocation etch pit. Cabrerass has considered the effect of the elastic portion of the internal energy on the formation of dislocation etch pits during evaporation. Some experimental evidences7 indicates that a core energy of a dislocation is more important than the elastic energy, but a comprehensive theoretical treatment of the problem has not yet been made. Repeated two-dimensional nucleation at a dislocation is not sufficient to fopm a visible etch pit. If the monomolecular steps sweep across the crystal too rapidly, the sides of the etch pit may slope gradually and not be visible. The shape and hence the visibility of an etch pit depends on the ratio of the rate at which the pit deepens, V,, and the rate V. a t which the steps move (Fig. 47). A very striking effect occurs if a disloca-

a

b

FIG. 47. (a) Etch pit formed by repeated nucleation of unit pita at a dislocation. V , = velocity of steps, and V , = nucleation rate times atomic spacing. (b) Shape assumed by the etch pit aft.er the dislocation has moved away.

tion is moved during dissolution because rapid nucleation of steps ceases at the original site but the steps continue to move. A flat-bottomed pit forms (Fig. 47b), as we have seen in a real crystal (Fig. 4). In order for preferential nucleation to occur, it is necessary for the nucleation rate to be negligible at places where dislocations are not present. This restricts the pitting process to lowenergy surfaces of crystals. Once N. Cabrera, The oxidation of metals in “Semiconductor Surface Physics” (R.H Kingston, ed.), p. 327. Univ. Penn. Press, Philadelphia, Pennsylvania, 1957. J. J. Gilman, in “The Surface Chemistry of Metals and Semiconductors” (H. C. Gatos, ed.), p. 136. Wiley, New York, 1960.

220

J. J. GILMAN AND W.

a.

JOHNSTON

a pit has been nucleated, the surface steps at its edges move rapidly when there is no inhibitor in the etching solution. Since the steps retreat from the center of each pit more rapidly than new monomolecular pits form at the center, the process results in very shallow macroscopic pits and the surface appears to be polished (Fig. la). 36. INHIBITION OF STEPMOTION In order to reduce the step velocity V,, it is necessary to add an inhibiting salt to the solvent. FeF3 was the first inhibitor that was tried. It was soon found that various iron salts, FeF3, FeC13,FeBr3, Fe(N03) 3, have quite similar effects, and it was concluded that the cation is the one that causes inhibition. Tests of some 25 different cations showed that Fe3+ and A13+ are the most effective. It is believed that this is related to their unusually strong tendency to combine with F- ions to form (MeFa)3+ complexes. The specific way in which inhibition is believed to take place is by Fe3+ions slowing down the motion of kinks by complexing with the three F- ions that appear at kinks on { 100f (001) surface steps. Evidence that this occurs rather than a reduction in the rate of kink nucleation is that etch pits become rounded at their corners when the concentration of inhibitor is increased above the optimum. The steps of rounded pits must contain an abundance of kinks that would straighten out the steps if they could move rapidly. That the steps do not straighten shows that the kinks can only move slowly. Etch pits deepen almost as rapidly in the presence of inhibitor ions 88 in their absence. This shows that the inhibitor ions have little effect on pit nucleation; probably because they can only absorb strongly at kink sites. Ives and HirthL2have studied the kinetics of pit formation by measuring the rates at which pits widen under various conditions. They found that the pits grow at a constant rate after an initial period of fast growth. Between 1" and 51°C the rate increases with temperature and decreases linearly with the concentration of LiF in the etching bath. When the LiF concentration reaches 25% of its saturation value, growth stops. These results are qualitatively consistent with the ideas expressed above about crystal dissolution. The slopes of etch pits (acute angles between pit sides and main surface) increase with increasing LiF concentration in the etchant, and with Fe3+ concentration to maximum values of 10-15" and then remain constant. This behavior is inconsistent with simple quantitative theories of crystal dissolution so a more complex theory is required.

DISLOCATIONS IN LITHIUM FLUORIDE CRYSTALS

221

FIG.48. Effect of heat treatment on the etching behavior of grown-in dislocations. An =grown crystal waa cleaved into two halves. (a) As-grown crystal, etched. Photograph reversed in printing. (b) Matching face of the second piece that was heated to 300°C and air-cooled prior to etching.

222

J. J. GILMAN AND W. G. JOHNSTON

37. EFFECT OF SEGREGATED IMPURITIES

It was shown in Fig. 2 that the etching action of an aqueous solution of FeF3 distinguishes between the fresh and aged dislocations in a slowly cooled crystal. We have concluded that the rate of etching is smaller a t aged dislocations because of segregated impurities." If a crystal is cooled rapidly from above 25OoC, impurity segregation does not have time to occur, and all the dislocations then etch like fresh ones. Figure 48 compares the etching behavior of slowly and rapidly cooled crystals. In purer crystals (such as those that can now be obtained from the Harshaw Chemical Company) aged and fresh dislocations etch almost identically. This further substantiates the idea that in impure crystals the difference in etching is caused by impurity segregation.