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## Material Information- Title:
- An interpretational study of field ion microscope images
- Added title page title:
- Field ion microscope images
- Creator:
- Sanwald, Roger Carl, 1941-
- Publication Date:
- 1967
- Language:
- English
- Physical Description:
- xiii, 164 leaves : ill. ; 28 cm.
## Subjects- Subjects / Keywords:
- Atoms ( jstor )
Coordinate systems ( jstor ) Eggshells ( jstor ) Electric fields ( jstor ) Energy ( jstor ) Geometric planes ( jstor ) Imaging ( jstor ) Ions ( jstor ) Modeling ( jstor ) Simulations ( jstor ) Dislocations in crystals ( lcsh ) Dissertations, Academic -- Metallurgical and Materials Engineering -- UF Image intensifiers ( lcsh ) Metallography ( lcsh ) Metallurgical and Materials Engineering thesis Ph. D - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis - University of Florida.
- Bibliography:
- Bibliography: leaves 161-163.
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- Manuscript copy.
- General Note:
- Vita.
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- University of Florida
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AN INTERPRETATIONAL STUDY OF FIELD ION MICROSCOPE IMAGES By ROGER CARL SANWALD A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, iW7 ACKNOWLEDGMENTS The author is deeply indebted to his advisory committee, Drs. Rhines, Reed-Hill, Conklin, and Smith, and particularly to his chairman Dr. Hren, who participated in many stimulating discussions during the course of the work. The writer is indeed grateful to National Aeronautics and Space Administration for the funds provided making the research possible. ii TABLE OF CONTENTS ACKNOWLEDGMENTS................. LIST OF FIGURES................. ABSTRACT .................... INTRODUCTION .................. Chapter I. REVIEW OF THEORY OF FIELD ION MICROSCOPY. II. REVIEW OF INTERPRETATIONAL WORK IN FIELD ION MICROSCOPY ......... III. RESULTS: INTREPRETATION OF FIELD ION IMAGES . . . . . . . . IV. CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY............ . Appendices I . . . . . . . . . . . Page 3 17 35 122 . 127 I . . . . . . . . . . II . . . . . . . . . . . I. . . . . . BIBLIOGRAPHY ................. BIOGRAPHICAL SKETCH .............. * . 132 * . 136 .137 .161 .164 iii LIST OF FIGURES Figure Page 1. Schematic diagram of field ion microscope .. 4 2. Field ion image of an iridium. specimen . 5 3. (a) Potential of free atom showing valence electron at depth VI; (b) potential of atom in the presence of an external field; (c) atom close to the surface of a specimen, illustrating the potential configuration at the critical distance............8 4. (a) Energy diagram for the removal of an. ion from the surface of the specimen (no field applied); (b) the same system as (a), with a field applied, showing the existence of the Schottky hump .........10 5. Surface of specimen with high field applied,causing positive atomic cores to protrude, as electron cloud is drawn back into the metal. Approximate variation of field with position is also shown......................12 6. Illustration of the imaging process in the field ion microscope ...........14 7. Current voltage characteristic for a typical field ion emitter. ..........16 8. Field ion image of an iridium specimen taken at 210K. Low index plane A (200), and a high index plane at B (931) are labeled.....................18 9. Relationship of neighboring atomic positions on (a) a high index plane, and (b) a low index plane ..................20 iv LIST OF FIGURES--Continued Figure Pg 10. The geometry of several projections: Pg9 (gnomonic projection), P (stereographic projection, P0 (orthographic projection), Pf (pseudo-stereographic) 11. Image of asymmetric Ir specimen, taken at liquid H2 temperature ....... 12. Defect in molybdenum interpreted as an edge dislocation...... .. .. .. 13. Single spiral at grain boundary in iridium interpreted as a dislocation 14. Change in the topographical features of a simulated field ion image (shell model) with a variation in radius of .05a,. .............. .. . 15. Simulated evaporation of the (420) pole in FCC using the shell model, caused by small change in radius ........ . . 22 . 24 28 * . 30 * . 34 36 16a. Variation of the 1st nearest neighbors over a stereographic triangle using the shell model (FCC, Radius = 350 X, shell thickness = .05a) ...............39 16b. Variation of the 2nd nearest neighbors over a stereographic triangle using the shell model (FCC, Radius = 350 X., Shell thickness = .05a0) .......... 16c. Variation of the 3rd nearest neighbors over a stereographic triangle using the shell model (FCC, Radius = 350 ~ Shell thickness = .05a0) ....... 17. Schematic diagram of the basic steps involved in the computer program to reconstruct the field ion image ... 18. The geometry of the projection showing the relative relationship with the specimen . . . . . . . . . . 40 41 43 48 V Page LIST OF FIGURES--Continued Figure Page 19. Comparison of the imaging criteria of the shell model and the "neighbor model" on ahigh index plane ................52 20. Micrograph of Pt................55 21. Determination of the local radius by graphically plotting atomic steps. .......58 22a. Shell model, (931) region. ..........60 22b. "Neighbor model," (931) region .........60 23a. Experimental image, (931) region ........63 23b. Relative values of A, the local sublimation energy in the (931) region based on the "neighbor model".. .. ...........63 24. Diagram of (111) plane showing how single and double spirals can be produced on a (220) plane with Burgers vectors of ao/2 [101] and ao/2 [1103 respectively . 67 25. Shell model; perfect FCC lattice, radius 0 1600A, shell thickness 05aof (111) projection..................70 26. Shell model of FCC lattice, 1600 AO, shell thickness 05ao, (111) projection, screw dislocation emerging at the center of the plane: (a) single leaved spiral; (b) double leaved spiral; (c) triple leaved spiral .................72 27. Pure screw dislocation causing single spiral on (204) planeoedges, b = ao/2 1110] radius = 1600 A (* = point of emergence of dislocation) ............73 28. Magnitude and direction of the displacements used to generate an edge dislocation from the perfect lattice .. 75 Vi LIST OF FIGURES-Cont-~nued Figure Page 29. Pure edge dislocation causin2 single spiral on (204) planeoedges b = ao/2 [110] radius = 1.600 A (* = point of emergence of dislocation) ............77 30 Two sets of plane edges visible in the region of a defect (radius =800A) Note that the g- criterion is satisfied for each set of planes (*= point of emergence of dislocation) ..........79 31. Schematic drawing indicating the relationship between the sense of the spiral on the image (clockwise, counterclockwise) with the direction of the normal component of the Burgers vecto (_+ad ~ represent unit vectors normal to the set of intersecting planes) ................81 32. (a) Two dislocations each producing single spirals with the same sense: the long range effect is a double spiral; (b) two dislocations each producing a single spiral of opposite sense yield no long-range effect; (c) two dislocations equidistant from pole producing spirals of opposite sense; (d) two dislocations producing spirals of opposite sense. Due to proximity of dislocations the effect is an extra plane segment .. ....... ...83 33a. Experimental image of Pt. Defects shown in the region of (204) pole (A and B); also shown is (002) pole (C)....................86 33b. Computer simulated image of defect configuration on (204) pole in Pt; radius 650 1k (x = point of emergence of dislocation ) ...... ..........86 vii LIST OF FIGURES--Continued Figure Page 33c. Computer simulated image of defect configuration on (002) pole in Pt; radius 900 AO( = point of emergence of dislocation).................87 34. Single spiral on {220} plane of an Ir specimen starting at point A. Points B and C indicate other disturbed regions probably due to the presence of dislocations................. .....91 35. Triple spiral on {1131 plane of an Ir specimen....................92 36. Simulated image of triple spiral caused by pair of noninteracting dislocations . 94 37. Single spiral on {220} plane in Ir specimen. Point A marks emergence of dislocation while point B shows another disturbance which cannot be interpreted from this single micrograph. .........95 38a. Horseshoe configuration on {100} plane of iron whisker .................98 38b. Simulation of defect configuration in 38(a) using pair of dislocations of mixed character normal to the surface (= point of emergence of dislocation) . 98 39a. Single spiral on {0ll} plane of tungsten specimen...................100 39b. Computer simulation of defect shown in Figure 39a with dislocation line normal to the surface and b = ao/2 [110] (* = point of emergence of dislocation)......100 40. Micrograph of Ru showing existence of double rings on poles A and B .........102 41. Geometrical origin of "double rings" due to "rippled" nature of certain planes in hexagonal materials .........103 viii LIST OF FIGURES--Continued Figure 42a. Simulation of perfect lattice, (1012) plane, c/a0 = 1.58, showing the double ring nature ............... 42b. Simulation of the perfect lattice-, (1011) plane, c/a0 1.58, showing the "double ring" nature. ............ .. . 43. Dislocation on (1011) plane of a HCP material with c/a0 = 1.58 (* = point of emergence of dislocation)... .. .. .. Page 105 105 107 44. Solid arrows correspond to partial dislocations which yield stacking fault between the dislocations of the form ABCBCA. Dotted arrows show incorrect sequence of P's giving rise to two "A" planes being stacked together. ...........110 45. Pair of Shockley partial dislocations on (220) plane; total 9= ao/2 [101] (* = point of emergence of dislocation)...........111 46. Pair of Shockley partial dislocations on (002) plane, total 8 = ao/2 [110] (* =point of emergence of dislocation)...........112 47. Micrograph of Pt taken at 210 K. Distortion around the edge of the image (A and B) may be due to the presence of stacking faults. ...... ...114 48. Schematic drawing of (111) planes in the vicinity of (a) a vacancy 1oop and (b) an interstitial 1oop. ........116 49. vacancy loop on (220) pole showing distortion in the direction normal to fault plane (* = point of emergence of dislocation)................117 ix LIST OF FIGURES--Continued Figure Pg 50. Extrinsic loop on (220) pole, showing distortion in the direction normal to the fault plane (* = point of emergence of dislocation) ................119 51. (a) Experimental image containing defect believed to be Frank vacancy loop; (b) computer simulation of (a) (* = point of emergence of dislocation)..................121 x Page Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AN INTERPRETATIONAL STUDY OF FIELD ION MICROSCOPE IMAGES By Roger Carl Sanwald August 12, 1967 Chairman: Dr. John J. Hren Major Department: Metallurgical and Materials Etigineering Two computer models, neighbor model and shell model, have been presented to aid in the interpretation of field ion micrographs of both perfect crystal structures and crystal structures containing defects. The development of the neighbor model, which uses the geometrical environment, or equivalently the atomic binding energy, as an imaging criterion; resulted in point for point agreement with an experimental image. It was shown that variation in intensity on a single plane correlates well with changes in local sublimation energy A; however, it appears that additional energy terms are required for intensity variations between planes. In order to apply the neighbor model to the dislocated crystal, an accurate knowledge of the interatomic xi potential function must be known, but this is, in general, not available. It was decided, therefore, that the dislocated crystal could more readily be interpreted using the shell model for simulation of the images. A physical basis for the shell model was established by using the neighbor model to show that those points which contribute to the image in the shell model are the atoms which are bound weakest, i.e., the atoms which protrude the most from the surface. The shell model was used to predict defect configurations in field ion images in the FCC, BCC, and HCP systems. Basically all line defects and defect configurations composed of total dislocations could be interpreted in terms of spirals by using the _*. criterion. Therefore, the important parameter in determining the resulting configuration in the micrograph is the component of the Burgers vector normal to the set of planes on which the dislocation emerges. A field evaporation sequence is usually helpful in determining a unique Burgers vector. Partial dislocations were studied in the FCC system. The stacking fault separating two Shockly partial dislocations causes a displacement of the rings along the intersection of the fault plane with the surface. The long-range effect due to the pair of dislocations is that predicted using the total dislocation from which the pair xii has dissociated. Secondary defects, vacancy loops and interstitial loops, were also simulated using a pair of dislocations, because the intersection of a loop with the surface can be described by a pair of dislocations of opposite sign (dipole). Identification of intrinsic vacancy loops was found to be a straightforward matter, intrinsic loops causing an inward collapse of the "rings." normal to the fault plane, while the opposite effect was observed for extrinsic loops. The results obtained allow one to interpret defects and defect configurations in field ion images with confidence, since both a physical basis and analytical means have been presented for the interpretation. Xiii INTRODUCTION The field ion microscope is the only instrument capable of atomic resolution andltherefore, offers many unique advantages for the study of defects on the atomic scale. Although the microscope was developed in 1951, Cl) its potentialities as related to metallurgical problems have only been realized in the past few years. Most of the early work was confined to an understanding of the principles of operation of the microscope,,with little quantitative work directed toward the interpretation of the images obtained. The potential value of the field iJon microscope as a metallurgical tool can be realized only if the problems of image interpretation can be elucidated. It was with this idea in mind, that the present research was undertaken. The role of defects is influential in determining material properties and an understanding of the defect structure is essential in practically any basic study. Therefore, the ultimate goal of the work was the interpretation of defects and defect configurations in field ion images. The geometrical problem of image simulation is difficult, with the numerous calculations involved requiring -l1 -2 - the aid of a computer. The problem of defect interpretation has, to date, only been attempted on a very qualitative level with many resulting ambiguities in the methods and results. Moore ()has shown that it is possible to simulate many characteristics of a field ion image of a perfect crystal, but the case of the dislocated crystal has never been investigated. The present work was aimed at an investigation of methods of image simulation, and using these methods, to investigate and interpret images containing dislocations. It was felt that this could be accomplished most convincingly by comparing simulated results with experimental images whenever possible. CHAPTER I REVIEW OF THEORY OF FIELD ION MICROSCOPY The Field Ion Microscope and Principles of Operation Introduc tion The field ion microscope was developed by Muller(1 in 1951. The microscope is capable of resolving atomic positions on the surface of a conducting specimen, and is basically a projection microscope in which the atomic positions on the surface of the specimen are projected onto a flat plane, thus producing the image. A schematic diagram of the microscope is shown in Figure 1 and a typical image of an iridium, specimen is shown in Figure 2. The specimen consists of a fine wire which has been electrochemically polished to a fine point. In order to obtain an image, the radius at the tip of the specimen should be in the region of 100-2000 A. The specimen is maintained at cryogenic temperatureswith a positive voltage applied, in the presence of a so-called "imaging gas." Due to the extremely small radius of the tip, very high fields can be obtained when a potential is applied to the specimen. Therefore, ions created at the surface of the specimen by a process known as "field ionization," are -3 - -4 - Cryofi --Liuipd H2 Genrator mgn a Liquid N2 Cryostat ---Liquid H2 To L_" Racofn High Voltage Raditio She~dLeodttTouqh Fibr Otic 'Aindowv Figure 1. Schematic diagram of field ion microscope. - 5 - Figure 2. Field iLon image of an iridium specimen. -6 - repelled toward the screen by the high positive potential, thereby striking the phosphor and forming the image. Before explaining the physics of image formation, it is necessary to understand two processes: (1) field ionization, and (2) field evaporation. Theory of FieZ-d Ionization One can consider a free atom as a potential well in which the valence electron is trapped at a depthV1(3 (Figure 3a). In the presence of an electric field, this potential can be modified, as shown in Figure 3b. According to wave mechanics it is possible for an electron to tunnel from A to B. (4) The stronger the field, the greater the slope of the perturbation and hence the width of the barrier through which the electron must tunnel is decreased. It can be shown that as the width of the barrier decreases, the tunneling probability increases ()and is given by an expression of the form: D =exp c.. /2-M (V-E) dx) , where M is the electron mass, ti is Planck's constant divided by 2H1, V is the potential energy, and E the total energy of the electron. (The limits of integration extend over the width of the barrier.) If the free atom is brought close to a positively charged metal surface - 7 - and the field is sufficiently high, it becomes possible for an electron to tunnel into the metal. This process is known as field ionization. ()However, when the energy of the tunneling electron decreases below the Fermi energy of the electrons in the specimen (the distance from the surface of the specimen at which this occurs is called the critical distance, x c) the tunneling probability decreases rapidly, since the number of unfilled states in the specimen decreases rapidly. Figure 3c illustrates the energy configuration of the gas atom and specimen at the critical distance, x c The analytical treatment of this problem is considerably more difficult than the case of an atom in free space. Interactions with the metal surface must be considered, since the electron can feel the force due to its "image" in the metal. The field required to ionize the gas atom will be dependent on its ionization potential, the 2 width of the barrier can roughly be taken as V I/F where F is the field and V Ithe ionization potential. Theory of Field Evaporation Field evaporation is different from field ionization in that ions are produced from atoms which are bound at the metal surface, these may be surface atoms of the specimen or absorbed impurities on the surface; however, the term field desorption is usually used to describe the C f de - V, CC Figure 3. (a) Potential of free atomn showing valence electron at depth VI; (b) potential of atom in the presence of an'external field; (c) atom close to the surface of a specimen, illustrating the potential configuration at the critical distance. A, I 00 I - 9- latter case. In the absence of an applied field, the energy that must be supplied to remove an ion from the surface is given by the desorption energy (3) QC = A + V I 0 (A is the vaporization energy required to remove a neutral atom, V I is the ionization potential, and (D is the work function, Figure 4a). However, in the presence of a field, an additional term due to the polarization energy must be added on. ()On an atomic scale, the vaporization energy A can vary depending on the number of broken bonds of the particular atom, i.e., an atom on a high index plane which has, say, 3 nearest neighbors, will have a lower vaporization energy than an atom at the center of a (111) plane which has 6 nearest neighbors. The application of a field (Figure 4b) causes a decrease in energy with distance, and a resulting "Schottky hump" of energy Q must be overcome in order for the ion to be removed. The mechanism is a direct ionic evaporation over this "hump" with time constant Q/kt T= T0e where T0= 1/v is the reciprocal Vibrational frequency of the bound particle. ()The size of the Schottky hump can be decreased if the field is increased, and when this hump is reduced to the magnitude of the vibrational energy, there is a finite probability that the ion can surmount this barrier and be removed from the surface. When this occurs, the atom has been "field evaporated." Field =o V, (D I Z A >% %.. Field= loo ?4/c "N, Q B Figure 4. (a) Energy diagram for the removal of an ion from the surface of the specimen (no field applied) ; (b) the same system as (a), with a field applied, showing the existence of the Schottky hump. 'I CD - 11 - Principles of ITmagre Formation in the FIM Applying a positive potential to the specimen causes the electron cloud to be drawn back slightly into the metal, thereby exposing regions of high positive charge which correspond to the atomic cores of the atoms in the specimen (Figure 5). Due to the very small tip radius, very high fields (400-600 MV/cm) can be obtained with applied voltages in the range of 0-30 kV. On an atomic scale, the field is highest where the positive cores protrude and decreases between atomic positions (see Figure 5). When an imaging gas atom approaches the vicinity of the specimen, it is polarized by the high field, and is attracted towards the specimen. Due to the polarization of the gas atoms in the field, the supply at the tip is "enhanced" above the value that would be predicted from kinetic gas theory. ()Its energy is lowered as it moves towards the specimen, and as the field increases, the tunneling barrier decreases. Finally, the gas atom reaches a point near the surface at which the energy of the valence electron corresponds to the Fermi energy of the electrons in the specimen. The distance from the specimen at which this occurs is the critical distance, and when the gas atom is closer to the surface than this distance, the tunneling probability decreases rapidly (Figure 3c). The reason for the sudden decrease in - 12 - Field Electron Gas Figure 5. Surface of specimen with high field applied, causing positive atomic cores to protrude, as electron cloud is drawn back into the metal. Approximate variation of field with position is also shown. - 13 - ionization probability is a result of the fact that below the Fermi energy, the number of unfilled states into which the electron can tunnel at cryogenic temperatures, decreases rapidly. There is a narrow region above the surface of the specimen where the ionization probability is appreciable. The region where most of the ionization occurs has been shown to lie in an energy band approximately .2 eV wide at a distance from 4 to 8 A above the surface, depending on the imaging gas. If the gas atom is not ionized on its first pass through the "ionization zone,"(8) it collides with the surface, loosing some energy. As its energy is decreased, therefore it spends more time in the ionization zone after each "bounce," while becoming thermally accommodated. The probability of ionization increases with each bounce, until finally it is ionized at a point over one of the protruding positive cores (Figure 6). Muller has calculated the number of "bounces" before ionization to be somewhere between 100 and 200. Even though the quantitative relationship between the equipotential lines and the underlying atomic structure is not known, it must vary qualitatively as shown in Figure 5, the field being the highest above the underlying atomic positions. Once the particle has been ionized, it is repelled towards the screen along a field line until it finally strikes the phosphor on the screen. - 14 - Polarized gas atomn I Cu~es of emitted ions Ionizationi in high field regiwis Figure 6. Illustration of the imaging process in the field ion microscope. - 15 - Figure 7 is a current voltage characteristic for a typical field ion emitter. ()As the voltage is increased initially, the ion current increases very rapidly (linear region AB); this is due to the rapid increase of the characteristic ionization time with field. At point B the ionization probability has saturated at a value corresponding to ionization of a gas atom near the surface, and the slow increase from B to C corresponds to the "enhanced" arrival of gas atoms at the tip with increasing field. The point C marks the onset of field evaporation of ions from the surface of the specimen. The field ion image can be thought of as a map of the electrostatic field over the surface of the specimen which has a one to one correspondence with the underlying atomic structure. - 16 - Ion Current Amps c Tip Radius x570A He Gas P-6 10-3 Tarr Tip Temperature z78'K 10-12($-Field 36 I 2 3 1 15 16 V Voltage Figure 7. Current voltage characteristic for a typical field ion emitter (after Brandon (9)). CHAPTER II REVIEW OF INTERPRETATIONAL WORK IN FIELD ION MICROSCOPY Perfect Crystal Interpretation Image Geometry Figure 8 is a field ion image of an iridium specimen, taken at liquid hydrogen temperature, where the dark circular region depicts a low index crystallographic pole of the type {002}. Atoms usually cannot be seen in the central regions of the low index poles, at the imaging voltage, due to the close spacing of the atoms on these planes. The close spacing of the atoms reduces the effective ionization field, as compared to the field in the vicinity of the atoms at the edge of the plane (Figure 6). In order to obtain the maximum amount of information from a photograph, images are usually recorded at the "best image voltage." ()This corresponds to imaging approximately one in every five atoms on the surface. When the voltage is increased above this value, the resolution in the higher index regions decreases, due to the increase in ionization probability over the entire plane. Theoretically, it is possible, however, to image atoms at the central regions of low index planes, by simply increasing - 17 - - 18 - Figure 8. Field ion image of an iridium specimen taken at 210K. Low index plane A (200), and a high index plane at B (931) are labeled. - 19 the field above the "best image voltage" until sufficient ionization occurs in the central region of the plane, thereby making the atoms visible. However, on a low index plane, at best image voltage, only atoms at the edge of the plane contribute to the image. This is not the case, however, for a high index plane, since adjacent atoms may be 5th or 6th nearest neighbors, and each atom on a high index plane can be thought of as protruding from the surface (Figure 9), with a large field being created above each atomic position in the plane. Thus, all atoms on the plane will contribute to the image. In general, the lower the index of the plane, the more prominent the plane will appear in the image. (1)This criterion is true if one uses the indices of a plane to fix both the atomic packing, and the spacing between planes. For example, in the FCC lattice the (110) plane does not exist, (the atomic packing on a (110), (220), etc., is the same, only the spacing between planes is changed if the indices are multiplied by an integer), since a FCC lattice could not be constructed in which the spacing of this particular plane wasa0 (a0 = lattice spacing). The correct labeling of the plane in question must be (220), since to construct the FCC lattice, the spacing of these planes must bea 18 - 20 - 4th or 5 th Neighbors Ist or 2nd Neighbors (b) Figure 9. Relationship of neighboring atomic positions on (a) a high index plane, and (b) a low index plane. - 21 - Projection Geometry The angular relationship between planes has been shown to be a pseudo-stereographic projection, the projection point being located at a distance 3R from the image plane, instead of the usual 2R as is the case for a stereographic projection (1)(Figure 10). One method commonly used to index field ion images is to plot the planes on a stereogram, and using this coupled with the knowledge of the symmetry of various planes, one can then determine the specific indices of the planes. Another method has been developed which employs only a linear measurement on the photograph, (1)and a knowledge of the optics used in recording the image. The equation is simply d =MRe, where d is the linear distance between two poles on the photograph, MR is a constant dependent on the recording optics and microscope geometry, while a is the angle between the planes in question. In practice, the product MR is computed for two known planes, then using this, the angle between any other planes can be determined by a single linear measurement. Specimen Endform Geometry In most systems (especially pure metals), determination of the various crystallographic poles in an image - 22 - Pq sf o Projection A Plane (X---X 0 B C z R R R Figure 10. The geometry of several proiections: P g(gnomonic projection), PS (stereographic projection), P0 (orthographic projection), Pf (pseudo-sterographic). - 23 - is a straightforward matter. The endform of the specimen will affect the local magnification, and hence the apparent spacing of poles in a image. The magnification at any region of the image is simply R/ r, ()where R is the specimen to screen distance, r is the local radius, and is the image compression factor, dependent on the microscope geometry. The radius of the tip is not constant, but can vary by a factor of 3 or 4 over the image. The local radius r, between any two poles can be computed by a method first proposed by Drechsler and Wolf. (13) It consists of counting the atomic steps between two poles; the radius is then given by: r = d 'cs here n is the number of step edges of a particular plane between the two planes, 0 is the angle between two planes, and "d" is the interplaner spacing of the planes being counted. The resulting r is the average local radius between the two planes. An image which appears elongated in one direction is sometimes observed. The endform of the specimen in this case is asymmetric (chisel shaped), and can be characterized by two radii, one being very much greater than the other. Thus, the magnification is greatly increased along the direction of the smaller radius. Figure 11 shows the resulting image from an asymmetric Ir specimen. - 24 - Figure 11. Image of asymmetric Ir specimen, taken at liquid H 2 temperature. - 25 - Defect Interpretation Po-nt Defects The vacancy is usually easy to recognize in a field ion image. Since atomic positions on the surface of the specimen are seen, a vacancy appears simply as a missing atom, and is most easily recognizable in the center of a high index plane. (14) There are many regions of the specimen where a vacancy is undetectable, for example, in the interior of any low index plane, i.e., (200) or (111). An apparent missing atom at the edge of a plane cannot be interpreted unquestionably as a vacancy, since this atom may have been preferentially field evaporated. Preferential field evaporation could, for example, be due to an underlying solute atom of larger size, causing the atom above it to protrude more from the surface than its neighboring atoms, and be field evaporated. Other point defects such as interstitials usually appear as randomly distributed single spots, which are much brighter than the ordinary image points. Not much work has been reported with regard to the interpretation of impurity atoms, and the effect on the image can be due to both size and electronic interactions with neighboring atoms. The electronic effects are difficult to determine, and the theory has not reached the point where - 26 - they can be predicted. Impurities have been observed along rows, and also as point defects, (15) inthe image; however, no quantitative work has been done in this area. Planar Defects Grain boundaries can be considered as a type of "defect" or "irregularity" in a field ion image. The crystallographic analysis of a grain boundary can be achieved by simply determining the orientation of each grain; however, in order to determine the plane of the boundary, a field evaporation sequence is required. In this way, the boundary configuration can be followed on an atomic scale. (16 It has been suggested that the appearance of stacking faults in the image can give rise to "streaks." (17) This work was very speculative, and no justification or proof for this interpretation has been shown. Linear Defects By far, the least understood, and probably most important defect, at least from the viewpoint of mechanical metallurgy, is the dislocation. Drechsler et-al. first reported seeing dislocations in field ion images in several materials. (18) The intersection of a dislocation with the surface was thought to yield an extra plane, as would be the case for the classical edge dislocation on a {112} plane. - 27 - Muller has also reported a pair of screw dislocation's in an iron whisker; (19) however, no quantitative basis is given for the interpretation. Brandon and Wald (20) interpreted the defect shown in Figure 12 as an edge dislocation. The analysis proposed by Brandon and Wald is that only the edge component of the dislocation can be analyz ed since all dislocations are normal to the surface they intersect in a field ion tip. Therefore, only lateral displacements can be detected, i.e., only edge character can be observed. The basis for defect interpretation was a simple geometrical model, in which the edge component was thought to cause the appearance of an extra plane, depending on its location in the image. For example, in a BCC, if the dislocation is assumed to be 2 0 (111], then an extra (111) plane should appear in the image, as a result of the dislocation. Several examples of dislocations in Pt have been seen; (21) however, their explanation is by no means unambiguous, and no attempt has been made to determine the 8 or line of the dislocation. More recently, a criterion has been proposed from which the effects.of total dislocations on an image can be predicted. It has been proposed that total dislocations will give rise to a spiral structure in the field ion image. (22,23) The number of leaves in the spiral configuration can be predicted by computing the component of - 28 - Figure 12. Defect in molybdenum interpreted as an edge dislocation (after Brandon and Wald(20)). - 29 - the in the direction of the normal to the plane under consideration. This component will always be an integral multiple of interplanar spacings, for that set of planes (this is because the tof a total dislocation must connect two atomic positions). If the 9 lies in the set of planes no spiral will occur in the image. This criterion is presented ad hoc, the reason given for this quantitative rule is very qualitative. Ranganathan (2)has compiled a table for BCC materials indicating the "number of leaves" to be expected on certain low index crystallographic planes. He has also shown several examples of dislocations in grain boundaries, and has used his criterion to interpret them. Figure 13 is an example of a "dislocation" in iridium, at a grain boundary. It should be noted that there has been no reported case in the literature of an isolated spiral configuration, i.e., not in a grain boundary, etc., as being interpreted using the criterion suggested by Ranganathan. A double spiral in Ruthenium has also been reported;(4 however, here again it is in a grain boundary. (It is of course very speculative to propose seeing individual dislocations in grain boundaries, since the structure of grain boundaries in general is not known.) It is possible that images next to grain boundaries may look like "dislocations," i.e., spirals, due simply to the geometrical orientation of the boundary. However, in small angle - 30 - 7%0YL U I kL, Figure 13. Single spiral at grain boundary in iridium interpreted as a dislocation (after Ranganathan(23)). - 31 - boundaries the possibility of seeing individual dislocations in the boundary cannot be discounted, since it has been shown that a small angle boundary can be constructed.,using individual dislocations. Ryan and Suiter (25) have reported seeing partial dislocations in W. The argument used in interpreting the defects is generally the same as that of Ranganathan; however, they make no attempt at any quantitative correlation, i.e., if the 8 lies out of the plane under consideration then some distortion will occur in the image. This argument is also used by them in the interpretation of dislocations in high angle boundaries in tungsten. Computer Simulation of FIM Images Image Simulation of' a Perfect Crystal: Shell Model The most rigorous attempt to date at interpretation of the geometrical aspects of a field ion image, has been the "shell model" proposed by A. J. W. Moore. (2) This model has been used by Moore to simulate many of the general characteristics of field ion images of both BCC and FCC materials. With the aid of a computer, Moore was able to obtain plots of the image, which were in very good agreement with an experimental image. The model, as presented, was used ove r the region of one complete stereographic triangle, and good correlation of the relative prominence of poles, - 32 - and the density of image points with an experimental image were shown. The program was written so that the coordinates of either BCC or FCC lattice points could be determined over the region of a stereographic triangle. A thin spherical shell is made to intersect the point lattice, the radius of the spherical shell corresponding to the local tip radius. The shell thickness is an empirical function of the radius, but is usually .02 .2a 0, where "ia 0'" is the lattice parameter. All points lying within this thin shell are considered image points; they are then projected onto a flat plane and the final image formed. Basically, this model takes atoms which are very close to a spherical surface, and projects them onto a plane. Muller has suggested that those atoms which "protrude" the most from the surface will contribute to the image. The "shell" is basically a measure of the protrusion of an atom from a hypothetical spherical surface. Moore has shown that the shell thickness must be decreased as the radius is increased. He determined this mainly by point counting, i.e., comparing the number of points in the experimental image with those obtained using the shell model,. He also found that in order to obtain good correlation with an experimental image over the region of a stereographic triangle, it was necessary to change the 33 shell thickness in various regions. This is due in part to the fact that the radius is not constant over a stereographic triangle. The computed image is extremely sensitive 0 to radius; a change in radius of say .05A can cause a marked change in the topographical features of a high index plane (Figure 14). Changing the radius by one interplanar spacing plane will cause an atomic plane of that species to be removed. Since the spacing of a set 0 of high index planes is of the same order as .05A, simulated field evaporation of this plane can occur, with a concomitant change in the topological features. The utility of the shell model is that it does show that it is possible to simulate at least some features of a field ion image by analytical means. SHELL MODEL Shell Thickness= e5A Radius= lug ~ '.~ 0 o% .0.. 0 0 0000 * 00 * 0.0 *0 0 0 % * 0.0 000 -' 00.. * 000% *. o~5 ~o~**** ** Oo* .00 0 0 00 .0 % % 0o~ **o 0000 o~ 0 0 *..* 0. o** o~ 0. 0 0 0 *o0* **.* *. 0. 00 *o 0 0 ** 0 0. 00 000 .0. 00 sO *Ooo 000 .:~. 0 *.%?%%.. 0 *o . *00 0 0 *~ .0...:. 0, -. *o *.*.: ... **** 0. 0~ * 0 00 .00 *0 0 %. 00.00.00 0 00 .00 0 0 0 0 0 00 %Iq1l~00. 0 0.00 00 00 *ooooo* UIJ~ 0000 000 0 0. 0.:..... ~ 0 ::.: (531)0. 0* 000 00 *0 00 0 S 000% .000 .00 0.0 00 000 00%o.. .0.0 0.0 0000... 00.0 #000. 0000 00000 000 0 00000000000 0000 00.0 .00 % 0 0000.0 .0 00000000.00 00 0 0 0 g.**,00o*. 0 3 .000 00~ ~ 1~1 g ; I' (110) .1~ Radius=ioo.SSA 0*** 0 0 0 0 0. *0~ 0 00 0 0 0 0 .0 *o ~. 0. ** *0 ** ** 000 0 0 0. 0 0 .0 .* 0.0 00 0 0 0 *0 00 00~. * 0 0 *0 0 .* 0 0 0 0 * *0.* 00 .0 0 *~* 0 0* ~* 0 0 0 0 %0~ 0 0 0 0 0 *0 *~: :: 00, *0 0* 0 0 0 0 0 0* 0 * 00, *. 0 o 0 0 %* ** 0 * 0 000 0% 0 0 *00 00 * 0 0 0 *00 0 0 .' ~ 0 0 00. .0 * *. .:: 0 Z(211) ~ ::. *. .(531) * *o 0 0 0 -~ 0 0 0o* 0 00 *.oo N .0 *0.. 0 0 * 0.. 0 0 00 ~. 5 0 0 * 0 0 0 ** 00 *0 0 0 0 .0 0. 0 0 5 .0 * 0 S *p.00, 0, S 0 *o 0 *0 0 0 '%. 0 0 0 * 0 *. * 0 0 ** 00 **. 0.0 * 0 0 0 ~ 0 0 0 * Figure 14. Change in the topogrzaphical features of a simulated field ion image model) with a variation in radius of .05a0 (shell CHAPTER III RESULTS: INTERPRETATION OF FIELD ION IMAGES A Physical Model for Field Ion Image Interpretation Introduction The shell model seems to lack direct physical basis, in that it is difficult, if not impossible to correlate the imaging criterion (shell thickness) with the interatomic parameters of the specimen. It is clear, that atoms near the surface of a metal will be less tightly bound than those deep in the bulk crystal. Therefore, it was felt, that a criterion from imaging might be related to the binding energy of an atom to the crystal. Analyzing the geometrical configuration of the atoms which contribute to the image, in the shell model, might shed some light on an imaging criterion which would have both physical basis and better agreement with an experimental image than the shell model. The computer program for the shell model was written so that it would be possible to look at any region of the image, since in many cases it is impractical to look at the entire stereographic triangle (both costly and time consuming). Figure 15 is a series of plots obtained, using - 35 - Radius = 576.1 Radius = 576.05 Radius= 576. 0 '6* 0 o 101 0 (a) Radrius: 575.95 I .* 1. t SO (d) ll it (b) * Ra.u =* 575 87. * .** ***..I 2 S *e* .:.::::.. *. : * ~. z. ii. 2* ::~..~2* *2 *: S.. 0* (f) Figure 15. Simulated evaporation of the (420) pole in FCC using the shell model, caused by small change in radius. - 37 - the shell model, which shows the removal in steps, of one monolayer on the (420) pole (simulated field evaporation), the total change in radius being 0.64 A. The Neighbor Model This work was directed toward developing a model that could give point for point agreement with an experimental image. The FCC lattice was chosen initially, since the main objective of the work was the study of defects in field ion images (in FCC there is less ambiguity as to the types and configuration of defects that can exist as compared to BCC or HCP). It was also hoped that the studies on interpretation of perfect crystals would shed more light on the general problem of defect interpretation. The region of the specimen over which a spherical surface will approximate the tip is dependent, of course, on the endform geometry. A good approximation to the surface can be achieved over an angle of about 100 with a single radius of curvature. Therefore, in the neighbor model the tip is assumed spherical over a small region only, possibly one or two poles. The geometrical environment of each of the surface atoms is computed (i.e., number and type of neighbors, e.g., 1st, 2nd, etc.), and based on this image points can be selected and projected. Using Moore's model (shell model) as a starting point, a shell thick enough to - 38 - assure more than enough points for the image was selected. There is no reason'to select more than a "thick" shell to begin with, since all the lower lying points within the specimen, are discarded anyway. The number and type of neighbors for each point lying within this thick shell (p = .la where p represents the shell thickness, and a0 the lattice parameter) were then analyzed with the result that most of the points in the shell have six 1st nearest neighbors and three 2nd nearest neighbors. (The total possible number of 1st neighbors is 12 and of 2nd neighbors is 8.) This is not surprising since it would be expected that approximately 1/2 the number of neighbors of a particular atom located at (a, b, c) would lie on either side of a plane passed through the point. Since the tip radius is much greater than the neighbor distances considered (3rd, 4th, 5th, etc.), the tip can be regarded as a plane over a very small region and the number of neighbors of any one type (1st, 2nd, etc.) lying on either side of the plane is thus approximately 1/2 of the total. Figures 16a, 16b, and 16c show the variation in the number of neighbors (1st, 2nd, and 3rd) over a stereographic triangle. Note on the lower index planes, that the number of neighbors of any particular order (1st, 2nd, or 3rd) are, in general, greater than on a high index plane. This is a result of the closer spacing on the low index planes, yielding a - 39 - FIRST NEIGHBORS (111) PROJECTION SHELL =Oar a -6 FIRST NEIGHBORS it it 91 99 0 0 0 0 0 0 *o* 0 0 0 0 * 0 0 0 00 0 0 0 0 0 S 0 *0 00 0 .0 0 @0 0 0 0 000 0 0 0 .0 0 0 0 * 0 @0 00 * 0 0 * 0 0 0 * *0 0 00 00 0 S sO 0 000 00 0 0 Oo Og 0 0 * 0 .0 0 00 0 0 0~ *0 ~0 0 0 0 0 0 0 ~(100) 0 0 00 0 060 0 0 0 00 %.0% % a*0 00 00 000 0 05 00 0 0 00 p 0 0 Figure l6a. Variation of 1st nearest neighbors over a stereographicotriangle using the shell model (FCC, Radius = 350 A, shell thickness =.05a ). 00~7 ~ -8 0 0 @0 0 0 0 4 0 0 0 I 0 0 * 0 0 0 ..:o 0. :: 00000 0. 00 0 000 0 O 0 .0 00 0 0 O 0 0 0 * 0 0 000 000 0 0 so 0 0 0 0 0 0 0 0 0 0 * 0 0 0 .000 OZ Ot t to ~ (ilo) % I, It - 40 SECOND NEIGHBORS SHELL = .05 a (W1) PROJECTION -2 SECOND NEIGHBORS 91 is 0 ~(100) 0 * 0 G S C o C G ee.o Ge.. Ge C *~ %* C G C G Gc *c . Ge G G G G C *CcO Ge C Ge. * ** *%% ** c.e.o Ge * C * *e G C G ~ C C e ~ C Ge c G e~* C C.. e e e C 0C. 0 C CC e 0 * C ~ e* 008 CGe~ ~ GO G e e Ce. C e e e ~ C G * C C * .e S. e 0 0 0 0 * G C G C G *e C 0 C GO GO CI I f* j*'(iiO) Figure l6b. Variation of the 2nd nearest neighbors over a stereographic triangle using the shell model (FCC, 00 * -4 v -5 c e e G e 0 Ge. G e C CC G C a e C C Ge Cc c C cC a ( 0 0C a G e C e e 0 00 0 0 0 C Ge 0 c. 00 e. 0 0 0 g o 0 e 0 ; 0 0 0 0 C. 0 ow O e0 0 0 0 1~ 0 soc C~ 0G 0 G Ce 0C 0..e. 0"e - 41 - THIRD NEIGHBORS (i11) PROJECTION SHELL =.o5 a --I THIRD NEIGHBORS X-4 0 0 0 0 000 0 0 @0 0 0 0 0 0 0 0 00 0. ~* 0 00 * 0 * 000 0 .0 00 00 .0 0 * 0. 0 0 00 0. .0 0 * .00 00. .0 000 00 O 000 00 00 0 0~0 e* it 4? (too) 4 0 0 0 0 0 * 0 * 0 * * 0 0 0 0.0 * 0 * 0 0 00 *. 0 * *. 00 *00*. 0 ** 0 0. ** 0 0 *o 0 0 ** 05 ** 000 05. 0 0 0 0 *o 0 0 0 0 0 ** 0 0 0 0 *%o, 0. *o 0 0 .0 0 0 ** .0 0 0 0 00 .0 ~ -- : 0 0 0 %* 00 0 0 i : ~ :** 00 0 0 * 0 00 0. I 05% 0. 05 1.000 0. - to0w 0.. 0 ~"*1 * 05.00 *.~o 0 0: : 0 .00 0 0 00.0 0 00~ 0. 0000.0 0 00 ** .0 0 0 0 .0. 0 00. 0 * 00 0 0 0 0 0.00.0.0.00. : 0000 0 0 O .0 o 0 0 gos 0 *~0 0 Og 0 0.0 0 *0* 0 00 0 0 00 0 0.00 0 0 00. ~** .0 .:: .0 0,0 .' 0 00 0 0 o 00 .0000 ~ :~ :0 0 Figure 16c. Variation of the 3rd nearest neighbors over a stereographic triangle using the shell model (FCC, Radius = 350 A, Shell Thickness = .05a ) 42 greater number of neighbors. A program was then written, in which the imaging criterion was the geometrical environment of an atom (the number of 1st neighbors, 2nd neighbors, etc.). A general outline of the actual simulation procedure is shown in Figure 17. The program is set up so that a solid rectangular parallelopiped containing points in an FCC array, can be generated anywhere in space, with respect to an orthogonal coordinate system (x, y, z). The size of this volume is fixed by six adjustable parameters which determine the maximum and minimum boundaries of the solid in space. The orientation of this volume in space can be adjusted, such that any desired poles can be obtained in the final image. This is easily done as follows: the position of the pole in space is computed, then based on the generating coordinate system (see Figure 17:1) and depending on the size of the region to be computed, the bounding values of the coordinates of the region are computed. This data is then used to generate a data tape, on which are stored the coordinates of the points in the FCC lattice lying-in the thick spherical shell. The advantage of the data tape is that data does not have to be generated each time the program is run, but can be used over and over. The reason for using a rather thick shell FIELD ION IMAGE RECONSTRUCTION 0 Coordinates of pt. in FCC lattice are determined. 1000 Va y 110 z r r r It r'< r pt. considered If r".r pt. discarded x I Environment determination Possible image pt. ~ ~~-., Ist neighbor shelf ~ .2nd neighbor shell r I' z Determination of coordinates z in rotated system x',y,'z' to give (a'b,c') Projection y' is projection direction Projection pt. peci m en Projection sphere Figure 17. Schematic diagram of the basic steps involved in the computer program to reconstruct the field ion image. I~ Image pt. Y ,1 z - 44 - (about "a 0" thick) is so that the radius can be varied slightly to permit some flexibility when correlating the image with experimental results. Having this information, a spherical surface corresponding in size to the local tip radius is passed through the points in the lattice (these are stored on the magnetic data tape) (Figure 17:2). The distance from the origin to each point must then be determined in order that the points which lie outside of the spherical surface can be discarded, since these are physically not part of the specimen. This then leaves points within the spherical surface to be examined in detail. Knowing the coordinates of a point inside the spherical surface (a possible image point).. the number of neighbors of any order (1st, 2nd, 3rd, etc.) can be computed (Figure 17:3). The coordinates of the various neighbors (1st through 6th) of a point in the FCC lattice, situated at the origin, are known. These were computed by finding all points in the lattice which are located at, say, the nearesst neighbor distance, 2nd nearest neighbor distance, or at any distance from the origin desired. A program has been written which gives the number and coordinates of these atoms situated at any distance from the origin. Knowing these coordinates, a simple linear transformation - 45 - gives the coordinates of any desired neighbors for a point in the lattice situated at any arbitrary position, x, y, z. For example, if the coordinates of a particular neighbor are xl, yl, and zl, and if the atom is situated at position (x, y, z) with respect to some origin, o, then the coordinates of the neighbor with respect to the same origin "o," are simply (x0,t y0 z 0) x =x + X o 1 YO y1 + y z 0 z 1+ z The distance from the origin to the neighbor under consideration is then computed; if this distance is less than the distance to the spherical surface, it is counted as a neighbor of that particular order, be it 1st, 2nd, 3rd, etc. This process is then repeated until the neighbors out to any desired order have been determined. The actual imaging criterion is now imposed; that is, if a point has the correct number of neighbors, it is then considered an image point, if not it is discarded. Determining the "correct" number of neighbors is done by comparing the resulting image with an actual experimental image; this process will be explained later in more detail. - 46 - At this stage, only the coordinates of the point are known with respect to the x, y, z, orthogonal coordinate system (refer to Figure 17:4). Obtaining the actual coordinates of the point on the image will now be explained in some detail. The x, y, z coordinate system is fixed in space, the x, y, and z axis corresponding to the [1103, [110], and [0013 directions in the FCC lattice respectively. On the other hand, the x', y', z' orthogonal coordinate system is adjustable in space (Figure 17:4), to the extent that it can be rotated while keeping the origins of the two coordinate systems coincident. The y' axis is adjusted such that it lies in the direction coincident with the pole of the projection desired, e.g., if a (111) projection is desired ((111) in the center of the image), then y' will lie along the [111] direction. It has been shown (11) that when the projection point (refer to Figure 17:5) is approximately 3R from the screen, where R is the specimen to screen distance, good correlation with an actual image is obtained. This, however, must-*be varied depending on the experimental conditions, and in the present program this projection point can be changed, so that actual imaging geometries can be closely approximated. The coordinates of the image point in the rotated coordinate system are now computed by using the standard rotation transformation equations. (6 - 47- The point is then projected radially onto the projection sphere by multiplying the coordinates of the point by the magnification, R/r, where r is the local specimen radius. The final step is then to project the point on the projection sphere onto the screen from a point located on the y' axis approximately 3R from the screen. Simple geometry gives the coordinates of the image point on the screen. This process is continued until all points which satisfy the imaging critericn have been found. These points are then plotted and the final image is obtained. A detailed description of the equations used in this analysis are given in Appendix I. Figure 18 shows the actual projection in more detail, the relation of the specimen surface and imaging atoms (dotted line in insert passes through image points) is shown. Note that the drawing is not to scale, R is of the order of magnitude of 10 cm, while the specimen radius is approximately 10- cm. Imaging Criterion Correlation between the image and the "neighbor model" is obtained by adjusting the imaging criterion, i.e., the geometrical environment of an atom. Assuming a monotonically decreasing potential to exist, (27) relative binding energies can be assigned to the various neighbor combinations. Since the distance to any neighbor is known, - 48 - /Spherical Screen I Projection mnage Pt. R Projection Sphere Pt. Figure 18. The geometry of the projection showing the relative relationship with the specimen. - 49 - a relative energy can be assigned to any neighbor, if an interatomic potential function is assumed, i.e., variation of energy with distance. It is assumed that the lowest energy configuration will always image, therefore it need only be decided where the high energy cutoff point is. The atoms which are the weakest bound have the greatest number of broken bonds and therefore are those atoms which protrude most from the surface, e,.g., an atom on a (1179) plane is much less tightly bound than an atom on a (110) plane (see Figure 16a, 16b, 16c). Since the atoms which contribute to the image have geometrical environments which are very similar (Figure 16a, 16b, and 16c), i t is expected that the effective binding energy of any atoms which contribute to the image will lie in a narrow energy band. It has been proposed (3,9) that the binding energy of an atom could be described by an equation of the form, Q = A + I n n D (where A is the sublimation energy from kink sites, I n is the n thionization potential, and (D is the local work function). Muller (28) has proposed an additional term p which must be added due to "field penetration polarization." The"field penetration polarization" is given by an expression of the form, Pa = a 01/2 d 1/6S 7/3 F 02/3 2n1 ergs/atom, where d = spacing of planes, a 0is the lattice constant, s is related to the density of atoms on the particular plane, and F 0is the - 50 - evaporation field. This term is suggested to account for the orientation dependent part of the field evaporation energy. In other words, assuming A and I~ to be constant, the variation in D does not account for the variation in field evaporation energy in different regions of the specimen; however, Muller suggests that the p a term may accoun t for this. The field penetration polarization energy is larger on a more open plane, i.e., high index plane, and is a measure of the amount by which the electron cloud is pulled back into the metal thereby exposing more or less of the positive ion cores. In the neighbor model, the differences in the geometrical environment of an imaging atom are related directly to local differences in A, the energy of sublimation, which has been considered to be a constant by most investigators. It is possible to calculate relative differences in A through the use of a suitable interatomic potential function. The work presented here, in effect, explains certain features of the image on the basis of local variations in A only, which will also effect the orientation dependence of the field evaporation energy. Results on the (113) plane indicate that at least some atoms with 7 nearest neighbors would have to be included to adequately simulate the image. This is not unexpected, since one would expect the work function and - 51 - the polarization energy to change in this region as compared to the (931) region. There is no reason to expect that the work function should be constant over these two regions. After analyzing the environment of atoms in the shell model, it is found in many cases that some atoms with the same environment as those within the shell lie outside of it (Figure 19, for example atoms 4 and 5). The shell model would image only atoms 1, 2, 6, 7, and 8, whereas the neighbor model proposed here would image all the atoms on this plane, namely 1 through 8. Experimental images correspond to this second criterion. It appears that the order of neighbors that must be considered is directly related to the distance between atoms on a particular plane, i.e., with a greater separation of imaging atoms, combinations of higher order neighbors must be considered in order to adequately simulate an image. Atoms at the edge of a plane will in general have a different environment than an atom in the center of the plane; this difference must be detected in order to simulate the image. Simulation of the (93Z) Region The (931) region was chosen to be investigated in some detail for the following reasons: - 52 - *- 2 Spherical Surface r 0 0 0 0 6 0 0 7 0 8 Shell Figure 19. Comparison of the imaging criterionof the shell model and the "neighbor model" on a high index plane. - 53 - 1. Good resolution in an experimental image is obtained. 2. The shell model does not give adequate results on a point to point basis. 3. Since it is a high index region, 5th or 6th neighbors would have to be considered to simulate the image, as compared to only 2nd or 3rd neighbors on a low index plane. Assuming a linearly decreasing relationship between bond strengths, i.e., 1st neighbors have a higher binding energy than 2nd neighbors, etc., all atoms with six Ist neighbors and various combinations of 2nd through 6th neighbors are plotted in the region of the (931) plane. Table I is a compilation of all combinations of neighbors that were required to simulate the image in this region. Note that not all possible combinations of neighbors exist, but rather only a relatively small number; however, all combinations with six 1st neighbors that did exist, were used in the simulation. The experimental image of this region was obtained from Prof. E. W. Muller, and the *author is very grateful to him for this micrograph. Figure 20 shows the entire micrograph. The (931) region (shown blocked in) was studied in more detail. TABLE I BOND COMBINATIONS REQUIRED FOR (931) REGION Total Neighbor Possible* Order Neighbors Combinations Used to Simulate (931) Region 1st 12 6 6 6 6 6 6 6 6 6 6 6 2nd 6 2 3 3 3 3 3 3 3 3 4 4 3rd 24 12 12 12 12 13 13 13 14 14 12 12 4th 12 6 6 6 6 6 6 6 6 6 6 6 5th 24 12 12 13 14 12 13 14 12 13 12 13 6th 8 4 4 4 4 4 4 4 4 4 4 4 Many combinations do not give any image points; only those combinations which contributed points to the image are listed. I, - 55 - Figure 20. Micrograph Qf, Pt Cphotoaraph courtesy of Prof. E. W. Muller, taken in 1959). - 56 - The radius in the region of the (931)_ p~tane was determined graphically, by counting (002) rings. A detailed explanation of the method used will now be given, since the results indicate that it is probably the most accurate method of local radii determination that I have found. First, the step width and height of the planes (731, 931, and 1131) are computed along the zone passing through the (002) pole, since (002) steps were used in the radius determination. This is done by employing the equations developed for building ball models. (29) The step height is simply a 0/2, which corresponds to stacking (002) planes in Pt. The step width is given by what is referred to as the "total horizontal translation." Since the three planes considered all lie in the same zone, the step height remains constant, and only the step width changes from plane to plane. It should be noted that the equations used to calculate the step width and height taken from Moore and Nicholas can only be used for (002) steps or (111) steps. The height and width of any other type of step would have to be computed. The number of stepsp" on each plane are determined from the-micrograph, and the corresponding number are drawn to scale on a twodimensional grid. A best fit radius can then be drawn through corresponding points on each step edge, e.g., select the radius such that it passes through the corner - 57 - of each step edge. Figure 21 shows graphically, the method used for the radius determination. It should be noted that the radius of curvature of this region does not necessarily have any correlation with other geome tric parameters in any other region of the tip; it is merely some radius which approximates the actual curvature of the tip very closely, over a small region. When the radius is measured by the usual method using the equations reported by Drechsler and Wolf, (13) i.e., r = (nd)/(l-cose) where n is the number of rings between two poles, d is the spacing of planes of the rings being counted, and D is the angle between the two poles for which the radius is desired, radii ranging from approximately 460 A to 750 A were obtained. Using a best fit radius of 445 A, determined from plotting the actual steps (as just described) excellent agreement between the number of steps in the simulated image, and the experimental image was obtained. In the simulated image, point for point agreement is obtained,except there is an extra row of atoms along a (113) step edge. This can be readily explained if a more detailed analysis of the local surface topography is performed; that is, the radius between the (931) and the (113) is smaller than that between the (001) and the (931). Decreasing the size of the area under consideration, thereby simulating - 58 - R (Best fit radius) Figure 21. Determination of the local radius by graphically plotting atomic steps. - 59 - one pole at a time, instead of all three, would result in the disappearance of the "extCra' row. Figures 22a and 22b are a comparison of the shell model and the "neighbor model" with the actual image (Figure 23b) of the (931) region. In the (931) region, using all combinations with six 1st neighbors gives excellent agreement with the image. Note that all possible combinations of neighbors do not exist, but rather only a limited number occur, as shown in the Table I. As can be seen (Figure 22a), the shell model does not give the complete net of atoms on~the (731) plane. If the shell thickness is increased in an attempt to fill out the net plane, nonacceptable, image points (atoms with 7 first neighbors) begin to appear before the (731) plane has been completely filled. Note also the absence of points in the shell model between the (931) and (731) plane. Points in the actual image cannot be resolved in this region; however, the bright zone that does appear seems to better correlate with the results of the "neighbor model," i.e., an unresolved row of atoms between the (731) and (931) plane. The relative brightness of many of the image points on each plane can also be explained. If the. bond energy is assumed to decrease linearly with distance between atoms, Shell Model (931) Region 0 Radius= 445 A 0 0 0 0 0 .0 0 .0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 gO 0 0 0 0 0 0 0 .0 * 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Neighbor Model (931) Region I Shell Thickness -Ax 0 00 0 00 0 0 0 00 0 0 0 0 -0 0 0 0 0 00 0 00 00 .0 0. 0 00 0 : 0:0 Figure 22a. Shell model (9311 region. F ,~u re 22b. region. "fNeighbor model," (9311 0 0.0. 0 ~ 0 0 0 S 0 S 0 0 0 0 ~0 * -. 0 ~ .0 * 00 0 0 0 0 0 00 0 0 .0 0 0 0 - 61 - then atoms near the "edges" of the plane will have a lower binding energy than those in the center of the plane, and therefore would appear brighter in the image. If an interatomic potential is used to compute the binding energy of each atom, then this may be used directly as an imaging criterion. The atoms which are the weakest bound must always contribute to the image; however, the adjustable parameter is now the value of binding energy above which atoms will not contribute to the image. In order to compute the relative binding energies, it is necessary to know the shape of the interatomic potential function. Computations of an interatomic potential function for platinum, however, were not available. An approximation of the relative binding energies of any nearest neighbors could be obtained by comparing platinum to elements with similar electronic structure and binding energy A, for which the calculations of an interatomic potential had been done. Comparison of the Morse potential function of Ni and Cu (30) (Ni has the same outer shell electron configuratibn as platinum) showed that the relative weighting of neighbors was the same for each. It was assumed, therefore, that the relative weighting of neighbors in platinum could be approximated by using the values for Ni and Cu. This was - 62 - done for the (931) region, and the results are exactly the same as for the case of using the geometrical environment criterion. Figure 23 (a arnd b) shows the correlation of relative intensity with the calculated binding energy. The image points are labeled such that the 0 is the weakest bound, while those bound the tightest are labeled 10. The neighbor model points up the fact that it is possible to simulate an image on a point for point basis if one looks at a sufficient number of neighbors. Even though the shell model does not give point for point agreement, the atoms which do contribute to the image are by and large those which protrude most, based on the relative numbers of neighbors of each atom. The neighbor model goes further since it shows that variations in intensity on a single plane, and changes in the local sublimation energy, A, can also be simulated; however, additional energy terms are apparently needed to account for variations in brightness between planes. Application of the Model The neighbor model does present certain limitations if it is to be used to simulate defects. In a dislocated crystal, atomic positions have been perturbed as compared Relative Intensities Neighbor Model (931) Region 101 0 3 S ItS~ lotoIs 1 10 IQ10 to 109 t o t 'DI 10 1 t 9 2 2, to 10 2 2 2 Decreasing Intensity 1 210 2 2 2 2 ,low] Figure 23a. Experimental image, (931) region (after Mdller). Figure 23b. Relative values ofA, the local sublimation energy in the' (931) region based on the "neighbor model." 3 3 3 3 3 /2 / 2 N2 ~ 2 2 /1 5 'N S 1 61 2' 2 -5 5 511 \ 2/2 5 s 5 ,iI 5 5 5 5 4)4 4 1 \5 5 5 5(4' 4 2, 1- 1 1 4 44 2 1 1 7 1 1 7 0 2 508 7 1 81 8 1 1 4 4 4 1 4 4 L~J - 64 - to a perfect crystal*, the distance between first neighbors, second neighbors, etc .no longer has any fixed meaning and it becomes essential to use an appropriate energy criterion to simulate images. This requires an accurate knowledge of the interatomic potential for the material under consideration, and in general this information is not known. It was shown in the (931) region that a difference in energy between 5th neighbors must be detectable; this is indeed a very small energy difference. Since the core structure of a dislocation is not known analytically for the general case, it would not be possible to use the neighbor model to give any useful information about the appearance of the core structure in an image, but rather only long range effects due to the defect could be analyzed. It appears, therefore, that the utility of the neighbor model is in the interpretation of effects in the perfect crystal, i.e., relative brightness, size of image points, etc. The model also gives a physical basis for using the shell model in further defect studies. Defect Simulation of Total Dislocations in FCC Materials, Usinq the Shell Model Introduction Results using the shell model indicate that the best agreement between the simulated and experimental images - 65 - can be obtained when the region under consideration is small enough so that the tip surface can be closely approximated. It has been proposed that a dislocation in a field ion image will result in a spiral configuration. (22,23) The nature of this configuration is given by the dot product, g- where -g represents the set of planes which the emerging dislocation intersects and 8 is the Burgers vector of the dislocation and represents a vector in real space. The dot product _* 8, for total dislocations, always yields an integer which represents the number of leaves of the spiral to be expected on the given set of planes (hkl) Since there are no diffraction phenomena involved in the present interpretations, the _*- criterion will be formulated in terms of real lattice vectors for the cubic lattice. It is hoped that this representation will more clearly illustrate the origin of the spiral structures. The reciprocal lattice vector ghk may be represented by: 4- n hkl ghk d hkl where nihk 1 is a unit normal vector to (hkl) and d hk 1 is the interplanar spacing. Using the identities - 66 - n thkl) nhkl (h 2 + k2+ 1 2 )1/2 (Eq. 1) a a ddh l (h 2+ k 2+ 1 2 )1/2 (Eq. 2) where a 0is the unit cell dimension, Equation 1 becomes -~ [hkl)* ghl a 0(Eq. 3) Since is given in terms of a 0, e.g., =a 0[uvw], the dot product g- yields (hu + kv + 1w) which is always either zero or an integer if 8 is a lattice vector, i.e., a total dislocation. The dot product g- may be thus thought of as being the projection of 8on n hk l in units of d hkl as illustrated in Figure 24. The component of the Burgers vector parallel to the set of intersecting planes causes only a very small displacement of the atoms in the image and the correlation of this effect with the image has not been attempted. A table of various spiral configurations to be expected on a few low index planes is shown in Appendix II. A considerable number of spiral defects have been observed experimentally but no serious prior attempt at a quantitative interpretation has been made. Furthermore, the g'- hypothesis is rather difficult to prove experimentally. For example, it would be necessary to have a - 67 fi 220 1[']=b 110 2D0 PLANE OF CUT (ii1) PERPENDICULAR TO (220) Figure 24. Diagram of (111) plane showing how single and double spirals can be produced on a (220) plane with Burgers vector of ao/2 [101] and ao/2 [110] respectively. - 68 - single dislocation inclined such that upon field evaporation through the specimen it would intersect several crystallographic planes. To date such a result has not been reported and it is clear that such a circumstance would be somewhat fortuitous. It is, however, possible to simulate a field ion image rather well by at least two procedures. While the "neighbor model" is more exact on a point by point basis, it has certain limitations when applying it to regions of the crystal where the interatomic distances have been perturbed. The shell model, on the other hand, provides a good qualitative simulation of an ion-emission image and can be applied straightforwardly to regions containing defects. It was decided therefore that a great deal of information involving defect interpretation could be obtained using the shell model. Before tackling the general problem of the mixed dislocation, it was decided to first look at the simpler cases of the pure edge and pure screw dislocations. Pure Screw Dislocation The perturbations in the lattice due to the presence of a screw dislocation are rather simple, in fact all displacements are parallel to the axis of the dislocation (for the classic isotropic case) and are given simply a:(1 AZ b ; Ax =Ay =O - 69 - The z axis is the axis of the dislocation, the Az being the perturbation of the atoms from their positions in the perfect crystal due to the presence of the screw dislocation. These perturbations are computed in a coordinate system which can be rotated to any orientation. For example, since the displacements of a screw dislocation are in a direction parallel to the dislocation line, the line of the screw dislocation can be along one of the axes of the rotatable coordinate system, i.e., the z axis. Thus a screw dislocation which is restricted to line in a <110> direction on a {111} plane can easily be produced. The 9- criterion could be tested using the pure screw dislocation, e.g., by looking at the effect of the dislocation on a particular family of planes on which the predicted configuration is different. This was done by simulating the image in three different regions containing poles of the type {420}. A local tip radius of 1600 A was used with a shell thickness of .05 a0, and in each case the dislocation emerged at the center of the pole. A large radius was selected in order to obtain more "rings" about each pole, and thus avoid any ambiguity in the interpretation. Figure 25 shows the (420) pole of the perfect lattice. The three poles chosen for the simulation were the (204), (402), and (420) poles, which should yield single, double, and triple spirals according to the - - 70 - (420) POLE .000* 0 *0.0 .00 00a. esOf o *o to "0a 0~ ** 0 0 .00.~ :0 Fiur 25 Shell!1 moel pefc FC atie radius 160 A,1 shl thcns 0 I 11 projection - 71 - criterion (Figure 26) These examples are indeed special cases since the probability that a dislocation will pass through the center of a pole is practically nil. Therefore, the dislocation was moved from the central region of the pole and the same simulation procedure followed (as just explained) for several other locations. The result on the (204) pole is shown in Figure 27, which shows a single spiral starting at the point of emergence of the dislocation. Note that in an actual image this configuration might be interpreted as an edge dislocation even, though the dislocation is of pure screw character. It is a relatively simple matter to include the anisotropy factor in the equation for the displacements aroun d a screw dislocation; the equation becomes: (32) Az =b 0 (tan A) 211 where A is the anisotropy factor. An anisotropy factor of A = 2.46 was used,(9 which is the value for platinum; however, the results obtained are remarkably similar to the isotropic case where A = 1. No noticeable elongation of the "rings" was obtained, and only a few isolated image points have been deleted or added. These simulated images should be accepted only qualitatively, i.e., no claim is being made at this point (402) POLE (204) POLE .0000 0 -0 Goo0. 0000 000 *s *. so000.* 0 0.0 0 00 *0 00 0. de we 0.0 .01t0 1*0 660 0.00% Oo 0 0 6 0% %, so0 a00 0 % 0o 4P* .06 .0. So% 00 0 1~0.0 1* 0 6* 40 0 0 eo 41 00 0 0 6 0.0 40 0. 0 1 0 00 ** 00 000 . %* 0 *00~ 00 a* 0% .0 0 0 0 0 0 00 a' '0 0so % 0 so 0s *0 1.0% % Poo*, 2 0 0 0 0 0. ~ 0B (420) POLE .00 104 900 O 0 0 00 @ 0 0 S..o 0 .000t% 000 0 .0 .00 *** 0' ** 0. 00 00 00 00 .:ol* t0.* 9. A6 )0 ~ 0 0- -O" :2 90 of0*:. a ) 0 00 0000 0*0 00 0 C Figure 26. Shell model of FCC lattice, 1600 A, shell thickness .05 a0, (111) projection, screw dislocation emerging at the center of the plane: (a) single leaved spiral; (b) double leaved spiral; (c) triple leaved spiral. I - 73 - .0 0 "'10. of 00 0 0Af go .go 0 of.0 00 0 .0 of*o o ff0 *0 0 g .000 0.0 @0 0 0 0 00 0. 0 00 0 goo 000o 0000 4* 0 00 00 0we of 0 e0 0 0 0 so 00,000 0000 be fe....0 00- of 00 000 0 0 o 0 0 0 go 0 ..000 o0 *q 00 0 0 0 G 40 o .0 .0 *4 0 0 *o ~0 .0 10 4 0 s 0 00 0 0 0 so 0 0iur 27. Pur sce dilcto casn singl 0 radiu 0 60Apito mrec fdso cat0i0000n). S - 74 - that exact agreement with experimental images will be obtained, although-the long-range effects should be correct. Pure Edge Dislocation Classically, the displacements due to an edge dislocation are computed in a plane normal to the dislocation line (here again isotropic elasticity is used since the anisotropic displacements have not been solved in the general case). If the dislocation line lies along the z direction, then the displacements are given by: (31) z= 0 -b 1 -2v ln+Cos (2(D) y Y 2 2(1 ) )n 4( b sin(20) x 20 4( + v where -1 s ff 1. These displacements are considerably more involved than for the case of a pure screw dislocation. In the case of an edge dislocation the displacements are in a {112} plane. The displacements used to "generate" the dislocation can be seen most easily by looking at a {112} plane (the dislocation line being normal to this plane) with the b lying in the plane. Figure 28 shows the resulting atomic positions on the (112) plane, and the magnitude and direction of the displacements used in order to "generate" the edge dislocation from the perfect crystal. a a a a U v '1 ~ V V\~ \a ~ThiIj~ JL~ I Ia I I a I..~.a a a a a a I CREATION OF EDGE DISLOCATION IN (112) PLANE All displacements (u'v) are in (112) plane Atom positions in perfect (112) plane Atom Positions in dislocated (112) plane Figure 28. Magnitude and direction of the displacements used to generate an edge dislocation from the perfect lattice. -I - 76 - The pure edge dislocation was simulated in the same regions on the {420} planes as the screw dislocation. The results on the (204) plane are shown (Figure 29) and they are qualitatively the same as for the screw dislocation. This is indeed very surprising and indicates that the component of the normal to the set of planes being considered is the only important factor determining the defect configuration. This criterion is tested on other poles, and the results are all qualitatively the same, i.e., the 8 criterion is always satisfied for both the case of the edge and the screw dislocation. The Mixed Dislocation Knowing the perturbations due to the edge and screw dislocations separately, the perturbations due to a mixed dislocation with any orientation of and dislocation line arethen a simple matter. The screw component will always lie parallel to the dislocation line and the edge component will lie normal to it. Then 8tal= 8srw+ 8eg'(3 where b total = a 0/2 <110>. The program is set up so that a dislocation lying in any orientation can be produced and either sessile or glissile dislocations can be investigated. Since only atom positions very close to the surface of the specimen can be seen in a field ion image, information about the orientation of the defect will not be discernible from a single micrograph. - 77- 4 0* o 0*6 0* *4 0 0 o *0 0 60 6% * .6 % 0~6 ~ % %. %* Fiue2. Pure edg dilcto casn a/ [10,rdis=100 A6 6onto emerenc of dilcto) - 78 - The results obtained using a single mixed dislocation also point up the fact that only the component of the b normal to the set of planes determines the longrange effect. As the orientation of the dislocation line is changed, keeping 8 constant, no detectable difference occurs in the image. There is an "image force" (34) on a dislocation, which tends to make the dislocation line lie normal to the surface it intersects. The image force results from the fact that the dislocation is near a free surface, i.e., the specimen surface, and the force increases as the dislocation approaches the surface. It would be possible, therefore, that as the dislocation line approaches the surface it could jog over a few atomic planes near the surface so that it would lie more nearly normal to the surface. It is assumed therefore that the dislocation line lies nearly normal to the surface. In order to determine the direction of the dislocation line, a field evaporation sequence of the defect would be required. The gross movement of the defect could then be followed and the direction of the line determined. It should be pointed out that in some cases two sets of plane edges from two poles (i.e., rings) can be seen in the region of the defect. Figure 30 is an example of a computed plot illustrating this point; however, in an - 79 - Dislocation Visible of Plane Edges on Two Sets .-Dislocation: bz 4 [110] 220) Rings Figure 30. Two sets of plane edges visible in the region of a defect (radius = 800 A). Note that the 4- criterion is satisfied for each set of planes (* = point of emergence of dislocation). - 80 - experimental image this effect would only be seen on higher index planes. (For example, {200}, f220}, and {111} planes have very close atom spacings on their edges, and other planes are extremely hard to detect in these regions.) Applying the -*- criterionto the displaced rings independently to determine the Burgers vector usually will not give a unique solution. For example, both a 0/2 [101] and a/2 [011] (also their negatives) will yield single spirals on a (002) plane. However, when two sets of plane edges can be seen, two simultaneous conditions on g must be satisfied and a unique 8 may be determined. Note that either a negative or positive result may be obtained for g-8 The significance of this manifests itself in the direction or sense of the spiral. If two dislocations can be seen in an image, then the directions of each Burgers vector relative to the other can be determined (Figure 31). Total Dislocation Configurations Introducing another dislocation allows one to analyze both dislocation dipoles, dislocation loops, and any other configurations resulting from two dislocations. The intersection of a dislocation loop with the surface shoul d be no different than that of a dipole of equivalent size. In order to determine what effects a dipole or a - 81 - Figure 31. Schematic drawing indicating the relationship between the sense of the spiral on the image (clockwise, counterclockwise) with the direction of the normal component of the Burgers vector (1q and f12 represent unit vectors normal to the set of intersecting planes). - 82 - loop would have on the image, the program was modified so that two mixed dislocations with any orientation and any position could be generated. Dislocation dipoles, and pairs of dislocations were simulated in various regions of the image and the results indicate that the long-range effects are given simply by the sum of the effects due to each individual defect. The possible cases which exist fall into four general categories which can all be explained. If two dislocations are visible on a set of plane edges and if the criterion for each is +1, then in the vicinity of both dislocations a single spiral will be observed. Furthermore, the effect far from the dislocations will be the sum of the two, that is, a double spiral (Figure 32a). On the other hand, if the -"- criterion of the two dislocations gives +1 and -1, respectively, then no long-range effect would be observed. A spiral of opposite sense would emanate from each dislocation',forming a smooth curve between them, whereas the plane edges inside and outside of the configuration would remain perfect. Figure 32b shows a schematic diagram of such a configuration. If instead, the pair of dislocations were approximately the same distance from the central pole, and say the criterion again predicted +1 and -1, the configuration shown in Figure 32c could result, where one of the plane edges appears to be horseshoe shaped. Now if the dot - 83 - Pair of Disocatioris Sam Burgers Vector A 0o Pair of Dislocations Opposite Burgers Vector C Pair of Dislocations Opposite Burgers Vector B Pair of Dislocations Opposite Burgers Vector D 0a Figure 32. (a) Two dislocationsleach producing single spirals with the same sense: the long-range effect is a double spiral; (b) two dislocations each producing a single spiral of opposite sense yield no long-range effect; (c) two dislocations equidistant from pole producing spirals of opposite sense; (d) two dislocations producing spirals of opposite sense. Due to proximity of dislocations the effect is an extra plane segment. Note: Some slight distortion is present in the region of the two dislocations. - 84 - product for each of the dislocations were reversed, a line of atoms or part of a plane edge would appear between the two dislocations (Figure 32d) In all cases there is some slight distortion in the plane edges in the immediate vicinity of the defects. Dislocations which are very close to one another (5-10 A) will interact strongly and the results may cause local blurring (or lack of resolution) in the image, resulting in a short streak between the two or a similar phenomenon. Correlation of' Simulated Defects with Experimental Images in the FCC System Introduction Correlation and interpretation of experimental images were obtained with images taken from the literature (previously unexplained) and images of iridium obtained experimentally by the author (see Appendix III for method of preparation). It should be pointed out that the volume of material under investigation in the field ion microscope is very small indeed; therefore in a material which has a dislocation density of say 10 12 /m2,onwulexcto see one dislocation in each specimen, assuming the dislocation to lie parallel to the wire axis. Thus it may be a very laborious task to find a dislocation; the method used was to take a photograph after evaporation of approximately 10 (002) layers. Evaporation of single layers is a very - 85 - tedious task, and care must be taken not to "flash" the specimen. Since the field and hence the stresses on the tip are extremely high, it is possible to literally destroy the specimen by increasing the field too much; this is known as "flashing." Simulation Process The first step in the simulation process for comparison with an experimental image is the calculation of the local radius of the specimen in the region of the defect (this is accomplished using the method previously explained on page 56. The 8 of the defect (or the various possible 8's) is then predicted based on the experimental image and the simulated image is computed. In Figure 33a two defect configurations can be observed on the (204) plane edges. They appear as an apparent collapsing or pinching together of the plane edges. Whatever the configuration may be, there is no long-range effect due to it, i.e., the plane edges close to the (204) pole are perfect, and those outside of the configuration are also perfect. This type of defect may be thought to be some sort of vacancy cluster, but a little reflection will show that this configuration may be treated as a small dislocation loop, and the intersection of a loop with the surface gives the same effect as a dislocation dipole. vo- 86 Figure 33a. Expe rimental image of Pt (courtesy of Prof". E. W. Muller).- Defects shown in the region of (204) pole (A and B) ; also shown is (002) pole (C). g000*.0 00 .0.. 0-00 A*-***V 00 0 .0 00 eg *0 0 .o 0 00 go0 0 Figure ~ ~ ~ ~ ~ 33.Cmue iuae mg fdfc configratio on (204 pol in Pt ais 5 (x~ = pon of emrgnc of diloaio) - 87 - 0 fe00 00O 0 o 0 0 0 e 0 0 0 e, 00 0.500060'e0 0 0 %* oOSSS 0. 00 0400 00. 00 S@ 000' *S *e do S* 0 0 0 e O 0 050 *~* of ~ 0 0 Fiue 3 ptrsmltd mg dfc 0 cofgrto on(0)pl 0nP;rdu 0 (0 = on5feegneo ilcto) |

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PAGE 1 AN INTERPRETATIONAL STUDY OF FIELD ION MICROSCOPE IlVIAGES By ROGER CARL SANW ALD A DIS SE RT ATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR 'IHE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA August, 1967 PAGE 2 ACKNOWLEDGMENTS The author is deeply indebted to his advisory committee, Drs. Rhines, Reed-Hill, Conklin, and Smith, and particularly to his chairman Dr. Hren, who partici pated in many stimulating discussions during the course of the work. The writer is indeed grateful to National Aeronautics and Space Administration for the funds provided making the research possible. ii PAGE 3 ACKNOWLEDGMENTS LIST OF FIGURES ABSTRACT INTRODUCTION Chapter TABLE OF CONTENTS . . . . . . I. REVIEW OF THEORY OF FIELD ION MICROSCOPY II. REVIEW OF INTERPRETATIONAL WORK IN III. IV. FIELD ION MICROSCOPY .. RESULTS: INTREPRETATION OF FIELD ION IMAGES CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY Appendices I. II. III. IV. BIBLIOGRAPHY BIOGRAPHICAL SKETCH. iii Page ii iv xi 1 3 17 35 122 127 132 136 137 161 164 PAGE 4 Figure 1. 2. 3. 4. LIST OF FIGURES Schematic diagram of field ion microscope Field ion image of an iridium specimen (a) Potential of free atom showing valence electron at depth VI; (b) potential of atom in the presence of an external field; (c) atom close to the surface of a specim e n, illustrating the potential configuration at the critical distance (a) Energy diagram for the removal of an ion from the surface of the specimen (no field applied); (b) the same system as (a), with a field applied, showing the existence of the Schottky hump 5. Surface of specimen with high field applied,causing positive atomic cores to protrude, as electron cloud is drawn back into the metal. Approximate vari ation of field with position is also 6. 7. shown . Illustration of the imaging process in the field ion microscope Current voltage characteristic for a typical field ion emitter .... 8. Field ion image of an iridium specimen taken at 21K. Low index plane A (200), and a high index plane at B (931) are Page 4 5 8 10 12 14 16 labeled. 18 9. Relationship of neighboring atomic positions on (a) a high index plane, and (b) a low index plane ................. 20 iv PAGE 5 Figure 10. 11. 12. 13. LIST OF FIGURES--Continued The geometry of several projections; Pg(gnomonic projection), Ps(stereo graphic projection, P 0 (orthographic projection), Pf(pseudo-stereographic) Image of asymmetric Ir specimen, taken at liquid H2 temperature Defect in molybdenum an edge dislocation. 1.nterpreted as Single spiral at grain boundary in iridium interpreted as a dislocation 14. Change in the topographical features of a simulated field ion image (shell model) with a variation in radius of Page 22 24 28 30 0 5 a 0 3 4 15. Simulated evaporation of the (420) pole in FCC using the shell model, caused by small change in radius ........ 16a. Variation of the 1st nearest neighbors 36 over a stereographic triangle using the shell model (FCC, Radius= 350 A, shell thickness= .05a 0 ) 39 16b. Variation of the 2nd nearest neighbors over a stereographic triangle using the shell model (FCC, Radius= 350 A, Shell thickness= .05a 0 ) 40 16c. 17. Variation of the 3rd nearest neighbors over a stereographic triangle using the shell ~odel (FCC, Radius= 350 A, Shell thickness = 05a 0 ) Schematic diagram of the basic steps involved in the computer program to reconstruct the field ion image .. 18. The geometry of the projection showing the relative relationship with the specimen V 41 43 48 PAGE 6 Figure 19. 20. 21. LIST OF FIGURES--Con tinued Comparison of the imaging criteria of the shell model and the "neighbor model" on a high inde x plane ...... Micrograph of Pt Determination of the local radius by graphically plotting atomic steps 22a. Shell model, (931) region. 22b. "Neighbor model," (931) region 23a. Experimental image, (931) region 23b. Relative values of A, the local subli mation energy in the (931) region based Page 52 55 58 60 60 63 on the "neighbor model" . . . . 63 24. Diagram of (lll) plane showing how single and double spirals can be produced on a (220) plane with Burgers vectors of a 0 /2 [IOI] and a 0 /2 (110] respectively ... 67 25. Shell model; perfect FCC lattice, radius 0 1600 A, shell thickness 05a 0 ( 111) projection . . . . . . 70 26. Shell model of FCC lattice, 1600 A, shell thickness .05a 0 (111) projection, screw dislocation emerging at the center of the plane: (a) single leaved spiral; (b) double leaved spiral; (c) triple leaved spiral . . . ....... 72 27. Pure screw dislocation causini single spiral on (204) plane edges, b = a 0 /2 [110], radius= 1600 A (*=point of emergence of dislocation) 73 28. Magnitude and direction of the displacements used to generate an edge dislocation from the perfect lattice vi 75 PAGE 7 LIST OF FIGURES--Continued Figure Page 29. Pure edge dislocation causin~ single spiral on (204) plane 0 edges b = a 0 /2 [110], radius= 1600 A (*=point of 30 emergence of dislocation) Two sets of plane edges visible in the region of a defect (radius= 800 A). Note that the gb criterion is satisfied for each set of planes (*=point of emergence of dislocation) .. 31. Schematic drawing indicating the relationship between the sense of the spiral on the image (clockwise, counterclockwise) with the direction of the normal component of the Burgers vector (N1 and N 2 represent unit vectors normal to the set of inter77 79 secting planes) ............. 81 32. 33a. 33b. (a) Two dislocations each producing single spirals with the same sense: the long range effect is a double spiral; (b) two dislocations each producing a single spiral of opposite sense yield no long-range effect; (c) two dislocations equidistant from pole producing spirals of opposite sense; (d) two dislocations producing spirals of opposite sense. Due to proximity of dislocations the effect is an extra plane segment .... Experimental image of Pt. Defects shown in the region of (204) pole (A and B); also shown is (002) pole (C) ......... Computer simulated image of defect configuration on (204) pole in Pt; radius 650 1 (x = point of emergence of dislocation ) . . . . . vii 83 86 86 PAGE 8 Figure 33c. 34. 35. 36. 37. 38a. 38b. LIST OF FIGURES--Continued Computer simulated image of defect configuration on (002) pole in Pt; 0 radius 900 A (+=point of emergence of dislocation) Single spiral on {220} plane of an Ir specimen starting at point A. Points Band C indicate other disturbed regions probably due to the presence of dislocations . . . . . . . . Triple spiral on {113} plane of an Ir specimen ........... Simulated image of triple spiral caused by pair of noninteracting dislocations Single spiral on {220} plane in Ir specimen. Point A marks emergence of dislocation while point B shows another disturbance which cannot be interpreted from this single micrograph ..... Horseshoe configuration on {100} plane of iron whisker Simulation of defect configuration in 38(a) using pair of dislocations of mixed character normal to the surface (*=point of emergence of dislocation) 39a. Single spiral on {011} plane of tungsten Page 87 91 92 94 95 98 98 specimen . 100 39b. Computer simulation of defect shown in Figure 39a with dislocation line normal + to the surface and b = a 0 /2 [110] (* = point of emergence of dislocation) 100 40. Micrograph of Ru showing existence of double rings on poles A and B 102 41. Geometrical origin of "double rings" due to "ripple&'nature of certairi planes in hexagonal materials 103 viii PAGE 9 Figure 42a. LIST OF FIGURES--Continued Simulation of perfect lattice, (1012) plane, c/a 0 = 1.58, showing the double ring nature 42b. Simulation of the perfect lattice (1011) plane, c/a 0 = 1.58, showing the "double Page 105 ring" nature . . . 105 43. Dislocation on (1011) plane of material with c/a 0 = 1.58 (* = of emergence of dislocation) a HCP point 44. Solid arrows correspond to partial dislocations which yield stacking fault between the dislocations of the form ABCBCA. Dotted arrows show incorrect sequence of 5 1 s giving rise to two "A" planes 107 being stacked together 110 45. Pair of Shockley partial dislo cations on (220) plane; total b = a 0 /2 [101] (*=point of emergence of dislocation) 111 46. Pair of Shockley partial dislo cations on (002) plane, total :+ b = a 0 /2 [110] (* = point of emergence of dislocation) 112 47. Micrograph of Pt taken at 21 Distortion around the edge of image (A and B) may be due to presence of stacking faults K. the the 48. Schematic drawing of (111) planes in the vicinity of (a) a vacancy loop 114 and (b) an interstitial loop 116 49. Vacancy loop on (220) pole showing distortion in the direction normal to fault plane (*=point of emergence of dislocation) ix 117 PAGE 10 Figure so. 51. LIST OF FIGURES--Continued Extrinsic loop on (220) pole, showing distortion in the direction normal to the fault plane (*=point of emergence of dislocation) .... (a) Experimental image containing defect believed to be Frank vacancy loop; (b) computer simulation of (a) (*=point of emergence of dislocation) ......... X Page 119 121 PAGE 11 Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AN INTERPRETATIONAL STUDY OF FIELD ION MICROSCOPE IMAGES By Roger Carl Sanwald August 12, 1967 Chairman: Dr. John J. Hren Major Department: Metallurgical and Materials E~gineering Two computer models, neighbor model and shell model, have been presented to aid in the interpretation of field ion micrographs of both perfect crystal structures and crystal structures containing defects. The development of the neighbor model, which uses the geometrical environment, or equivalently the atomic binding energy, as an imaging criterion; resulted in point for point agreement with an experimental image. It was shown that variation in intensity on a single plane corre lates well with changes in local sublimation energy A; however, it appears that additional energy terms are re quired for intensity variations between planes. In order to apply the neighbor model to the dislo cated crystal, an accurate knowledge of the interatomic xi PAGE 12 potential function must be known, but this is, in general, not available. It was decided, therefore, that the dislo cated crystal could more readily be interpreted using the shell model for simulation of the ima~es. A physical basis for the shell model was established by using the neighbor model to show that those points which contribute to the image in the shell model are the atoms which are bound weakest, i.e., the atoms which protrude the most from the surface. The shell model was used to predict defect configu rations in field ion images in the FCC, BCC, and HCP systems. Basically all line defects and defect configu rations composed of total dislocations could be interpreted in terms of spirals by using the gb criterion. Therefore, the important parameter in determining the resulting configuration in the micrograph is the component of the Burgers vector normal to the set of planes on which the dislocation emerges. A field evaporation sequence is usually helpful in determining a unique Burgers vector. Partial dislocations were studied in the FCC system. The stacking fault separating two Shockly partial dislocations causes a displacement of the rings along the intersection of the fault plane with the surface. The long-range effect due to the pair of dislocations is that predicted using the total dislocation from which the pair xii PAGE 13 has dissociated. Secondary defects, vacancy loops and interstitial loops, were also simulated using a pair of dislocations, because the intersection of a loop with the surface can be described by a pair of dislocations of opposite sign (dipole). Identification of intrinsic vacancy loops was found to be a straightforward matter, intrinsic loops causing an inward collapse of the "rings~ normal to the fault plane, while the opposite effect was observed for extrinsic loops. The results obtained allow one to interp~et defects and defect configurations in field ion images with confi dence, since both a physical basis and analytical means have been presented for the interpretation. xiii PAGE 14 INTRODUCTION The field ion microscope is the only instrument capable of atomic resolution and,therefore, offers many unique advantages for the study of defects on the atomic scale. Although the microscope was developed in 1951, (l) its potentialities as related to metallurgical problems have only been realized in the past few years. Most of the early work was confined to an understanding of the principles of operation of the microscope~with little quantitative work directed toward the interpretation of the images obtained. The potential value of the field ion microscope as a metallurgical tool can be realized only if the problems of image interpretation can be elucidated. It was with this idea in mind, that the present research was undertaken. The role of defects is influential in determining material properties and an understanding of the defect structure is essential in practically any basic study. Therefore, the ultimate goal of the work was the interpre tation of defects and defect configurations in field ion images. The geometrical problem of image simulation is difficult, with the numerous calculations involved requiring 1 PAGE 15 2 the aid of a computer. The problem of defect interpre tation has, to date, only been attempted on a very quali tative level with many resulting ambiguities in the methods and results. Moore( 2 ) has shown that it is possible to simulate many characteristics of a field ion image of a perfect crystal, but the case of the dislocated crystal has never been investigated. The present work was aimed at an investigation of methods of image simulation, and using these methods, to investigate and interpret images containing dislocations. It was felt that this could be accomplished most convincingly by comparing simulated results with experimental images whenever possible. PAGE 16 CHAPTER I REVIEW OF THEORY OF FIELD ION MICROSCOPY The Field Ion Microscope and Principles of Operation Introduction The field ion microscope was developed by Muller(l) in 1951. The microscope is capable of resolving atomic positions on the surface of a conducting specimen, and is basically a projection microscope in which the atomic positions on the surface of the specimen are projected onto a flat plane, thus producing the image. A schematic diagram of the microscope is shown in Figure 1 and a typical image of an iridium specimen is shown in Figure 2. The specimen consists of a fine wire which has been electrochemically polished to a fine point. In order to obtain an image, the radius at the tip of the specimen 0 should be in the region of 100-2000 A. The specimen is maintained at cryogenic temperatures,with a positive voltage applied, in the presence of a so-called "imaging gas." Due to the extremely small radius of the tip, very high fields can be obtained when a potential is applied to the specimen. Therefore, ions created at the surface of the specimen by a process known as "field ionization," are 3 PAGE 17 4 ----------------------1 To +I Fiber ~tic Wird<:M -------Figure 1. Schematic diagram of field ion microscope. PAGE 18 5 Figure 2. Field ion image of an iridium specimen. PAGE 19 6 repelled toward the screen by the high positive potential, thereby striking the phosphor and forming the image. Before explaining the physics of image formation, it is necessary to understand two processes: (2) field evaporation. Theory of Field Ionization (1) field ionization, and One can consider a free atom as a potential well in which the valence electron is trapped at a depth v 1 ( 3 ) (Figure 3a). In the presence of an electric field, this potential can be modified, as shown in Figure 3b. According to wave mechanics it is possible for an electron to tunnel from A to B. ( 4 ) The stronger the field, the greater the slope of the perturbation and hence the width of the barrier through which the electron must tunnel is decreased. It can be shown that as the width of the barrier decreases, h 1 . < 3 ) d . b t e tunne ing probability increases an is given y an expression of the form: x2/8M D = exp (f h. 2 (V-E) dx), xl where Mis the electron mass, tis Planck'$ con$tant dt vided by 2rr, Vis the potential energy, and Ethe total energy of the electron. (The limits of integration extend ,~ over the width of the barrier.) If the free atom is brought close to a positively charged metal surface PAGE 20 7 and the field is sufficiently high, it becomes possible for an electron to tunnel into the metal. This process is k f ld .. t ()) nown as 1e 1on1za ion. However, when the energy of the tunneling electron decreases below the Fermi energy of the electrons in the specimen (the distance from the surface of the specimen at which this occurs is called the critical distance, x) the tunneling probability decreases C rapidly, since the number of ~nfilled states in the specimen decreases rapidly. Figure 3c illustrates the energy configuration of the gas atom and specimen at the critical distance, x The analytical treatment of this problem is C considerably more difficult than the case of an atom in free space. Interactions with the metal surface must be considered, since the electron can feel the force due to its "image" in the metal. The field required to ionize the gas atom will be dependent on its ionization potential, the 2 width of the barrier can roughly be taken as V 1 /F where Fis the field and v 1 the ionization potential. Theory of Field Evaporation Field evaporation is different from field ioni zation in that ions are produced from atoms which are bound at the metal surface, these may be surface atoms of the specimen or absorbed impurities on the surface: however, the term field desorption is usually used to describe the PAGE 21 A 8 C Figure 3. (a) Potential of free ato~ $hawing v~lence e1ect~on at depth Yr; (b) potential of atom in the presence of an external field; (c) atom close to the surface of a specimen 1 illustrating the potential configuration at the critical distance. 00 PAGE 22 9 latter case. In the absence of an applied field, the energy that must be supplied to remove an ion from the surface is given by the desorption energy( 3 ) Q 0 =A+ VI (A is the vaporization energy required to remove a neutral atom, VI is the ionization potential, and ~ is the work function, Figure 4a). However, in the presence of a field, an ad ditional term due to the polarization energy must be added on. (S) On an atomic scale, the vaporization energy A can vary depending on the number of broken bonds of the par ticular atom, i.e., an atom on a high index plane which has, say, 3 nearest neighbors, will have a lower vapor ization energy than an atom at the center of a (111) plane which has 6 nearest neighbors. The application of a field (Figure 4b) causes a decrease in energy with distance, and a resulting "Schottky hump" of energy Q must be overcome 0 in order for the ion to be removed. The mechanism is a direct ionic evaporation over this "hump" with time constant Q/kt =, e where, = 1/v is the reciprocal vibrational 0 0 frequency of the bound particle. (G) The size of the Schottky hump can be decreased if the field is increased, and when this hump is reduced to the magnitude of the vibrational energy, there is a finite probability that the ion can surmount this barrier and be removed from the surface. When this occurs, the atom has been "field evaporated." PAGE 23 Field =o A B Figure 4. (a) Energy diagram for the removal of an ion from the surface of the specimen (no field applied)~ (b) the same system as (a), with a field applied, showing the existence of the Schottky hump. PAGE 24 11 P r incipl es of Image Formation in the FIM Applying a positive potential to the specimen causes the electron cloud to be drawn back slightly into the metal, thereby exposing regions of high positive charge which correspond to the atomic cores of the atoms in the specimen (Figure 5). Due to the very small tip radius, very high fields (400-600 MV/cm) can be obtained with applied voltages in the range of 0-30 kV. On an a t omic scale, the field is highest where the positive cores protrude and decreases between atomic positions (see Figure 5). When an imaging gas atom approaches the v i cinity of the specimen, it is polarized by the high f i eld, and is attracted towards the specimen. Due to the polarization of the gas atoms in the field, the supply at the tip is "enhanced" above the value that would be pre d i cted from kinetic gas theory. ( 3 ) Its energy is lowered as it moves towards the specimen, and as the field in creases, the tunneling barrier decreases. Finally, the gas atom reaches a point near the surface at which the energy of the valence electron corresponds to the Fermi energy of the electrons in the specimen. The distance from the specimen at which this occurs is the critical distance, and when the gas atom is closer to the surface than this distance, the tunneling probability decreases rapidly (Figure 3c). The reason for the sudden decrease in PAGE 25 12 r ---------------------------Field Electron Gas -------------------Figure 5. Surface of specimen with high field ap plied, causing positive atomic cores to protrude, as electron cloud is drawn back into the metal. Approximate variation of field with position is also shown. PAGE 26 13 ionization probability is a result of the fact that below the Fermi energy, the number of unfilled states into which the electron can tunnel at cryogenic temperatures, decreases rapidly. There is a narrow region above the surface of the specimen where the ionization probability is appreci able. The region where most of the ionization occurs has been shown to lie in an energy band approximately .2 eV wide at a distance from 4 to 8 i above the surface,(?) depending on the imaging gas. If the gas atom is not ionized on its first pass through the "ionization zone,"(S) it collides with the surface, loosing some energy. As its energy is decreased, therefore it spends more time in the ionization zone after each "bounce," while becoming thermally accommodated. The probability of ionization increases with each bounce, until finally it is ionized at a point over o n e of the protruding positive cores (Figure 6). Muller has calculated the number of "bounces" before ionization to be somewhere between 100 and 200. Even though the quantitative relationship between the equipotential lines and the underlying atomic structure is not known, it must vary qualitatively as shown in Figure 5, the field being the highest above the underlying atomic positions. Once the particle has been ionized, it is repelled towards the screen along a field line until it finally strikes the phosphor on the screen. PAGE 27 Polarized gas atom j 14 C ones of e mitted i ons Io nization i n high fi eld regi ons Figure 6. Illustration of the imaging process in the field ion microscope. PAGE 28 15 Figure 7 is a current voltage characteristic for a typical field ion emitter. ( 9 ) As the voltage is increased initially, the ion current increases very rapidly (linear region AB); this is due to the rapid increase of the characteristic ionization time with field. At point B the ionization probability has saturated at a value corre sponding to ionization of a gas atom near the surface, and the slow increase from B to C corresponds to the ''enhanced" arrival of gas atoms at the tip with increasing field. The point C marks the onset of field evaporation of ions from the surface of the specimen. The field ion image can be thought of as a map of the electrostatic field over the surface of the specimen which has a one to one correspondence with the underlying atomic structure. PAGE 29 Ion Current Amps 10-" 10-1 2 .,_ If B I 9 16 I I 10 I ,, F,etrJ I C T,p Raa,us = 570A He Gas p=6 10-3 Torr Tip Tempuature = 78'K 5 6 IVA I I I I I 12 13 1, 15 16 kV Voltage Figure 7. Current voltage characteristic for a typical field ion emitter (after Brandon(9)). PAGE 30 CHAPTER II REVIEW OF INTERPRETATIONAL WORK IN FIELD ION MICROSCOPY Perfect Crystal Int e rpr e t a tion Imag e G e o me try Figure 8 is a field ion image of an iridium speci men, taken at liquid hydrogen temperature, where the dark circular region depicts a low index crystallographic pole of the type {002}. Atoms usually cannot be s e en in the central regions of the low index poles, at the imaging voltage, due to the close spacing of the atoms on these planes. The close spacing of the atoms reduces the ef fective ionization field, as compared to the field in the vicinity of the atoms at the edge of the plane (Figure 6). In order to obtain the maximum amount of infor mation from a photograph, images are usually recorded at the "best image voltage."( 3 ) This corresponds to imaging approximately one in every five atoms on the surface. When the voltage is increased above this value, the resolution in the higher index regions decreases, due to the increase in ionization probability over the entire plane. Theo retically, it is possible, however, to image atoms at the central regions of low index planes, by simply increasing 17 PAGE 31 18 -_-------; -. E'igure 8. Field ion image of an iridium specimen taken at 21K. Low index plane A (200), and a high index plane at B (931) are labeled. PAGE 32 19 the field above the "best image voltage" until sufficient ionization occurs in the central region of the plane, thereby making the atoms visible. However, on a low index plane, at best image voltage, only atoms at the edge of the plane contribute to the image. This is not the case, however, for a high index plane, since adjacent atoms may be S~h or 6th nearest neighbors, and each atom on a high index plane can be thought of as protruding from the surface (Figure 9), with a large field being created above each atomic position in the plane. Thus, all atoms on the plane will contribute to the image. In general, the lower the index of the plane, the more prominent the plane will appear in the image. (lO) This criterion is true if one uses the indices of a plane to fix both the atomic packing, and the spacing between planes. For example, in the FCC lattice the (110) plane does not exist, (the atomic packing on a (110), (220), etc., is the same, only the spacing between planes is changed if the indices are multiplied by an integer), since a FCC lattice could not be constructed in which the spacing of this particular plane was a 0 /{2' (a 0 = lattice spacing). The correct labeling of the plane in question must be (220), since to construct the FCC lattice, the spacing of these planes must be a /{a'. 0 PAGE 33 20 4th or 5 th Neighba-s I-! (a) 1st or 2nd Neighbors ' ( b) Figure 9. Relationship of neighboring atomic po sitions on (a)~ high index plane, and (b) a low index plane. PAGE 34 21 Projection G e om e t r y The angular relationship between planes has been shown to be a pseudo-stereographic projection, the pro jection point being located at a distance 3R from the image plane, instead of the usual 2R as is the case for a stereoh . (11) ( 10) grap 1c pr0Ject1on Figure One method commonly used to inde x field ion images is to plot the planes on a stereogram, and using this coupled with the knowledge of the symmetry of various planes, one can then determine the specific indices of the planes. Another method has been developed which employs only a linear measurement on the photograph, (l 2 ) and a knowledge of the optics used in recording the image. The equation is simply d = MR0, where dis the linear distance between two poles on the photograph, MR is a constant dependent on the recording optics and microscope geometry, while 0 is the angle between the planes in question. In practice, the product MR is computed for two known planes, then using this, the angle between any other planes can be determined by a single linear measurement. Specimen Endform Geometry In most systems (especially pure metals), determi nation of the various crystallographic poles in an image PAGE 35 22 i C ......r_ _ z ---.... ------------Figure 10. The geometry of several pro jections: Pg(gnomonic projection), Ps(stereographic projection), P 0 (ortho graphic projection), Pf(pseudo-stero. graphic). PAGE 36 23 is a straightforward matter. The endform of the specimen will affect the local magnification, and hence the apparent spacing of poles in a image. The magnification at any region of the image is simply R/Sr, ( 3 ) where R is the specimen to screen distance, r is the local radius, and Sis the image compression factor, dependent on the microscope geometry. The radius of the tip is not constant, but can vary by a factor of 3 or 4 over the image. The local radius r, between any two poles can be computed by a method first proposed by Drechsler and Wolf. (l 3 ) It consists of counting the by: atomic steps between two poles: the radius is then given nd r = 1 0 where n is the number of step ed ges of a -cosparticular plane between the two planes, 0 is the angle between two planes, and "d" is the interplaner spacing of the planes being counted. The resulting r is the average local radius between the two planes. An image which appears elongated in one direction is sometimes observed. The endform of the specimen in this case is asymmetric (chisel shaped), and can be character ized by two radii, one being very much greater than the other. Thus, the magnification is greatly increased along the direction of the smaller radius. Figure 11 shows the resulting image from an asymmetric Ir specimen. PAGE 37 24 Figure 11. Image of asymmetric Ir specimen, taken at liquid H 2 temperature. PAGE 38 25 Defe c t Int e rpr e tation Point Defects The vacancy is usually easy to recognize in a field ion image. Since atomic positions on the surface of the specimen are seen, a vacancy appears simply as a missing atom, and is most easily recognizable in the center of a high index plane. (l 4 ) There are many regions of the specimen where a vacancy is undetectable, for example, in the interior of any low index plane, i.e., (200) or (111). An apparent missing atom at the edge of a plane cannot be interpreted unquestionably as a vacancy, since this atom may have been preferentially field evaporated. Preferential field evaporation could, for example, be due to an under lying solute atom of larger size, causing the atom above it to protrude more from the surface than its neighboring atoms, and be field evaporated. Other point defects such as interstitials usually appear as randomly distributed single spots, which are much brighter than the ordinary image points. Not much work has been reported with regard to the interpretation of impurity atoms, and the effect on the image can be due to both size and electronic interactions with neighboring atoms. The electronic effects are difficult to determine,and the theory has not reached the point where PAGE 39 26 they can be predicted. Impurities have been observed along rows, and also as point defects, (lS) in the image; however, no quantitative work has been done in this area. Planar Def e cts Grain boundaries can be considered as a type of "defect" or "irregularity" in a field ion image. The crystallographic analysis of a grain boundary can be achieved by simply determining the orientation of each grain~ however, in order to determine the plane of the boundary, a field evaporation sequence is required. In this way, the boundary configuration can be followed on an atomic scale. (lG) It has been suggested that the appearance of stacking faults in the image can give rise to "streaks."(l?) This work was very speculative, and no justification or proof for this interpretation has been shown. Linear Defects By far, the least understood, and probably most important defect, at least from the viewpoint of mechanical metallurgy, is the dislocation. Drechsler et ai. first reported seeing dislocations in field ion images in several materials. (lB) The intersection of a dislocation with the surface was thought to yield an extra plane, as would be the case for the classical edge dislocation on a {112} plane. PAGE 40 27 Muller has also reported a pair of screw dislocation's in an iron whisker;(l 9 ) however, no quantitative basis is given for the interpretation. Brandon and Wald( 2 0) in terpreted the defect shown in Figure 12 as an edge dis location. The analysis proposed by Brandon and Wald is that only the edge component of the dislocation can be analyied since all dislocations are normal to the surface they intersect in a field ion tip. Therefore, only lateral displacements can be detected, i.e., only edge character can be observed. The basis for defect interpretation was a simple geometrical model, in which the edge component was thought to cause the appearance of an extra plane, depending on its location in the image. For example, in a 0 BCC, if the dislocation is assumed to be 2 (111], then an extra (111) plane should appear in the image, as a result of the dislocation. Several examples of dislocations in Pt have been seen; < 21 ) however, their explanation is by no means unambiguous, and no attempt has been made to determine the b or line of the dislocation. More recently, a criterion has been proposed from which the effects of total dislocations on an image can be predicted. It has been proposed that total dislocations will give rise to a spiral structure in the field ion image. < 22 23 ) The number of leaves in the spiral con figuration can be predicted by computing the component of PAGE 41 28 Figure 12. Defect in molybdenum interpreted as an edge dislocation (after Brandon and wa1a( 2 0)). PAGE 42 29 the bin the direction of the normal to the plane under consideration. This component will always be an integral multiple of interplanar spacings, for that set of planes (this is because the b of a total dislocation must connect two atomic positions). If the b lies in the set of planes no spiral will occur in the image. This criterion is presented ad hoc, the reason given for this quantitative rule is very qualitative. Ranganathan( 2 J) has compiled a table for BCC materials indicating the "number of leaves" to be expected on certain low index crystallographic planes. He has also shown several examples of dislocations in grain boundaries, and has used his criterion to interpret them. Figure 13 is an example of a "dislocation" in iridium, at a grain boundary. It should be noted that there has been no reported case in the literature of an isolated spiral configuration, i.e., not in a grain boundary, etc., as being interpreted using the criterion suggested by Ranganathan. A double spiral in Ruthenium has also been reported; < 24 ) however, here again it is in a grain boundary. (It is of course very speculative to propose seeing individual dislocations in grain boundaries, since the structure of grain boundaries in general is not known.) It is possible that images next to grain boundaries may look like "dis locations," i.e., spirals, due simply to the geometrical orientation of the boundary. However, in small angle PAGE 43 30 .-. ------------------~~ -----------. -_ ----. ---~ ------------------Figure 13. Single spiral a~ grain boundary in iridium interpreted as a dislocation {after Ranga~athan(23)). PAGE 44 31 boundaries the possibility of seeing individual dislocations in the boundary cannot be discounted, since it has been shown that a small angle boundary can be constructed,using individual dislocations. Ryan and Suiter( 2 S) have reported seeing partial dislocations in W. The argument used in interpreting the defects is generally the same as that of Ranganathan; however, they make no attempt at any quanti tative correlation, i.e., if the b lies out of the plane under consideration then some distortion will occur in the image. This argument is also used by them in the interpre tation of dislocations in high angle boundaries in tungsten. Computer Simulation of PIM Images Image Simulation of a Perfect Crystal: Shell Model The most rigorous attempt to date at interpretation of the geometrical aspects of a field ion image, has been (2) the "shell model" proposed by A. J. W. Moore. This model has been used by Moore to simulate many of the general characteristics of field ion images of both BCC and FCC materials. With the aid of a computer, Moore was able to obtain plots of the image, which were in very good agreement with an experimental image. The model, as presented, was used over the region of one complete stereographic triangle, and good correlation of the relative prominence of poles, PAGE 45 32 and the density of image points with an e x perimental image were shown. The program was written so that the coordinates of either BCC or FCC lattice points could be determined over the region of a stereographic triangle. A thin spherical shell is made to intersect the point lattice, the radius of the spherical shell corresponding to the local tip radius. The shell thickness is an empirical function of the radius, but is usually .02 .2a, where 0 "a" is the lattice parameter All points lying within 0 this thin shell are considered image points; they are then projected onto a flat plane and the final image formed. Basically, this model takes atoms which are very close to a spherical surface, and projects them onto a plane. Muller has suggested that those atoms which "protrude" the most from the surface will contribute to the image. The "shell" is basically a measure of the protrusion of an atom from a hypothetical spherical surface. Moore has shown that the shell thickness must be decreased as the radius is increased. He determined this mainly by point counting, i.e., comparing the number of points in the experimental image with those obtained using the shell model~ He also found that in order to obtain good corre lation with an experimental image over the region of a stereographic triangle, it was necessary to change the PAGE 46 33 shell thickness in various regions. This is due in part to the fact that the radius is not constant over a stereo. graphic triangle. The computed image is extremely sensitive 0 to radius~ a change in radius of say .OSA can cause a marked change in the topographical features of a high index plane (Figure 14). Changing the radius by one interplanar spacing plane will cause an atomic plane of that species to be removed. Since the spacing of a set 0 of high index planes is of the same order as .OSA, simulated field evaporation of this plane can occur, with a concomitant change in the topological features. The utility of the shell model is that it does show that it is possible to simulate at least some features of a field ion image by analytical means. PAGE 47 Figure model) SHELL MODEL Shell ~0) Thickness: osA Radius= m A . \ \ . .. . . . \ . . . \ .. . . . ...... .. .. .... .. ... . ... .. . .. .. . .. ... .. .. .. ,"' ....... ........ . . . .. .. .. . ... .... . .... ... .. .. . . .. . .. : . . ... : ... : ..... ... .:: ..~,, . . .. .. .. . . . .. : . . . .. . .. . . ..: .. ... .. .. . : .. .. : ... .... . : : .: ... .... : ... :: . . ... 41 ,,.:... .... '. ::: ... : -~. .. .. . . . --... .. .. .. . . :: ....... .. ,, ................. '(211",,.; ( 0 531} .. . .. . . . .. . ..... .. . . . .. ... .. 14. with .. .. ..._ . . . ... .. ..... . . . . ... .. .. ... ' . . .. ... .. .. .. ,.. . ...... ... .. ..... ........... . .. "' .. . .. . . ........... ........ . .. . . . . . . .. I ... ,. ...... .... .... ,, .. If 1"11f'JJ f J 1' (110) Change in the topog~aphical features in radius of .05a 0 a variation Radius:m 15A . . .. . . . . . . . . . . .. .. '. ... .. . . .. : ..... . . . . ... ....... .. . . .. .... . .. ._ .. : .... .. .... ..... . .... . ... ... . . .. . . ... . .. . . .. .. ..... ... : : .. . . ..... .. . : : .... ........ : : -... ... . . . ". . . : : (211) .. tsm . . -. . . . . ._. . . . . . .. . . . ....... ,':::== -.: .. ... ' . .... . . .., .. ,' .. . .. ..... : . . . . . . . . : .. .. I t t i of a simulated field ion .. ::, image (shell PAGE 48 CHAPTER III RESULTS: INTERPRETATION OF FIELD ION IMAGES A Physical Model for Field Ion Image Interpretation Introduction The shell model seems to lack direct physical basis, in that it is difficult, if not impossible to correlate the imaging criterion (shell thickness) with the interato~ic parameters of the specimen. It is clear, that atoms near the surface of a metal will be less tightly bound than those deep in the bulk crystal. Therefore, it was felt, that a criterion from imaging might be related to the binding energy of an atom to the crystal. Analyzing the geometri cal configuration of the atoms which contribute to the image, in the shell model, might shed some light on an imaging criterion which would have both physical basis and better agreement with an experimental image than the shell model. The compu~er program for the shell model was written so that it would be possible to look at any region of the image, since in many cases it is impractical to look at the entire stereographic triangle (both costly and time con suming). Figure 15 is a series of plots obtained, using 35 PAGE 49 Figure caused 15. by Radius = 576. I Radius= 5 76. 05 Radius= 576. 0 : (a} Radius= 575. 95 (d) Simulated ev a poration small change in radius. I. : : .. .. .. (b} I .. \' s .: .1 . Rad i us= 5 75. 90 (e} of the ( 4 20) pole ... . .. ... ,....... : : : Ii i ,: : s it : i t: :, r . i 1 i i i . ii .i t 0 ... i I : : 1:: : is .: :: : ::: .. :.. .... :. . (c) Rad i us = 575 87 .. .. .. . ... : . . .. ... .. in FCC ( f) using . . ... .... the shell model, PAGE 50 37 the shell model, which shows the removal in steps, of one monolayer on the (420) pole (simulated field evaporation), 0 the total change in radius being 0.64 A. The Neighbor Model This work was directed toward developing a model that could give point for point agreement with an experi mental image. The FCC lattice was chosen initially, since the main objective of the work was the study of defects in field ion images (in FCC there is less ambiguity as to the types and configuration of defects that can exist as compared to BCC or HCP). It was also hoped that the studies on interpretation of perfect crystals would shed more light on the general problem of defect interpretation. The region of the specimen over which a spherical surface will approximate the tip is dependent, of course, on the endform geometry. A good approximation to the surface can be achieved over an angle of about 10 with a single radius of curvature. Therefore, in the neighbor model the tip is assumed spherical over a small region only, possibly one or two poles. The geometrical environment of each of the surface atoms is computed (i.e., number and type of neighbors, e.g., 1st, 2nd, etc.), and based on this image points can be selected and projected. Using Moore's model (shell model) as a starting point, a shell thick enough to PAGE 51 38 assure more than enough points for the image was selected. There is no reason to select more than a "thick'' shell to begin with, since all the lower lying points within the specimen, are discarded anyway. The number and type of neighbors for each point lying within this thick shell (p = .la where p represents the shell thickness, and a 0 0 the lattice parameter) were then analyzed with the result that most of the points in the shell have six 1st nearest neighbors and three 2nd nearest neighbors. (The total possible number of 1st neighbors is 12 and of 2nd neighbors is 8.) This is not surprising since it would be expected that approximately 1/2 the number of neighbors of a particular atom located at (a, b, c) would lie on either side of a plane passed through the point. Since the tip radius is much greater than the neighbor distances con sidered (3rd, 4th, 5th, etc.), the tip can be regarded as a plane over a very small region and the number of neighbors of any one type (1st, 2nd, etc.) lying on either side of the plane is thus approximately 1/2 of the total. Figures 16a, 16b, and 16c show the variation in the number of neighbors (1st, 2nd, and 3rd) over a stereographic triangle. Note on the lower index planes, that the number of neighbors of any particular order (1st, 2nd, or 3rd) are, in general, greater than on a high index plane. This is a result of the closer spacing on the low index planes, yielding a PAGE 52 6 -7 x-a a-9 39 FI R ST N E I GHBORS FI RS T (111) P R OJ E CTI ON SH E LL =.os a NE I GHBOR S " " ,, . : ..... e e I . ... ...... -. . : .. ... ... .... o. o ... o. o. 0 o 4':, ... .... .,_ .,_ 3 ti,,:,..,. .. & .. (111} l g .. .. o C. . . : . 8 : I I I 0 I 0 I .: I c 110 > Figure 16a. Variation of stereographic 0 triangle using Radius= 350 A, shell thickness nearest neighbo~s over shell model (FCC, =.OSa). 0 1st the a PAGE 53 40 -----, SECOND NEIGHBORS SHELL= .os a (111) PROJECTION -2 S ECOND NEIGHBORS -3 " -4 " n-5 " 0 0 -~ .. (100) .. 0 0 0 0 0 .. . 0 . 0 0 0 0 .. . . . .. o .... ...... .. . 0 0 .o 0 .. ... ... .. . 0 . 0 .. ... ..... . .. . . .. . . .... eo ..o .. eo .. e o -. e o . . . 00 .... 00 0 0 . : ... . ... .. 0 .. ... .. ..... ,.-~ .. ... s s I ..o .... .. .. .. : 0 r r I I I : i I 0 000 0 00 00 0 0 oo 0 0 0 0 0 0 0 00 ..0 (110) 2nd nearest neighbors over Figure 16b. Variation of the a stereographic triangle using the shell model (FCC, Radius= 350 A, Shell thickness = 05a ) 0 PAGE 54 41 T HIRD NE I GHBORS -1 T HIRD (111} P ROJECTION S HE LL = .os a NE I GHBORS -2 -3 -4 . . 0. ,oo " . . .. .. .... (100) . . . . ......... ... . . .. .. ..... .. .. . . . . . . 0. 0 . . . . .. 0..., ... -:: ..... .... .... ,::.,:": . .. . : . . 0 .. .. 0. ...... .. . . : . . . 0. .. .. .. ..... .. ... .. .. .. . 0 ...... ... .. .. .. 0 0 0 ,.o o 00 0 0 0 0 0 0 0 . ~ 0 .. 0 : . I t I : : : I ( 110) the Figure 16c. Variation of stereographic triangle a Radius 3rd nearest neighbors over the shell rnod~l (FCC, using Thickness 0 = 350 A, Shell = 05a ) 0 PAGE 55 42 greater number of neighbors. A program was then written, in which the imaging criterion was the geometrical environ ment of an atom (the number of 1st neighbors, 2nd neighbors, etc.) A general outline of the actual simulation procedure is shown in Figure 17. The program is set up so that a solid rectangular parallelopiped containing points in an FCC array, can be generated anywhere in space, with respect to an orthogonal coordinate system (x, y, z). The size of this volume is fixed by six adjustable parameters which determine the maximum and minimum boundaries of the solid in space. The orientation of this volume in space can be adjusted, such that any desired poles can be obtained in the final image. This is easily done as follows: the position of the pole in space is computed, then based on the generating coordinate system (see Figure 17:1) and depending on the size of the region to be computed, the bounding values of the coordinates of the region are computed. This data is then used to generate a data tape, on which are stored the coordinates of the points in the FCC lattice lying in the thick spherical shell. The advantage of the data tape is that data does not have to be generated each time the program is run, but can be used over and over. The reason for using a rather thick shell PAGE 56 CD FIELD ION IMAGE RECONSTRUCTION Coordinates of pt in FCC lattice are determined !00 z , Q , ., , , ~:,,,, C l Determination of coordinates in rotated system x y ; z to give (a',b ,c ) y y X z Environment determination JJl1e::f =-r' <_r_p-t._c_o_n_s-id-e~,:d J If r"> r pt. discarded Projection y is projection direct ion z Projection sphere Possible image pt '=~-xl 51 neighbor shell I I "2 nd neighbor shell Image pt y Figure 17. Schematic diagram of the basic steps involved in the computer program to reconstruct the field ion image. PAGE 57 44 (about "a" thick) is so that the radius can be varied 0 slightly to permit some flexibility when correlating the image with experimental results. Having this information, a spherical surface corresponding in size to the local tip radius is passed through the points in the lattice (these are stored on the magnetic data tape) (Figure 17:2). The distance from the origin to each point must then be determined in order that the points which lie outside of the spherical surface can be discarded, since these are physically not part of the specimen. This then leaves points within the spherical surface to be examined in detail. Knowing the coordinates of a point inside the spherical surface (a possible image point~ the number of neighbors of any order (1st, 2nd, 3rd, etc.) can be computed (Figure 17:3). The coordinates of the various neighbors (1st through 6th) of a point in the FCC lattice, situated at the origin, are known. These were computed by finding all points in the lattice which are located at, say, the nearesst neighbor distance, 2nd nearest neighbor distance, or at any distance from the origin desired. A program has been written which gives the number and coordinates of these atoms situated at any distance from the origin. Knowing these coordinates, a simple linear transformation PAGE 58 45 gives the coordinates of any desired neighbors for a point in the lattice situated at any arbitrary position, x, y, z. For example, if the coordinates of a particular neighbor are x 1 y 1 and z 1 and if the atom is situated at position (x, y, z) with respect to some origin, o, then the coordi nates of the neighbor with respect to the same origin "o," are simply (x, y, z): 0 0 0 The distance from the origin to the neighbor under consider ation is then computed: if this distance is less than the distance to the spherical surface, it is counted as a neighbor of that particular order, be it 1st, 2nd, 3rd, etc. This process is then repeated until the neighbors out to any desired order have been determined. The actual imaging criterionis now imposed~ that is, if a point has the correct number of neighbors, it is then considered an image point, if not it is discarded. Determining the "correct" number of neighbors is done by compa~ing the resulting image with an actual experlmental image; this process will be explained later in more detail. PAGE 59 46 At this stage, only the coordinates of the point are known with respect to the x, y, z, orthogonal coordi nate system (refer to Figure 17:4). Obtaining the actual coordinates of the point on the image will now be explained in some detail. The x, y, z coordinate system is fixed in space, the x, y, and z axis corresponding to the [110], {110], and [001] directions in the FCC lattice respectively. On the other hand, the x', y', z' orthogonal coordinate system is adjustable in space (Figure 17:4), to the extent that it can be rotated while keeping the origins of the two coordinate systems coincident. They' axis is adjusted such that it lies in the direction coincident with the pole of the projection desired, e.g., if a (111) projection is desired ((111) in the center of the image), then y' will lie along the [111) direction. It has been shown(ll) that when the projection point (refer to Figure 17:5) is approximately 3R from the screen, where R is the specimen to screen distance, good correlation with an actual image is obtained. This, however, must be varied depending on the experimental conditions, and in the present program this projection point can be changed, so that actual imaging geometries can be closely approximated. The coordinates of the image point in the rotated coordinate system are now computed by (26) using the standard rotation transformation equations. PAGE 60 47 The point is then projected radially onto the projection sphere by multiplying the coordinates of the point by the magnification, R/r, where r is the local specimen radius. The final step is then to project the point on the projection sphere onto the screen from a point located on they' axis approximately 3R from the screen. Simple geometry gives the coordinates of the image point on the screen. This process is continued until all points which satisfy the i m aging critericnhave been found. These points are then plotted and the final image is obtained. A de tailed description of the equations used in this analysis are given in Appendix I. Figure 18 shows the actual pro jection in more detail, the relation of the specimen surface and imaging atoms (dotted line in insert passes through image points) is shown. Note that the drawing is not to scale, R is of the order of magnitude of 10 cm, while the -6 specimen radius is approximately 10 cm. Imaging Criterion Correlation between the image and the "neighbor model" is obtained by adjusting the imaging criterion, i.e., the geometrical environment of an atom. Assuming a monotonically decreasing potential to exist, ( 27 ) relative binding energies can be assigned to the various neighbor combinations. Since the distance to any neighbor is known, PAGE 61 r 48 .. . -.. .... .. . . .. . ~ Projection Pt -------Figure 18. The geometry of the projection showing the relative relationship with the specimen. PAGE 62 49 a relative energy can be assigned to any neighbor, if an interatomic potential function is assumed, i.e., variation of energy with distance. It is assumed that the lowest energy configuration will always image, therefore it need only be decided where the high energy cutoff point is. The atoms which are the weakest bound have the greatest number of broken bonds and therefore are those atoms which protrude most from the surface, e.g., an atom on a (1179) plane is much less tightly bound than an atom on a (110) plane (see Figure 16a, 16b, 16c). Since the atoms which contribute to the image have geometrical environments which are very similar (Figure 16a, 16b, and 16c~ it is expected that the effective binding energy of any atoms which contribute to the image will lie in a narrow energy band. It has been proposed( 3 9 ) that the binding energy of an atom could be described by an equation of the form, Q =A+ I n~ (where A is the n sublimation energy from kink sites, In is the n th ionization potential, and~ is the local work function). .. (28) Muller has proposed an additional term p which must be added due to "field penetration polarization." The"field penetration polarization" is given by an expression of the form, pa= a 0 1 1 2 d 1 / 6 s 7 / 3 F 0 2 / 3 2TI ergs/atom, where d = spacing of planes, a 0 is the lattice constant, sis related to the density of atoms on the particular plane, and F 0 is the PAGE 63 50 evaporation field. This term is suggested to account for the orientation dependent part of the field evaporation energy. In other words, assuming A and I to be constant, n the variation in does not account for the variation in field evaporation energy in different regions of the specimen; however, Muller suggests that the p term may a account for this. The field penetration polarization energy is larger on a more open plane, i.e., high index plane, and is a measure of the amount by which the electron cloud is pulled back into the metal,thereby exposing more or less of the positive ion cores. In the neighbor model, the differences in the geometrical environment of an imaging atom are related directly to local differences in A, the energy of subli mation, which has been considered to be a constant by most investigators. It is possible to calculate relative differences in A through the use of a suitable interatomic potential function. The work presented here, in effect, explains certain features of the image on the basis of local variations in A only, which will also effect the orientation dependence of the field evaporation energy. Results on the (113) plane indicate that at least some atoms with 7 nearest neighbors would have to be included to adequately simulate the image. This is not unexpected, since one would expect the work function and PAGE 64 51 the polarization energy to change in this region as compared to the (931) region. There is no reason to expect that the work function should be constant over these two regions. After analyzing the environment of atoms in the shell model, it is found in many cases that some atoms with the same environment as those within the shell lie outside of it (Figure 19, for example atoms 4 and 5). The shell model would image only atoms 1, 2, 6, 7, and 8, whereas the neighbor model proposed here would image all the atoms on this plane, namely 1 through 8. Experimental images correspond to this second criterion. It appears that the order of neighbors that must be considered is directly related to the distance between atoms on a particular plane, i.e., with a greater separation of imaging atoms, combinations of higher order neighbors must be considered in order to adequately simulate an image. Atoms at the edge of a plane will in general have a different environment than an atom in the center of the plane: this difference must be detected in order to simulate the image. Simulation of the (93l) Region The (931) region was chosen to be investigated in some detail for the following reasons: PAGE 65 52 . --2 -. -~ ,, ' e \ 4 \ \ ., 5\ I I :6 Spherical Surface I r ----------------Figure 19. Comparison of the imaging criterionof the shell model and the "neighbor model" on a high inde x plane. PAGE 66 53 1. Good resolution in an experimental image is obtained. 2. The shell model does not give adequate results on a point to point basis. 3. Since it is a high index region, 5th or 6th neighbors would have to be considered to simu late the image, as compared to only 2nd or 3rd neighbors on a low index plane. Assuming a linearly decreasing relationship between bond strengths, i.e., 1st neighbors have a higher binding energy than 2nd neighbors, etc., all atoms with six lst neighbors and various combinations of 2nd through 6th neighbors are plotted in the region of the (9~1) plane. Table I is a compilation of all combinations of neighbors that were required to simulate the image in this region. Note that not all possible combinations of neighbors exist, but rather only a relatively small number: however, all combi nations with six 1st neighbors that did exist, were used in the simulation. The experimental image of this region was obtained from Prof. E.W. Miiller, and the author is very grateful to him for this micrograph. Figure 20 shows the entire micrograph. The (931) region (shown blocked in) was studied in more detail. PAGE 67 TABLE I BOND COMBINATIONS REQUIRED FOR (931) REGION Total Neighbor Possible Order Neighbors Combinations Used to Simulate ( 931) Region 1st 12 6 6 6 6 6 6 6 6 6 6 6 2nd 6 2 3 3 3 3 3 3 3 3 4 4 3rd 24 12 12 12 12 13 13 13 14 14 12 12 u, 4th 12 6 6 6 6 6 6 6 6 6 6 6 5th 24 12 12 13 14 12 13 14 12 13 12 13 6th 8 4 4 4 4 4 4 4 4 4 4 4 Many combinations do not give any image points; only those combinations which contributed points to the image are listed. PAGE 68 55 Figure 20. Micrograph of ,.Pt (photoc;:rraph courtesy of Prof. E. w. Muller, t~ken in 1959). PAGE 69 56 The radtus in the region o~ tn~ l93ll r.lane was determined graphically, by counting (002) rings. A de tailed explanation of the method used will now be given, since the results indicate that it is probably the most accurate method of local radii determination that I have found. First, the step width and height of the planes (731, 931, and 1131) are computed along the zone passing through the (002) pole, since (002) steps wer~ used in the radius determination. This is done by employing the equations developed for building ball models. < 29 ) The step height is simply a /2, which corresponds to stacking 0 (002) planes in Pt. The step width is given by what is referred to as the "total horizontal translation." Since the three planes considered all lie in the same zone, the step height remains constant, and only the step width changes from plane to plane. It should be noted that the equations used to calculate the step width and height taken from Moore and Nicholas can only be used for (002) steps or (111) steps. The height and width of any other type of step would have to be computed. The number of "steps" on each plane are determined from the micrograph, and the corresponding number are drawn to scale on a two dimensional grid. A best fit radius can then be drawn through corresponding points on each step edge, e.g., select the radius such that it passes through the corner PAGE 70 57 of each step edge. Figure 21 shows graphically, the method used for the radius determination. It should be noted that the radius of curvature of this region does not necessarily have any correlation with other geometric parameters in any other region of the tip~ it is merely some radius which approximates the actual curvature of the tip very closely, over a small region. When the radius is measured by the usual method using the equations reported by Drechsler and Wolf, (l 3 ) i.e., r = (nd)/(1-cos 0 ) where n is the number of rings between two poles, d ~s the spacing of plahes of the rings being counted, and t is the angle between the two poles for which the radius is desired, radii ranging 0 0 from approximately 460 A to 750 A were obtained. Using a 0 best fit radius of 445 A, determined from plotting the actual steps (as just described), excellent agreement between the number of steps in the simulated image, and the experimental image was obtained. In the simulated im~ge, point for point agreement is obtained,except there is an extra row of atoms along a (113) step edge. This can be readily explained if a more detailed analysis of the local surface topography is performed; that is, the radius between the (931) and the (113) is smaller than that between the (001) and the (931). Decreasing the size of the area under consideration, thereby simulating PAGE 71 58 (002) Step ( Best fit radius ) --------------------------~ ------------Figure 21. Determination of the local radius by graphically plotting atomic steps. PAGE 72 59 one pole at a time, instead of all three, would result in the disappearance of the "extra" row. Figures22a and 22b are a comparison of the shell model and the "neighbor model" with the actual image (Figure 23b) of the (931) region. In the (931) region, using all combinations with six 1st neighbors gives excel lent agreement with the image. Note that all possible combinations of neighbors do pot exist, but rather only a limited number occur, as shown in the Table I. As can be seen (Figure 22a), the shell mqdel does not give the complete net of atoms on the (731) plane. If the shell thickness is increased in an attempt to fill out the net plane, nonacceptable image points (atoms with 7 first neighbors) begin to appear before the (731) plane has been completely filled. Note also the absence of points in the shell model between th~ (931) and (731) plane. Points in the actual image cannot be resolved in this region; however, the bright zone that does appear seems to better correlate with the results of the "neighbor model," i.e., an unresolved row of atoms between the (731) and (931) plane. The relative brightness of many of the image points on each plane can also be explained. If the bond energy is assumed to decrease linearly with distance between atoms, PAGE 73 Shell (931) Model Region 0 Radius:445 A Shell Thickness =.1! l -------.--.-.-+-.:....i-r------lr-..;;.a;.---------! i I i I .. Figure 22a. Shell model (9311 region, I -~---Neighbor Model (931) Region 0 = . 0 ft<,1u:re 22b. region. ''Neighbor model," (931) I 1 I I I 0\ 0 PAGE 74 61 then atoms near the "edges" of the plane will have a lower binding energy than those in the center of the plane, and therefore would appear brighter in the image. If an interatomic potential is used to compute the binding energy of each atom, then this may be used directly as an imaging criterion. The atoms which are the weakest bound must always contribute to the image: however, the adjustable parameter is now the value of binding energy above which atoms will not contribute to the image. In order to compute the relative binding energies, it is necessary to know the shape of the interatomic potential function. Computations of an interatomic potential function for platinum, however, were not available. An approximation of the relative binding energies of any nearest neighbors could be obtained by comparing platinum to elements with similar electronic structure and binding energy A, for which the calculations of an interatomic potential had been done. Comparison of the Morse potential function of Ni and cu< 3 o) (Ni has the same o uter shell electron configuration as platinum) showed that the relative weighting of neighbors was the same for each. It was assumed, therefore, that the relative weighting of neighbors in platinum could be approximated by using the values for Ni and Cu. This was PAGE 75 62 done for the (931) region, and the results are exactly the same as for the case of using the geometrical environ ment criterion. Figure 23 (a and b) shows the correlation of relative intensity with the calculated binding energy. The image points are labeled such that the O is the weake~t bound, while those bound the tightest are labeled 10. The neighbor model points up the fact that it is possible to simulate an image on a point for point basis if one looks at a sufficient number of neighbors. Even though the shell model does not give point for point agreement, the atoms which do contribute to the image are by and large those which protrude most, based on the relative numbers of neighbors of each atom. The neighbor model goes further since it shows that variations in intensity on a single plane, and changes in the local sublimation energy, A, can also be simulated; however, additional energy terms are apparently needed to account for variations in brightness between planes. Application of the Model The neighbor model does present certain limitations if it is to be used to simulate defects. In a dislocated crystal, atomic positions have been perturbed as compared PAGE 76 Figure 23a. Experimental image, (93l). region (after Miiller). ,j T ---------------i I Relative Intensities Neighbor Model (931) Reg i on Decreasing Intensity 1 9 9 19 99 D ,, It ID ID lC 9 9 10 1 ID 1D 10 1 IQ 10 ID ID 10 ID 10 1 1 2 2 .,, 10 10 )3 ' 1 2 I ID ID I 1 '2 1 I '2 I '2 z 1 2 lD 2 I / 1 1 3 / 3 Z I 2, z 1 3 ',, 4 1 4 4 4 4 ) 0 10 Ftgure 23b. Relative value~ of A, the local sublimation energy in the (931) region based on the "neighbor model." 0\ w PAGE 77 64 to a perfect crystal~ the distance between first neighbors, second neighbors, etc.,no longer has any fixed meaning and it becomes essential to use an appropriate energy criterion to simulate images. This requires an accurate knowledg e of the interatomic potential for the material under con sideration, and in general this information is not known. It was shown in the (931) region that a difference in energy between 5th neighbors must be detectable~ this is indeed a very small energy difference. Since the core structure of a dislocation is not kno w n analytic~lly for the general case, it would not be possible to use the neighbor model to give any useful information about the appearance of the core structure in an image, but rather only long range effects due to the defect could be analyzed. It appears, therefore, that the utility of the neighbor model is in the interpretation of e~fects in the perfect crystal, i.e., relative brightness, size of image points, etc. The model also gives a physical basis for using the shell model in further defect studies. Defeat Simulation of Total Dislocations ~n FCC Materials 1 Using th e Shell Model Introduction Results using the shell model indicate that the best agreement between the simulated and experimental images PAGE 78 65 can be obtained when the region under consideration is small enough so that the tip surface can be closely appro x imated. It has been proposed that a dislocation in a field . lt . 1 f' t' (22,23) ion image wi resu in a spira con 1gura ion. The nature of this configuration is given by the dot -+ ,+ -+ product, gb where g represents the set of planes which the emerging dislocation intersects and bis the Burgers vector of the dislocation and represents a vector in real -+ ,+ space. The dot product ghklb, for total dislocations, always yields an integer which represents the number of leaves of the spiral to be expected on the given set of planes (hkl). Since there are no diffraction phenomena involved in the present interpretations, the gb criterion will be formulated in terms of real lattice vectors for the cubic lattice. It is hoped that this representation will more clearly illustrate the origin of the spiral structures. -+ The reciprocal lattice vector ghkl may be represented by: A where nhkl is a unit normal vector to (hkl) and dhkl is the interplanar spacing. Using the identities PAGE 79 66 nhkl = [hkl] (h2 + k2 + 12)1/2 (Eq. 1) a and dhkl 0 = (h2 + k2 + 12)1/2 (Eq. 2) where a 0 is the unit cell dimension, Equation 1 becomes = [hkl] a 0 (Eq. 3} Since bis given in terms of a e.g., b = a [uvw], the 0 0 + >dot product gb yields (hu + kv + lw} which is always either zero or an integer if bis a lattice vector, i.e., a total dislocation. + ,+ The dot product gb may be thus ,+ A thought of as being the projection of b on nhkl in units of dhkl as illustrated in Figure 24. The component of the Burgers vector parallel to the set of intersecting planes causes only a very small displacement of the atoms in the image and the correlation of this effect with the image has not been attempted. A table of various spiral configurations to be expected on a few low index planes is shown in Appendix II. A considerable number of spiral defects have been observed experimentally but no serious prior attempt at a quantitative interpretation has been made. Furthermore, the gb hypothesis is rather difficult to prove experi mentally. For example, it would be necessary to have a PAGE 80 n 220 f [110] nm 67 < ~non= b \ \ A PLAN E OF CUT Ott) PER P E N DICU L AR TO (220) Figure 24. Diagram of (111) plane showing how single and double spirals can be produced on a (220) plarie with Burgers vector of a 0 /2 [101] and a 0 /2 [110] respectively. .. PAGE 81 68 single dislocation inclined such that upon field evaporation through the specimen it would intersect several crystallo. graphic planes. To date such a result has not been reported and it is clear that such a circumstance would be somewhat fortuitous. It is, however, possible to simulate a field ion i m age rather well by at least two procedures. While the ''neighbor model" is more exact on a point by point basis, it has certain limitations when applying it to regions of the crystal where the interatomic distances have been perturbed. The shell model, on the other hand, provides a good qualitative simulation of an ion-emission image and can be applied straightforwardly to regions containing defects. It was decided therefore that a great deal of information involving defect interpretation could be obtained using the shell model. Before tackling the general problem of the mixed dislocation, it was decided to first look at the simpler cases of the pure edge and pure screw dislocations. Pure Screw Dislocation The perturbations in the lattice due to the presence of a screw dislocation are rather simple, in fact all displacements are parallel to the axis of the dislocation (for the classic isotropic case) and are given simply as:(Jl) b = 2rr 0 ; tx = ty = O PAGE 82 69 The z axis is the axis of the dislocation, the 6Z being the perturbation of the atoms from their positions in the perfect crystal due to the presence of the screw dislocation. These perturbations are computed in a coordinate system which can be rotated to any orientation. For e x ample, since the displacements of a screw dislocation are in a direction parallel to the dislocation line, the line of the screw dislocation can be along one of the axes of the rotatable coordinate system, i.e., the z axis. Thus a screw dislocation which is restricted to line in a <110 > direction on a {111} plane can easil~ be produced. The gb criterion could be tested using the pure screw dislo cation, e~g., by looking at the effect of the dislocation on a particular family of planes on which the predicted configuration is different. This was done by simulating the image in three different regions containing poles of 0 the type {420}. A local tip radius of 1600 A was used with a shell thickness of .05 a, and in each case the 0 dislocation emerged at the center of the pole. A large radius was selected in order to obtain more "rings" about each pole, and thus avoid any ambiguity in the interpre tation. Figure 25 shows the (420) pole of the perfect lattice. The three poles chosen for the simulation were the (204), (402), and (420) poles, which should yield -+ + single, double, and triple spirals according to the gb PAGE 83 70 (420) POLE . I ,.. .. s. s . .. ., ' ,. I I I \ \ zU I 1 1 a ~=1 J l l I t l l' .ii ::. I l I r 1 1 \ r ,r :n ., 1 1 : I J: ::\! ,S i i I : '!:!!:::!: ,.,: ,,, _.; :: : . I I .1 : ,, .,,. ............. .... .. . 'I... .................. .. .... ... .... .... .. ... .... Figure 25. Shell model; perfect FCC lattice, radius 1600 A, shell thickness .05 a 0 (111) projection. PAGE 84 71 criterion (Figure 26). These examples are indeed special cases since the probability that a dislocation will pass through the center of a pole is practically nil. Therefore, the dislocation was moved from the central region of the pole and the same simulation procedure followed (as just explained) for several other locations. The result on the (204) pole is shown in Figure 27, which shows a single spiral starting at the point of emergence of the dislo cation. Note that in an actual image this configuration might be interpreted as an edge dislocation even.though the dislocation is of pure screw character. It is a relatively simple matter to include the anisotropy factor in the equation for the displacements around a screw dislocation; the equation becomes: < 32 ) b ~z = 2 Il 0 (tan A) where A is the anisotropy factor. An anisotropy factor of A= 2.46 was used, ( 9 ) which is the value for platinum: however, the results obtained are remarkably similar to the isotropic case where A= 1. No noticeable elongation of the "rings" was obtained, and only a few isolated image points have been deleted or added. These simulated images should be accepted only qualitatively, i.e., no claim is being made at this point PAGE 85 (204) POLE ,,, .... ,,, ., .. .. . .... .. .,I'. ,. ... . .... .... ... ,. .. .. ,,,. ,,, .... ... "' ,,:,.,. ..... ,.. .. .. I "' ,_. .. ... t ,.,,,,. :: : .. : .. ;:~. ~ .. : \ .. ... -'f -~ i ,-::\ !\ -. : . -; -="I'#. : : . ,. ~ ... : ....... ..... ... . :. .,. ,,,,,.:, ... ,;.t;,' -: ;. : .. .. :--...-~ .. .. .. :. .. ~.. .,,, . ..... .... ,. ,.,.: : ...... .,. .... ..,,. .,.. :. ...... .. .. "' .. "" ..... .,... .. ". ...,.-er .,, . . .. ... .... . / ........ A I (402) POLE ............. ... . ... .. .. .. .,,. . .. .. _.,. ...... "'"' ..... . ... .... \. .. _.. ~,.. .. .. . _.,,,,. . . .. ,. ... .. .. .. .. .. .. ... ... . . ._-,:.. :.... : ::,r ..... .. . : .. _.,,.. ... .. .. .. .. ..... .. : I : \ : : , ' : ' I : : ; .. .:--::. .. : ~::. ,_ : .. :=:=: .. .. --:: ':::: : :. :=::.-; .. :: :: .: : : : . .,. .... ; .: : : . .. .. -~.. . . . .. .. . ..... ...... .JI. .. .. . .,.. _,, .. ._ ,fll _.. .. ,._ ..... .. ... : .. ... / .. .. .. --... ...... .,.,,... ,. .. .. .,. ... ......... ,. ,... .... .. B I ___ 0 (420) POLE ...... .. . .. .. .. ... . ... .. .. .. .. .. .. : .... ..... .. . .. ; .. ......... ., . ' I,: I : : := .:== :,:::-:-.,u.:: ::, ,. : i s a= = ; .J i 1 it I i f! 1 :i .. i !\ l a \ i !, ~ :, t \ J !t } i i : :,. l:s:,zt 1 1 1 I : ., -i -.. .-I :... .: 1 .. : I :. \:, .... .: . ... .. .. -i:, . C Figure 26. Shell screw dislocation (b) double leaved model of FCC lattice, 1600 A, shell thickness .OS a 0 (111) projection, plane: (a) single leaved spiral; emerging at spiral; (c) of the the center triple leaved spiral. PAGE 86 .. .. ... 73 ... .. .. .. .. \ .. J> ... .... . . -~ ... \ -::.. ~... -~ .. ~-~ .. -.~ ... ,: :: . r \ : ... : ..... ..... ..~ .. ............ ,:. .,,. .. .. .. :. .. ...... .. .. .. .. .. .. . .. .. . .. ti' ... Figure 27. spiral radius= cation) Pure screw on (204) plane 0 dislocation causing :+ edges, b = a 0 /2 [110] 1600 A(*= point of emergence of single dislo PAGE 87 74 that exact agreement with experimental images will be obtained, although the long-range effects should be correct. Pure Edge DisZocation Classically, the displacements due to an edge dislocation are computed in a plane normal to the dislo cation line (here again isotropic elasticity is used since the anisotropic displacements have not been solved in the general case). If the dislocation line lies along the z direction, then the displacements are given by: ( 3 l) z = 0 -b 1 2v cos(2 0 ) y = 2II 2(1 v) ln + 4(1 v) b 0 + sin(20) X = 20 4 (1 v) where -IT~ 0 rr. These displacements are considerably more involved than for the case of a pure screw dislocation. In the case of an edge dislocation the displacements are in a {112} plane. The displacements used to "generate" the dislocation can be seen most easily by looking at a {112} plane (the dislocation line being normal to this plane) with the b lying in the plane. Figure 28 shows the re sulting atomic positions on the (112) plane, and the magnitude and direction of the displacements used in order to "generate" the edge dislocation from the perfect crystal. PAGE 88 u ........ CREATION or EDGE DISLOCATION IN (112) PLA N E All displacements (u,v) are in (112) plane Atom positions in perfect (112) p l ane Atom positions in dislocated (112) p l ane I I l I .. J Figure 28. Magnitude and direction of the displacements used to generate an edge dislocation from the perfect lattice. -...J Ln PAGE 89 76 The pure edge dislocation was simulated in the same regions on the {420} planes as the screw dislocation. The results on the (204) plane are shown (Figure 29) and they are qualitatively the same as for the screw dislocation. This is indeed very surprising and indicates that the component of the b normal to the set of planes being considered is the only important factor determining the defect configu ration. This criterion is tested on other poles, and the results are all qualitatively the same, i.e., the gb criterion is always satisfied for both the case of the edge and the screw dislocation. The Mixed Dislocation Knowing the perturbations due to the edge and screw dislocations separately, the perturbations due to a mixed dislocation with any orientation of band dislocation line arethen a simple matter. The screw component will always lie parallel to the dislocation line and the edge component will lie normal to it. Th b = b + b ( 33 ) en total screw edge' where btotal = a 0 /2 <110>. The program is set up so that a dislocation lying in any orientation can be produced and either sessile or glissile dislocations can be investigated. Since only atom positions very close to the surface of the specimen can be seen in a field ion image, information about the orientation of the defect will not be discernible from a single micrograph. PAGE 90 77 .. .. ... . . . \ .. ...... .. ...... \ \ .. .... .. \ ...... .. .. . . .. \ . .. \ .. .. ... ...... , \, \ . .. .... \. .. \ -'' ... ... .. .. .... . ..., . . ... ": ... \ ', ... .. \ .. ~ .. . ~ .. .... .. -: ~ --=-''~"~~ \ \ = : -! : ..~ ~~:: : : .. S" ~ .... .... '\',,: \ \._,-:o , , : ,.. .... .. .. .. .. \ ... : l '\ .. \ ., .. ... .... ..... ... .. \ .. .. .. ... .. \ \ ... .. .. . \ \ ..... ... : \ . .. ... .. . . \\, ... .... ... ... . . ...... .. Figure 29. Pure edge dislocation causing -+ single spiral on (204) plane edges b = a 0 /2 [110], radius= 1600 A (*=point of emergence of dislocation). PAGE 91 78 The results obtained using a single mixed dislo cation also point up the fact that only the component of the b normal to the set of planes determines the longrange effect. As the orientation of the dislocation line is changed, keeping b constant, no detectable difference occurs in the image. There is an "image force''( 34 ) on a dislocation, which tends to make the dislocation line lie normal to the surface it intersects. The image force results from the fact that the dislocation is near a free surface, i.e., the specimen surface, and the force increases as the dislocation approaches the surface. It would be possible, therefore, that as the dislocation line approaches the surface it could jog over a few atomic planes near the surface so that it would lie more nearly normal to the surface. It is assumed therefore that the dislocation line lies nearly normal to the surface. In order to determine the direction of the dislocation line, a field evaporation sequence of the defect would be required. The gross movement of the defect could then be followed and the direction of the line determined. It should be pointed out that in some cases two sets of plane edges from two poles (i.e., rings) can be seen in the region of the defect. Figure 30 is an example of a computed plot illustrating this point; however, in an PAGE 92 79 Dislocation Visible on Two Sets of Plane Edges ..-Dislocation : b= + (110) 220) Rings Figure 30. Two sets of plane edges visible in the region of a defect (radius= 800 A). Note that the gE criterion is satisfied for each set of planes (*=point of emergence of dislocation). PAGE 93 80 experimental image this effect would only be seen on higher index planes. (Fo~ example, {200}, {220}, and {111} planes have very close atom spacings on their edges, and other planes are extremely hard to detect in these regions.) Applying the gb criterionto the displaced rings inde pendently to determine the Burgers vector usually will not give a unique solution. For example, both a /2 [101] and 0 a 0 /2 [011] (also their negatiyes) will yield single spirals on a (002) plane. However, when two sets of plane edges can be seen, two simultaneous conditions on g.b must be satisfied and a unique b may be deter~ined. Note that either a negative or positive result may be obtained for gb. The significance of this manifests itself in the direction or sense of the spiral. If two dislocations can be seen in an image, then the directions of each Burgers vector relative to the other can be determined (Figure 31). Total Dislocation Configurations Introducing another dislocation allows one to analyze both dislocation dipoles, dislocation loops, and any other configurations resulting from two dislocations. The intersection of a dislocation loop with the surface should be no different than that of a dipole of equivalent size. In order to determine what effects a dipole or a PAGE 94 81 Figure 31. Schematic drawing indicating the re lationship between the sense of the spiral on the image {clockwise, counterclockwise) with the direction of the normal component of the Burgers vector {N 1 and N2 represent unit vectors normal to the set of intersecting planes). PAGE 95 82 loop would have on the image, the program was modified so that two mixed dislocations with any orientation and any position could be generated. Dislocation dipoles, and pairs of dislocations were simulated in various regions of the image and the results indicate that the long-range effects are given simply by the sum of the effects due to each individual defect. The possible cases which exist fall into four general categories which can all be explained. If two dislocations are visible on a set of plane edges and if the gb criterion for each is +l, then in the vicinity of both dislocations a single spiral will be observed. Furthermore, the effect far from the dislocations will be the sum of the two, that is, a double spiral (Figure 32a). On the other hand, if the gb criterion of the two dislo cations gives +land -1, respectively, then no long-range effect would be observed. A spiral of opposite sense would emanate from each dislocation,forming a smooth curve between them, whereas the plane edges inside and outside of the configuration would remain perfect. Figure 32b shows a schematic diagram of such a configuration. If instead, the pair of dislocations were approximately the same distance from the central pole, and say the gb criterion again predicted +land -1, the configuration shown in Figure 32c could result, where one of the plane edges appears to be horseshoe shaped. Now if the dot PAGE 96 ; Pair of Di slo cations Same Burgers Vector A Pair of Dislocations Opposrte Burgers Vector C .___ __ 83 Pair of Dislocations Opposite Burgers V e ctor B Pair of Dislocations Opposite Burgers Vector D Figure 32. (a) Two dislocations, each producing single spirals with the same sense: the long-range effect is a double spiral; (b) two dislocations each producing a single spiral of opposite sense yield no long-range effect; (c) two dislocations equidistant from pole producing spirals of opposite sense; (d) two dislo cations producing spirals of opposite sense. Due to proximity of dislocations the effect is an extra plane segment. Note: Some slight distortion is present in the region of the two dislocations. PAGE 97 84 product for each of the dislocations were reversed, a line of atoms or part of a plane edge would appear between the two dislocations (Figure 32d). In all cases there is some slight distortion in the plane edges in the immediate vicinity of the defects. Dislocations which are very close 0 to one another (5-10 A) will interact strongly and the results may cause local blurring (or lack of resolution) in the imagetresulting in a short streak between the two or a similar phenomenon. Correlation of Simulat e d Defects ~ith Experim en tal Images in the FCC System Introduction Correlation and interpretation of experimental images were obtained with images taken from the literature (previously unexplained) and images of iridium obtained experimentally by the author (see Ap pendix III for method of preparation). It should be pointed out that the volume of material under investigation in the field ion microscope is very small indeed~ therefore in a material which has a 12 2 dislocation density of say 10 /cm, one would expect to see one dislocation in each specimen, assuming the dislo cation to lie parallel to the wire axis. Thus it may be a very laborious task to find a dislocation: the method used was to take a photograph after evaporation of approximately 10 (002) layers. Evaporation of single layers is a very PAGE 98 85 tedious task, and care must be taken not to "flash'' the specimen. Since the field and hence the stresses on the tip are extremely high, it is possible to literally destroy the specimen by increasing the field too much; this is known as "flashing." Simulation Process The first step in the simulation process for comparison with an experimental image is the calculation of the 16cal radius of the specimen in the region of the defect (this is accomplished using the method prviously explained on page 56. The b of the defect (or the various possible b's) is then predicted based on the experimental image and the simulated image is computed. In Figure 33a two defect configurations can be observed on the (204) plane edges. They appear as an apparent collapsing or pinching together of the plane edges. Whatever the configuration may be, there is no long-range effect due to it, i.e., the plane edges close to the (204) pole are perfect, and those outside of the configuration are also perfect. This type of defect may be thought to be some sort of vacancy cluster, but a little reflection will show that this configuration may be treated as a small dislocation loop, and the inter section of a loop with the surface gives the same effect as a dislocation dipole. PAGE 99 86 pj gure 33a. Experimental i mage of Pt (courte sy of Pro f E.W. Miiller). Defects shown in the region of (204) pole (A and B); also shown is (002) pole (C). .. . . .. . . .. . . . .. A . .,,-../4"T--. -... ... ..... ......... ..... : : :: : 1-~ ... -~ =-! :/: . . : . ...: ...,.. : . ... : ... ..... : : . . .. : : : : .. . . : . . .. .. . . . .. :: :. . : . . .. .. .: ... : : = : --=::::::&___~~~~w,/"::-;. .,.: .: ~~ 1/ B . ----------Figure 33b. Computer simulated image of defect 0 configuration on (204) pole in Pt; radius 650 A (x = point of emergence of dislocation). PAGE 100 87 .. . . .. , JI .. ............ _/ ... -~ ...,,.,,..,~ .... + ~ .. -,.-,.... ...... ---. .. .... C -..-----............. ----Figure 33c. Computer simulated image of configuration on (002) pole in Pt; radius (+=point of emergence of dislocation). defect 900 0 A PAGE 101 88 In one case two of the planes appear "pinched" together (Figure 33a:A), while the configuration on the opposite side of the pole clearly shows a "pinching off" of three planes (Figure 33a:B). Both of these configu rations can be interpreted as a pair of spirals of opposite sense, producing no long-range effect. The configuration at B can be considered as a pair of triple-leaved spirals of opposite sense arising from the Burgers vectors a /2 0 [101] and a /2 [101]. The results of the simulation are 0 shown in Figure 33b. For the configuration at A, two possible b's which produce opposite double spirals are a 0 /2 [011] and a 0 /2 [011]. There 1s another combination of b's which could give rise to the same configuration, namely a /2 [011] and 0 a /2 [011], however, this combination corresponds to two 0 separate "isolated" dislocations, i.e., not part of a loop or dipole. The author wishes to point out that in the micrograph shown, the specimen had been flashed, or annealed and quenched, thus any vacancies present would tend to agglomerate and the chances of finding small dislocation loops would be higher than normal~ therefore, the dipole configuration was used to simulate the image at A. In the simulated image (Figure 33b) there are some extra points in the region between the first step edge of (204) and the defects (A). These could be eliminated if the shell thickness PAGE 102 89 were varied slightly or possibly if the disloca,t;Lons were moved by a very small amount. It should be remembered that the shell model is very sensitive to changes such as this. It is not known from this one micrograph, ho w ever, if the configuration is really a dipole, a loop, or the two distinct dislocations just cited. Once again a field evaporation sequence would be required in order to determine this. If it were a loop its apparent size would change as successive layers were removed. Figure 33c shows another configuration (or loop) visible on (002) rings. Here again the separation of the defects is enough so that the defects can be treated individually. On looking at the micrograph, a single spiral of clockwise sense is observed on (002) rings, and farther from the (002) a spiral of opposite sense is observed. The step edges near the (002) and those outside of the configuration are perfect. It is readily seen that there are several possibilities which give a single spiral on an (002) plane, thus from this configuration a unique b cannot be determined for either of the dislocations~ how ever, it is known that the sense of each must be opposite. If the configuration is not a loop, it may be possible to rule out some possibilities based on the forces between the two dislocations. Certain combinations would repel each other or be attracted, and based on their separation these PAGE 103 90 combinations may be impossible. It i?. impo~siple to see any step edges other than the (002) planes; if this were possible a unique b could probably be determined. The results of the simulation shown in Figure 33c would indeed :+ be the same for other combinations of b's, pointing up the fact that from this one particular image a complete analy sis cannot be made. However, this was not the case for the (204) pole where a unique set of b's could be determined. Figure 34 shows a single spiral on a (220) plane of an iridium specimen starting at point A. This is clearly a total dislocation with S = a /2 [101) or [101) (or their 0 negatives). Note that some local disturbance is caused by the dislocation in the adjacent ring. There is also some disturbance in the image points at Band c. It is hard, however, to say exactly what this may be, since the complete rings cannot be seen in this region . The three regions of disturbance might be a dislocation tangle; the separation 0 between A and C is approximately 100 A. There is no reason to believe that they should be partial dislocations, since the stacking fault energy of Ir is approximately 700 ergs/ 2 cm. Figure 35 shows an apparent triple spiral on a (113) plane of an Ir specimen. A little thought will show that :+ it is not possible for ab of a /2 <110> to produce a 0 triple spiral on this set of planes. It is possible, however, for two dislocations to give a total of three PAGE 104 91 Figure 34~ Single spiral on {220 } plane of an Ir specimen starting at point A. Points Band c indicate 6th2r disturbed regions probably due to the presence of dislocations. PAGE 105 92 Figure 35. Triple spiral on {113} plane of an Ir specimen. PAGE 106 93 spirals. For example, on the (113) pole the dislocations with b 1 = a 0 /2 [011] and another with b 2 = a 0 /2 {011] would give the required triple spiral. These could be separated 0 by 10 A in the specimen, and would be essentially noninteracting since their b's are perpendicular. Using this set of b's, the image was s i mulated in the region of this pair of defects, and the results are shown in Figure 36. Another single spiral on a (220) plane in iridium is shown in Figure 37, and here the possible b's are a /2 0 [101] or a /2 [101]. The emergence of the dislocation is 0 at point A, and note also the distortion in the adjacent planes caused by the strain field of the dislocation. Note some disturbance around the point B: however, it is so close to the edge of the image that the nature of the disturbance cannot be determined. Defects in general cannot be interpreted without ambiguity on the low index planes from a single micrograph; however the analysis presented here does allow one to obtain a great deal of useful information from the atomic positions observed in a single micrograph, both on low and high index planes. PAGE 107 94 . . . .. .. . . . . .. .. ---.. ...... ,,---____:;, .... I: I : : : I I / .r r 1 / I 1 : II I, :I : . l : ,, .. ,,. 'I.: I I \ I I'/ 6 I I : .. '' ,,. : : 1: . .. : ;' I: I ( i (. :: :, I I I t 1 ' f I l . .: I I I .. .. . . : I . .. z .. . .. ...... ... .. Figure 36. Simulated spiral caused by pair dislocations. . . . image of triple of noninteracting PAGE 108 95 Figure 37. Single spiral on {220} plane in Ir specimen. Point A marks emergence of dislocation while point B shows anothe~ disturbance which cannot be interpreted from this single micrograph. PAGE 109 96 Defect SimuZation of Tot a l Dislocations ~n BCC Materials 1 Using the Sh e ll Mode l Introduction There is a great deal of uncertainty as to the nature of the dislocations to be expected in BCC materials. Several dislocation reactions have been proposed, a few of which can be summarized as follows: < 35 ) a a a 0 [111) 0 [112) + 0 [111) 2 -+ 3 6 a a a 0 [111) 0 [111] + 0 [111) 2 -+ 6 3 a a a a 0 [111) 0 [111] + 0 [133] + 0 [111] 2 -+ 66 6 a a a a 0 [111] 0 [111] + 0 [ 113] + 0 [111] 2 -+ 6 6 6 a a a a 0 [111] 0 [ 011] + 0 [211] + 0 [011]. 2 -+ 84 8 There has not been, however, any positive identification of these reactions. The interpretational problems involved in BCC materials, therefore, have less physical basis as compared to the FCC materials. Results of the Simulation The same methods were used as for the FCC materials in simulating th~ image, and the results are very similar to those reported for FCC. For an isolated dislocation a PAGE 110 97 spiral configuration is always obtained, except for the case of gb = 0. The gb criterion can again be used to predict the number of leaves in the spiral. It should be pointed out that the spiral configuration is not always obvious, but again may appear as a 3 over 2 or 2 over 1 edge dislocation, even though the b has screw character. This occurs when the dislocation line emerges at the edge of or between crystallographic poles. The direction of the dislocation line does not cause any noticeable change in the simulated image, only the direction and magnitude of the bare sufficient. Correlation with Experimental Images Muller has reported a pair of screw dislocations emerging on a (001) plane in an iron whisker. This image was simulated using a pair of mixed dislocations (the line of the dislocations were made normal to the surface) with opposite b's. The horseshoe configuration (Figure 38a) can be interpreted as two spirals of opposite sense, each starting from the "end" of the horseshoe. The long-range effect of a pair of dislocations is equal to the sum of the individual effects and 1 therefore the long-range f h f h b~'s e feet is zero, since t e sum o t e is zero. A dislocation with b = a /2 [111], which is expected to 0 occur more frequently in BCC than say a dislocation with PAGE 111 Figure 38a { 1 0 0 } plane Mtiller( 8 )). Ho r seshoe confi g urat i on on of iron whisker (afte r . .. .. ... .. .. . . .. . . . .. . . . . . . . . . 9 .. . .. ... ... .. ... .. . . . ....... . . . .. . . '*' .. . . . . . . . . . . . . . . Fi g ur e r a tion o f 38b S im ulation of de f ect conf ig in 38( a) us ing p air o f disloca ti ons mix e d c hara ct er riorma l to t he s urfa c e ( = p o i nt o f emergen c e o f dis loc ati o n ). \.0 00 PAGE 112 99 ~=a [100], etc., will yield a single spiral on the (001) 0 plane. The simulated image is shown in Figure 38b_,with b 1 = a 0 /2 [111] and b 2 = a 0 /2 (111]. It is clear that the dipole configuration is necessary in order to simulate the image: a single dislocation would not give rise to the horseshoe configuration. Figure 39a is a micrograph of a tungsten image taken at 77 K. A single spiral starts on the first step edge of the (011) plane. The computer simulation of this defect is shown in Figure 39b with b = a /2 (111]. The 0 density of image points could be increased by increasing the shell thickness~ however it was felt that this was not necessary in this case, since the long-range effect would remain the same. Note that there are several <111 > di rections which make the same angle with the (011) pole. However, this is not the case in general, and usually there will be only one or two possible b's which could give the observed configuration. Interpretation of both single dislocations and multiple dislocation configurations is possible in BCC and much information is available if one looks at the images, with the gb criterion in mind, and notes that in the g~neral case a unique b can be determined. If two se t s of rings are visible, the gb criterion will be PAGE 113 i I i 'I I ,j I I I I Figure 39a. Single spiral on {011} plane of tungsten specimen (after Hren). i I i .. ... .. . .. .. .. . .. ,, ... ... ,._ .. I .,. .e1-.. \\ ,, ,;I r\ \.. ': : :: : . . : : \ \ \...._ /I : ; .. = \ ,. '1 ......... .... .. ._ ._ .... .... , .. .. .. Figure 39b. Computer $imulation of defect shown in Figure 39a with dislocation line normal to the surface and$= a 0 /2 IllO] (*=point of emergence of dislocation line) I-' 0 0 PAGE 114 lOl :+ satisfied on each set of planes, and a unique b can usually be determined in this way. Simulation of the Perfect and Dislocated Hexagonal Lattice Introduction Rhenium was the first hexagonal material to be d h f. ld ( 3 6 ) image int e 1e ion microscope, and Muller has suggested that the alternation in intensity on the (0001) planes was due to the ABAB stacking of these planes. Later it was suggested by Melmed, < 37 ) after looking at field ion images of ruthenium in which "double rings" appeared on certain planes, that the reason for the "double ring" (see Figure 40) was due to the fact that in hexagonal materials some planes are "rippled." This means that not all of the atoms associated with a crystallographic plane physically lie in the plane. In fact it can be shown that for a given plane (hkil), if the expression 2h + 4k + 31 does not yield a multiple of 6, then the plane is rippled. < 24 ) Figure 41 shows schematically the nature of a "rippled" plane in a hexagonal material, and the geometric reason for the appearance of the "double rings." Note in the micrograph (Figure 40) that as the distance from the center of the pole is increased, the double rings can no longer be resolved. PAGE 115 ) 102 C:-----~ --.-, -. ----.=--.-:::T-------_ -,~ -r,:, i :_ ___ __ _j E'igu.re 40. Micrograph of Ru showing existence cf double rings on poles A and B (after Melmed(37)). PAGE 116 Double Ring I 103 ocal Tip Radius X Relative Position of Atoms in Image Figure 41. Geometrical origin of "double rings~ due t6 "rippled" nature of cer~ain planes in hexagonal materials. PAGE 117 104 Simulation of th e P e rf e ct HCP Lattic e It was hoped that,using a computer simulation tech nique similar to that used for both FCC and BCC, the geometric origin of these "double rings'' and the nature of the possible defects in HCP materials could be shown. Several poles on which the rippled nature was expected were simulated using the shell model. Figure~ 42 a and b show the results for a perfect lattice on both the (lOI2) and (lOil) poles. In both cases, the "double ring" nature appears, and as the distance from the center of the pole is increased, the spacing of these "double rings" decreases. The effect is purely geometrical in nature. Changing the c/a ratio has little effect on the image; 0 it simply increases or decreases the distance between pairs of rings. A plane on which no rippling was expected was also simulated; here the rings are equally spaced. The reason for the disappearance of the "double rings" as the distance from the central pole is increased is due simply to the fact that the resolution of the microscope is insufficient to resolve the separation between the two "rings ";therefore, only a single ring can be seen in the image: PAGE 118 . ...... Figure 42a. Simulation of perfect lattice, {1012) plane, c/a 0 = 1.58, showing the double ring nature. ............ . . ......... .... ... .... . ... .. . .. . . . ........... perfect 1.58 Figure 42b. S ;t mula,t;i:.on o~ the lattice, {1011) plane, c/ao = showing the "double ring" nature. I-' 0 \J1 PAGE 119 106 Defect In te r pre tatio n of HCP M at e ri a ls Ranganathan and MelmedC 24 ) have reported a "dislo cation" in a grain boundary in ruthenium and also predicted that on a "rippled'' plane, the normal spiral configuration expected from the gb criterion must be multiplied by two. Using the shell model, dislocations on s e veral planes were simulated; a typical result is shown in Figure 43. It is proposed here that the position of the dislocation with respect to the central region of the pole, will determine whether the product of gb must be multiplied by two. For example, if on the (1010) plane (rippled) a dislocation intersects the pole close to the central region, a double spiral will exist. If, however, it intersects the pole in a region where the "double ring" nature cannot be detected, a single spiral will result, this, in short, is the criterion that must be us e d on any pole to predict the resulting configuration. Of course, on a smooth plane the results can be interpreted straightforwardly by the same criterion as that used for FCC crystals. There have been no cases reported in the literature of an isolated dislocation in a HCP material which could be interpreted by the gb criterion. (Muller has reported a Rh micrograph which has a large region of deformation, however, it is impossible to isolate any dislocations in the photograph.) PAGE 120 107 .. ... .. : ......... . .. : : .. .... ... ----Figure 43. Dislocation on of a HCP material with c/a 0 (1011) plane = 1.58 (* = point of emergence of dislocation). PAGE 121 108 Partial Di s locations in FCC Mat e ri a ls Intro d uction There has been a considerable amount of speculation as to the nature of the appearance of partial dislocations in field ion images. It has been suggested that partial dislocations would give rise to some sort of streaking in the image, (l?) however, no basis has been given. The gb criterion does not appear to yield much useful inf o r w. a tion, since values of 1/3 and 2/3 can be obtained, however, this really has no meaning based on the previouswork where gb always gave rise to an integer. It was decided~ therefore, that computer simulation of partial dislocations could give ma:1y useful results. Method of Sim u lation and R e sults: Shockley Partial Dislo cations The displacement equations used to generate total dislocations could also be used for partial dislocations, the program being the one used for a pair of dislocations7 only in this case each Burgers vector corre~ponded to that of the appropriate partial dislocation. Using this program, it was possible to generate partial dislocations with any orientation and Burgers vector. For a given b there is only one set of partial dislocations which give rise to a stacking fault of the PAGE 122 109 form ABCBCA, as shown in Figure 44. Using the improper sequence of b's (dotted lin e s in Figure 44) corresponds to an unrealistic physical situation in which two "A'' planes are stacked together. Several Shockley partial dislocations were simu lated on various poles and the results are always quali tatively the same in that there is always a displacem e nt of the rings along a great circle joining the two dislo cations, corresponding to the intersection of the stacking fault plane with the surface. For dislocations separated by small distances, the disturbance will appear along a straight line joining the two dislocations. Figure 45 shows a pair of Shockley partial dislocations on a (011) plane, the total b being a /2 (101]. The effect far from 0 the dislocations is always that expected from the total dislocation from which the pair has dissociated. Figure 46 is another example showing the effect of a pair of Shockley partials on the image; note that even though the dislocation lies in the plane (no long-range spiral expected) 1 the distortion in the image due to the presence of the stacking fault is still observed. Figure 47 is an experimental image of a platinum specimen. There are several distorted regions around the edge of the image in which a boundary appears to terminate; note particularly the regions labeled A and B. It is PAGE 123 110 ~Positions of Atoms on Next Plane ~ ---------~ -----.. ---~ --------Figure 44. Solid arrows correspond to partial dislocations which yield stacking fault between the dislocations of the form ABCBCA. Dotted arrows show incorrect sequence of ~'s giving rise to two "A~ planes being stacked together. PAGE 124 111 Stocking Fault m (220) Pole .. . .. .. .,.._.. .. .. .. .. _..... -... .. r.,. ..... ........... ... ~. -. .~ ,r --... .... ... r ... . \ I ,,_ _ ,_ .,,, \ .. ,'f "r .... . \C U ;) \ r,,.,_ .,,:r'.. .. ~ ... .. .. ---...... _., .. ,. .. .: 9\ .... __ .... .. .. ............. ~ \ -\. ... -.. i \. -----------------------Figure 45. ~air of Shockley partial dislo cations on (220) plane; total E a 0 /2 (101] (*~point of emergence of dislocation). PAGE 125 112 -----. ------Stacking Fault on (002) Pole -.---------------------Figure 46. Pair of Shockley partial dislocations on (002) plane, total ,+ b = a 0 /2 [110] (*=point of emergence of dislocation). PAGE 126 113 suggested that most of the distortion present, especially at A, is due to the presence of a stacking fault. The stress state in a field ion tip has not been calculated. In general, however, it does appear as though it has a definite effect on the dislocations present in the tip. Muller has shown( 3 ) that there is a hydrostatic stress due to the high fields present, and more recently Hren has shown that as a result of the "image force" on a dislo cation, it is possible that dislocations would tend to slip out away from the wire axis. These effects cannot be neglected, and they are probably the reasons for the apparent preponderance of dislocations around the edge of the image, as is clearly seen in Figure 47. Streaks may arise in an image due to the presence of a stacking fault since streaks have been observed in heavily deformed specimens, and their origin has not been accounted for. A "streak'' would not show up as such in a purely geometrical model, but would probably arise in the region of a stacking fault due to the difference in electronic density in the region of the stacking fault as compared to the unfaulted regions. Method of Simulation and Results: Frank Partial Dislocations Another defect of considerable importance in the FCC system is the vacancy loop with b = a /3 <111>. This fault 0 PAGE 127 114 -a._ ---------~--~. --------------Figure 47. Micrograph of Pt taken at 21 K. Dis t6rtion around the edge of the image (A and B) may be due to the presence of stacking faults. PAGE 128 115 is pure edge character in nature, and is usually considered a secondary defect, since it is formed by the asglomeration of vacancies on a (111) plane. The (111) planes then collapse in slightly forming a stable dislocation loop. A schematic diagram of the vacancy loop is shown in Figure 48a, which depicts the collapsed (111) planes in the r~gion of the defect. A similar defect, usually formed as a result of radiation damage, is the inter stitial loop, here again the bis a /3 <111>~ however, in 0 this case the atomic configuration can be explained by the insertion of a (111) plane over a limited region. A schematic drawing of the (111) planes in the region of the defect is shown in Figure 48b. Both the vacancy and interstitial loop could be simulated in a field ion image with a pair of dislocations since the intersection of the loop with the surface can be described with a pair of dislocations of opposite sign (dipole). Both loo~s are of pure edge character and 1 as with the case of Shockley partials, both loops are confined to lie in (111) planes. The vacancy loop was simulated on several poles; a typical result on the (110) pole is shown in Figure 49. This is a rather special case, since the loop is right at the center of the pole. It is used here only as an illustrative example. Note that the distortion in the image appears to be normal to the fault PAGE 129 116 ------------.-----------------------.. -------(a) (b) Figure 48. Schematic drawing of (111) planes in the vicinity of (a) a vacancy loop and (b) an interstitial loop. PAGE 130 117 .. .. .. ..... ---... .. .. .r- . .. ~. ... .. ..... / ..... \ .... ., -.. ..... \ /(( v. ~, )\\\ It ih t w I l: -ii. ., ..,, r:,' .. .. .. Je ,. . -,._ .. ._ ~ ._er . ... .. ...... .. .. .... ..-: .. 9'L.. .. -. .... e e e Na .. .. .. .. .. Figure 49. Vacancy loop on (220) pole showing distortion in the direction normal to fault plane (*=point of emergence of dislocation). PAGE 131 118 plane: this might be exp e cted since the principl e distortion in the lattice due to the presence of the defect lies along this direction. Note that in a field ion image, small displacem e nts normal to the "sph e rical surface" produce a relatively large change in the apparent ring size in that region. This is a direct result of the sensitivity of the topological features with radius, as illustrated in Chapter II. The distortion in the image app e ars to be mainly on one side of the emerging loop (asymmetric)~ the side on which this is greatest is the side on which the projected compon e nt of the Burgers vector onto the surface is largest. Even though the displacements due to the presence of either the vacancy or interstitial loop are symmetric if one looks at a {112} plane, there is no reason to expect the distortion in the lattice to be symmetric on any other planes. On changing the loop from intrinsic to extrinsic, the distortion remains on the same side of the emerging loop (since b of an extrinsic loop is simply the + negative of the b for the vacancy loop). In the case of the vacancy loop, there appears to be an inward collapse of the segments of the planes between the dislocations (see Figure 49): however, when an extrinsic loop is simulated, the plane segments between two dislocations are "pushed out" (Figure 50). It is therefore a simple matter PAGE 132 119 .. .. . .... . .. a e -,ona re e-t.... .. -... .... .. r .... .... . r ,zr -~ ..... "' I F .... \. : I' 15\..,. : I (!~~ \ \ 0 t t )1 l ~f t~~ l \ \. ,~ "I\,,..,.. _ J ~' J .... -....... ... .. .. .. .. .... .. _,.. ... .. .. -_. .. .. -.. .... .. .. \ ... .... .. I -----. -----. --------Figure 50. Extrinsic loop on (220) pole, showing distortion in the direction normal to the fault plane (*=point of emergence of dislocation). Note: "pushing out" of planes. PAGE 133 l20 to distinguish the difference between extrinsic and in trinsic loops. An experimental image which was believed to contain a vacancy loop was simulated to t~st this hypothesis. The results of the simulation and the image are shown in Figure 51. It is seen that the overall ring configuration is the same for both the simulated and experimental images, and it is thought that the streak may lie along the dis torted region, which in this case would be normal to the fault plane. (Note in this case the distortion normal to the fault plane is difficult to detect.) The possibility of a prismatic loop cannot be discounted, since it could yield the same overall ring configuration (see Figure 33c). Since the dislocations are closely spaced, the streak may arise as a result of interactions between the two dislo cations,causing some local blurring and giving rise to a streak. It would be a simple matter to determine whether the defect was either a Frank loop or a prismatic loop if an evaporation sequence containing the defect could be seen, i.e., if it were a Frank loop the streak would move in a direction normal to the axis of the streak. PAGE 134 r .. I L -----., 'l\ Frank Loop on (002) Pole i I I I I ) 1,,11.,, J 0 I '.,.:'' .......... : .. ... ___ .,J'I.,, .,r I : : .' : : . _,.... / . . / . .... J .., --~ .. , . . "'/ ... . / ., ..... ~-I I / .I I .. .. -., ..,._ .. / ..,._.,,I' I' ., v r.. . . ,, .. / .. .. J'. .. .. f; f -. . . . : --..... I', ,,, .. ,. .......... ,,,. .. .. . .. .. .,. . e, ,t/ . . . . .. B Figure 51. (a) Experimental image containing defect believed to be Frank vacancy loop; (b) computer simulation of (a) (*=point of emergence of dislocation). PAGE 135 CHAPTER IV CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY Two computer models, neighbor model and shell model, have been presented to aid in the interpretation of field ion micrographs of both perfect crystal structures and crystal structures containing defects. The development of the neighbor model, which uses the geometrical environment, or equivalently the binding energy, as an imaging criterion, resulted in point for point agreement with an experimental image. It was shown that variations in intensity on a single plane correlate well with changes in local sublimation energy A; however, it appears that additional energy terms are required for intensity variations between planes. The utility of the neighbor model was shown to lie primarily in the interpretation of effects in the perfect crystal, i.e., relative brightness, size of image points, etc. The computation of the relative intensity, for example, could be computed by relating the binding "energy" of an atom "Q" to the image int~nsity. The 0 expression for Q is thought to vary as: Q =A+ VI ~, 0 0 where A is the atomic sublimation energy, the latter being directly calculable from the neighbor model. The work 122 PAGE 136 123 function~ can be determined from electron emission studies. Then, the variation in relative image intensity can be correlated with the variation in Q It may be possible, 0 in this way, to use known information for~ and In coupled with the variation in image intensity, to obtain a relative measure of the atomic sublimation energy, and hence the shape of the interatomic potential curve. Results such as this would indeed have far-reaching effect . There are certain limitations inherent in this model's applicability to regions of the crystal which have been perturbed. In order to apply the neighbor model to the dislocated crystal, an accurate knowledge of the interatomic potential function must be known, but is, in general, not available. It was decided, therefore, that the dislocated crystal could more readily be interpreted, using the shell model for simulation of the images. A physical basis for the shell model was established by using the neighbor model to show that those points which contribute to the image in the shell model are the atoms which are bound weakest, i.e., the atoms which protrude the most from the surface. The shell model was used to predict defect configu rations in field ion images in the FCC, BCC, and HCP systems. Basically all line defects and defect configurations composed of total dislocations could be interpreted PAGE 137 124 in terms of spirals by using the gb criterion. Therefore, the important parameter in determining the resulting con figuration in the micrograph is the component of the Burgers vector normal to the set of planes on which the dislocation emerges. A field evaporation sequence is usually helpful in determining a unique Burgers vector. Partial dislocations were studied in the FCC system. The stacking fault separating two Shockley partial dislo cation causes a displacement of the rings along the intersection of the fault plane with the surface. The long-range effect due to the pair of dislocations is that predicted using the total dislocation from which the pair has dissociated. Secondary defects, vacancy loops and interstitial loops, were also simulated using a pair of dislocations, because the intersection of a loop with the surface can be described by a pair of dislocations of opposite sign (dipole). Identification of intrinsic, or vacancy, loops was found to be a straightforward matter: intrinsic loops causing an inward collapse of the ''rings" normal to the fault plane while the opposite effect was observed for extrinsic loops. There has been no reported work in the literature of a positive identification of either extrinsic or intrinsic loops in a field ion image. Extrinsic loops can be obtained with radiation damager therefore irradiating PAGE 138 125 a wire specimen (the conditions can be determined from previous electron microscopy experiments~ and imaging this sample would then show a convincing verification of the results presented here. The structure of grain boundaries has to date been an unsolved problem. Using the simulation procedure for the generation of several dislocations, it should be possible to construct a small angle grain boundary,using the computer. The extension of this work to include a model for a high angle grain boundary might then be possible. An extension of the work on partial dislocations to include the BCC system could result in the positive identification of some of the dislocation mechanisms which have been proposed in the BCC system. The results obtained allow one to interpret defects and defect configurations in field ion images with confidence, since both a physical basis and ana lytical means have been presented for the interpretation. The neighbor model, on the other hand, affords a method for the interpretation of effects in the perfect crystal. PAGE 139 APPENDICES PAGE 140 APPENDIX I These equations were used to obtain the final coordinates on the plot of a point located in the specimen at (xyz). First the coordinates of the point must be projected radially onto the projection sphere: z y The two surfaces shown are spherical in nature, representing the specimen with local radius r, and the projection sphere (specimen to screen distance) with radius R. Note that the drawing is not to scale, since 127 PAGE 141 128 r~lo6 cm while R~lO cm. The coordinates of the image point are known (xr' yr, zr), and the coordinates of this point projected radially onto the projection sphere must be computed. Now oa = r and using similar triangles, ob = R z (~) oa r (r) ZR z = = = ob ZR R r r similarly r Yr YR (~) = = Yr R y r R and X (~) r r then XR R = = X YR r r Thus, the coordinates of the point on the projection sphere (xR, YR' ZR) are known. Before this point can be projected onto the image plane (screen), it must first be rotated into a coordinate system which corresponds crystallo. graphically to the specimen axis. This flexibility really allows the specimen to be oriented in any direction in space, and is done so that a simulated image can be compared directly with experiment. The rotation of coordinate systems is accomplished through use of the standard rotation transformation equations. PAGE 142 That is: X / / /X: r / / 129 _____ y --B2 X 1 = X cos al+ y COS Sl + Z COS yl y' = x cos a 2 + y cos s 2 + Z cos y 2 Z' = x cos a 3 + y cos s 3 + Z cos y 3 They' axis was chosen as the projection axis (the choice was arbitrary since any axis can be made the pro jection axis). PAGE 143 130 Having the coordinates of the point in the (x', y', z') coordinate system, the point can now be projected onto the image plane. Projection Point --~ 0 -Z' The coordinates (Z Y X) are known, and the coordinates p p p of the point on the image (XF, ZF) must be determined. The projection point is situated at 3R from the screen; however, this distance can easily be varied. Using similar triangles: TI 03 and 23 03 = = 46 06 56 06 X 2R + y or _E. = e XF 3R PAGE 144 131 3R XF = (2R + Y) X p p (Eq. 1) Now ZF is computed: 03 13 = 06 46 2R + y (X 2 + Z 2)1/2 or ..E. = I2 12 3R 2 Z 2)1/2 (XF + F squaring: X 2 + z 2 = ( 3R )2 (X 2 + z 2) F F 2R + Y p p r z 2 3R ) 2 (X 2 + z 2) X 2 = (2R + F y p p F p 3R ) 2 (X 2 + z 2) X 2]1/2 (Eq. ZF = [(2R + y p p F p Therefore, the value of XF is computed from equation (i); then,using this value, ZF can be computed from equation (2) . 2) PAGE 145 APPENDIX II Expectancy and Nature of Spirals, due to dislo cations in a field ion image, are predicted by the gb criterion in (1) FCC, (2) BCC and (3) HCP materials. 132 PAGE 146 TABLE 1 EXPECTANCY AND NATURE OF SPIRALS BASED ON gb t, 0 CRITERION: FCC LATTICE b -+ a /2 [110) a /2 [110) a /2 (101) a /2 [101) a /2 [ 011) a /2 (011) g 0 0 0 0 0 0 220 2 0 1 1 1 ... 022 1 1 1 1 0 2 202 1 1 0 2 1 1 202 1 1 2 0 1 1 022 1 1 1 1 2 0 I-' w w (111) 1 0 1 0 1 0 (111) 0 1 1 1 1 0 (111) 0 1 1 0 0 1 (lll) 1 0 0 1 0 1 200 1 1 1 1 0 0 020 1 1 0 0 1 1 002 0 0 1 1 1 1 Note: O = No spiral; 1 = Single-leaved; 2 = Double-leaved; 3 = Triple leaved. PAGE 147 TABLE 2 EXPECTANCY AND NATURE OF SPIRALS BASED ON g.b r O CRITERION: BCC LATTICE b + a /2 [111) a /2 [111) a /2 [111) a /2 [111] g 0 0 0 0 110 1 0 0 1 011 0 0 1 1 101 0 1 0 1 101 1 0 1 0 I-' 011 1 1 0 0 w ,i,. 121 0 1 2 1 112 0 1 1 2 211 1 1 0 2 211 2 0 1 1 112 2 1 1 0 121 2 1 0 1 020 1 1 1 1 200 1 1 1 1 002 1 1 1 1 Note: 0 = No spiral; 1 = Single..:.leaved; 2 = Double-leaved. PAGE 148 TABLE 3 -+ ,+ EXPECTANCY AND NATURE OF SPIRALS BASED ON gb r O CRITERION: HCP b -+ 1/3 [2110) 1/3 [1210) 1/3 [1120) [0001) g Ri:e:eled Plane (1010) 2 0 2 0 (0110) 0 2 2 0 (1100) 2 2 0 0 Smooth Plane (1120) 1 1 2 0 (2110) 2 1 1 0 (1210) 1 2 1 0 Ri:e:eled Plane (1012) 2 0 2 4 (0112) 0 2 2 4 (1102) 2 2 0 4 Ri:e12led Plane (1122) 2 2 4 o_ (2112) 4 2 2 0 (1212) 2 4 2 a Note: The values listed for the rippled planes assume ideal conditions, i.e., that resolution of all atoms at the edge of the plane is possible. However, in practice the number of spirals seen may be one half the value listed in the table. I-' w IJ1 PAGE 149 APPENDIX III Iridium specimens were prepared by electrochemically polishing a 5 mil wire. The polish used was suggested by Ahlers and Balluffi for polishing platinum specimens for transmission electron microscopy. A thin coating of stop-off lacquer was put at the end of the wire, localizing the attack of the polish to the vicinity of the interface, causing that area to thin preferentially. An initial voltage of about 8 volts A.C. was used~ this was then decreased to about 3-4 volts until the region containing the lacquer finally dropped off, leaving a specimen tip. The time to polish was rather rapid, usually less than a minute. Since it was desired to obtain a high density of dislocations in the specimen, the wire to be used in preparation of specimens was heated resistively under a load to about 1200 C (MP. 2500 C), so that failure occurred in approximately 5 minutes. Tips were then polished as close to the necked down region as possible. 136 PAGE 150 APPENDIX IV Generation of the FCC Lattic e This program will generate points in the perfect FCC lattice in any rectangular parallelepiped in space. A spherical shell, the thickness of which can be varied, is passed through this solid, and the coordinates of the points lying in the volume defined by the inter section of the spherical shell with the rectangular parallelepiped are stored on a "magnetic tape." The figure on the following page shows~schematically, the region over which the coordinates of points are stored. The shell thickness used is usually of the order of "a ,'' 0 this is, of course, much thicker than required to simulate the image; however, this allows some flexibility in varying the radius, thereby eliminating the need of generating a new data tape for every run. 137 PAGE 151 138 z Rectangular --.-, Pa r a! ie lopiped X PAGE 152 139 GENERATION OF FCC LATTICE C DATA GENERATION FOR FIM C FCC LATTICE IS GENERATED DIMENSION A(500),B(5CO),C(500l,D(500),E(500),F(500),FA ( 5CO) CB ( 50 10),X(3000),Y(30CO),Z(30CO) PAUSE REAC INPUT TAPE 5,16,HA,R,EB REWIND 4 REAC INPUT TAPE 5,17,KA,NP,NPA,NQ,NQA,ND 1 7 FORr-'AT(6llO) C NEED ONLY MAKE ND SUCH THAT ND*l.414 IS APPROX. N C N IS THE NU M BER OF ATOMS IN A ROW C KA us~o TO POSITION THE ATOMS IN THE z PLANE C KA GIVES PLANE THROUGH Z=O I.E. KA= 10,11,12, ETC. C EB ALSO USED TO POSITION Z PLANE I.E. EB= 1.5 ,2.5,3.5 R2=R**2.0 HAB=(R**2.0)-(R-HA)**2.0 NC=O NK= C CO 71 I=l,NP A(I)=NPA-I CO 71 J=l,NQ B(J)=~ lQA-J 00 71 K=l,ND CB(K)=FLOATF(KA-K) C(K)=CB(K)*l.41421 GA=A(I)*2+8(J)**2+C(K)*2 C HA IS THE SrELL THICKNESS C R IS THE SPbERE RADIUS FB2=R2-GA IF(F82)71,7,7 7 IF ( IABF B 2) 7 1 10, 10 23 FOR~AT(3E20.8) 16 FOR~AT (3ElC.6) 10 NC=NC+l 4 X(NC)=A(I) Y(NC)=8(J) Z(NC)=C(K) IF(~C-2000)71,8,71 ETC. 8 WRITE TAPE 4,(X(IB),YCIB),Z(IB),IB=l,2000) NC=O NK="K+l 71 CONTINUE IF(~C) 521,522,521 521 WRITE TAPE 4,(X(IB),Y(IB),Z(IB),IB=l,NC) 522 NE=O NKK=C DO 72 I=l,NP PAGE 153 140 C ( I l =FL CAT F ( NP AI ) + 5 DO 72 J=l, N Q E(J)=FLOATF(NQA-J)+.5 DO 72 K=l,NC FA(K)=FLO A TF(K) F(K)=(EB-FA(K))*l.41421 GB=D(I)**2+E(J)*2+F(K)**2 FC2=R2-GB IF( FC2)72, 15, 15 15 IF(rA B -FC2)72,18,18 18 NE=I\E1 l 30 X( N E l=D( I l Y( N E)=E(J) Z(NE)=F(K) IF( I\ E-2000)72,32,72 32 WRITE T A PE 4,(X(IC),Y(IC),Z{IC),IC=l,2000) NE= O !\KK=NKK+l 72 CONTI N UE IF(NE) 523,524,523 523 WRITE TAPE 4,(X(IC),Y(ICJ,Z(IC),IC=l,NE) 524 NT=2COO* N K+2000NKK+NE+NC WRITE OUTPUT TAPE6,106,NK,NKK,NE,NC,~T 106 FOR M AT(6H NK= ,Il0/7H NKK= ,Il0/6H NE= ,Il0/6H NC= ,I10/27H N lT=TCTAL NUMBER UN TAPE= ,110) C NA IS THE NUMBER OF PTS. IMAGED C D 0\-/N TO F IRS T \WT S TATE M E NT A TOM S I N OD D ( E VE N ) LAY E RS OETERfJI N ED C SECCNO HALF IMAGING ATOMS IN EVEN LAYERS (ODD) OETER~I NED END FILE 4 REWIND 4 PRI!\T 242 242 FOR ~ AT(33HlSAVE A4 MARK SANWALD-2 HIT START) PAUSE CALL EXIT END GENERATION OF A PAIR OF DISLOCATIONS 92 FOR~AT{3E20.8) C SCALE IS DIVIDING FACTOR GIVES COORDS. IN INCHES TO BE PLOTTED C ALL MEASURE~ENTS WERE IN ANGSTROMS UNTIL THIS POINT READ INPUT TAPE 5,707,XS,ZS 707 FOR~AT(2El0.6) DO 708 I=l,NC XX(I)=XX(I)+XS 708 ZX(Il=ZX(I)+ 7 S: DO 91 l= l /\ 1 C 7V '. ., .;;_ !.. \ \I) /SCALE ';1 1 ;...x( Il=XX(I)/SCALE WRITE OUTPUT TAPE 6,62,R 62 FOR~AT(l9H R=SPHERE RADIUS= ,E20.8l PAGE 154 141 WRITE OUTPUT TAPE 6,101,AB(l),AB(2),COSA3,COSB1,AB(3), C I SB 3 AB ( 6 ) 1AB(4),AS(5),RA 101 FOR~AT(20H AB(l)=COSF(A{l)}= ,E20.8/20H AB(2}=COSF(A (2))= ,E20.8 l/9H COSA3= ,E20.8/9H COSBl= ,E20.8/20H A8(3}=COSFCB (2))= ,E20.8 2/9H CISB3= ,E20.8/20H AB(6)=COSF(C(l))= ,E20.8/19H AB(4)=COSFCC 31))= ,E20.8/20H AB(5)=COSF(C(3))= ,E20.8/34H RA=SPEC lr-'.EN TO SCRE 4EN CISTANCE= ,E20.8) WRITE OUTPUT TAPE 6,102,SCALE,AMULT,ASUBT 102 FOR~AT(9H SCALE~ ,E20.8/8H AMULT= ,E20.8/8H ASUBT= ,E 20.8) WRITE OUTPUT TAPE 6,710,XS,ZS 710 FORMAJ(25H XS IS XX COORD. SHIFT= ,E20.8/25H ZS IS Z X COORO. SHI lFT= ,E20.8) CALL PLOTS(BUFFER(500),500l CALL PLOT (14.5,14.5,-3) C ABOVE RESETS THE ORIGIN TO THE CENTER OF THE PAPER DO 722 I=l,NC IE(I)=(E(l)-ASUBT)~AMULT XK=BCDIN ( IE CI)) IF(ABSF(ZX( Il)-14.0)721,720,720 721 CALL SY~ol4(XX (1),ZX {I),.10,XK,0.0,3) GO TC 722 720 WRITE OUTPUT TAPE 6,723,ZX{I),I 723 FOR~AT(E20.8,Il0) 722 CONTINUE REAC INPUT TAPE 5,602,NL 602 FORMAT(4Il0) C NL AND NLL ARE NU~BER OF LINES OF BCD TC BE PRINTED(lS C NEIGf-'BORSl CO 829 I=l,NL REAC INPUT TAPE 5,603,XA,YA,XWORD 603 FOR~AT(2El5.6,3A6l T AND 2ND C XA AND YA ARE COORDS OF LINE (LOWER LEFT) 829 CALL SYMBL4(XA,YA,.25,XWORD(3J,0.0,+18) CALL PLOT (-14.5,-14.5,-3) CALL AXISB(0.0,14.59,28.0,0.0,28.0) CALL AXISB(l4.56,0.0,28.0,90.C,28.0l 5 RETURN ENO PAGE 155 142 Neighbor Model This program will image atoms based on their geometrical environment. A schematic diagram of the basic steps involved in the simulation is shown in the figure on page 138. Having selected an image point, the projection and final coordinates of the points in the image are computed, using the equations shown in Appendix I. PAGE 156 143 NEIGHBOR MODEL C IMAGE INTERPRETATION Oil"EI\SION A(2000),B(2000),C(2COO),X(l000l,Y(l000l,Z(l0 00) ,AX(20), l AY ( 20), AZ ( 20), BX ( 20), BY ( 20 l, BZ ( 20 l 2 ,CX(30),CY(30),CZ(30l,DX(30),0Y(30),DZ(30),EX(30),EY( 30),EZ(30), 3FX(3Cl,FY(30),FZ(30),E{lCOOl COM MON X,Y,Z,HA,R,NC,E REWINC 4 PRII\T 321 321 FOR~AT(48HlLOAD DATA TAPE ON A4 MARKED SANWALD-6 HITS TART) PAUSE C I\BC IS TOTAL NUNBER OF RECORDS WRITTEN NC=O REAC INPUT TAPE 5,222,NBC 222 .FORl"AT( I 10) REAC INPUT TAPE 5,16,R 16 FORtJAT(El0.6) REAC INPUT TAPE 5,1,(AX( Il,AY(I),AZ(I),I~l,12) 1 FORl"AT{6El0.6) R2=R**2.0 C CRITERIA OF PTS LYING OUTSIDE OF SHELL NEED BE CCNSIDE RED C FIRST NEAREST NEIGHBORS AX(IJ, ETC. REAC INPUT TAPE 5,2,(BX(IJ,BY(l),BZ(I),1=1,6) REAC INPUT TAPE 5,62,(CX(I),CY(I),CZ(I),I=l,24} REAC INPUT TAPE 5,62,(0X(Il,OY(IJ,DZ(Il,I=l,12) REAC INPUT TAPE 5,62,(EX(Il,EY(Il,EZ(Il,I=l,24) REAC INPUT TAPE 5,62,{FX(I),FY(ll,FZ(Il ,I=l,8) REAC INPUT TAPE 5,2,EMIN,E~AX REAC INPUT TAPE 5,2,El,E2,E3,E4,E5,E6 62 FORl"AT(6El0.6l 2 FORfJAT(6El0.6) CO 71 IAA=l,NBC REAC INPUT TAPE 5,501,NDD 5 0 1 F O R r 1 A T ( 6 I l C ) RE AC TAPE 4 ( A { 1B ) B ( I B l C ( I 8 l I B = 1 ND D l CO 6 I=l,NDO NCl=O ND2=C f'iC3=C NC4-=C NC5=C ND6=C GA=fi ( I ):it-it2+e ( I l+C ( I l **2 IF{R2-GAl6,2C0,200 C T~ESE TWC STATE M ENTS CAUSE CNLY THOSE PTS. WHICH ARE I NSIDE GF R PAGE 157 1,44 C TO BE COMPUTED 200 LC=LC+l co 3 J=l,12 AXX=A( I )+AX(J) AYY=B(1 )+AY(J) AZZ=C(I)+AZ(Jl GA1=AXX**2+AYY**2+AZZ**2 IF(R2-GA1)3,4,4 4 NDl=NDl+l 3 CONTINUE 00 5 K=l,6 El XX= A ( I ) + BX ( K ) BYY=B(I)+BY(K) BZZ=C( I )+BZ(K) GZ2=BXX**2+8YY**2+8ZZ**2 IF(R2-Gl2l5,8,8 8 ND2=N02+1 5 CONTINUE DO 63 L=l,24 CXX=A( I )+CX(L) CYY=BCI)+CY(L) CZZ=C( I )+CZ(L) GZ3=CXX**2+CYY**2+CZZ**2 IF(R2-GZ3)63,152,152 152 ND3=NC3+1 63 CONTINUE CO 13 JA=l,12 CXX=A ( I H:DX ( JA) CYY=B( I )+CY(JAl OZZ=C ( I l+DZ ( JA) GZ4=DXX**2+CYY**2+DZZ**2 IF(R2-GZ4)13,18,18 13 N04=N04+1 13 CONTINUE DO 14 JB=l,24 EXX=A( I )+EX(JB) EYY=B( I )+EY(JBl EZZ=C( I )+EZ(JB) GZ5=EXX**2+EYY*2+EZZ**2 IF(R2-GZ5)14,19,19 19 NC5-=N05+1 14 CONTINUE CO 15 JC=:1,8 FXX=A( I }+FX(JC) FYY=B( I )+FY(JC} FZZ=C(Il+FZ(JC) GZ6=FXX**2+FYY**2+FZZ IF(R2-GZ6)15,20,20 20 NC6=N06+l 15 CONTINUE EA =(FLOATF(NDll )El+(FLOATF(ND2) lE2+(FLOATF(N03) lE 3+(FLCATF(NO 14))*E4+(FLGATF(N05}lE5+(FLOATF(ND6ll*E6 IF(EA -EMIN)6,848,848 PAGE 158 145 848 IF(EA -EMAX)l61,161,6 161 NC=NC+l X(NC)=A(I) Y( N C)=B(ll Z(NC)=C(l) E(NC)=EA 6 CON1I N UE 71 CCN1I N UE REWIND 4 PRI N T 532 532 FOR ~ AT(38HlSAVE DATA TAPE A4 SA NW AL0-6 HIT START) PAUSE WRITE OUTPUT TAPE 6,851,(E(ll,I=l,NC) 851 FOR~AT{lOX El5.6) CALL PROJ CALL EXIT END SUBRCUTINE PROJ C PROJECTIC N C FIRST CALC. CIR. COS. C POSITIO N PTS. IN THE ROTATED SYSTEM Cir-'.ENSIONX(lCOO),Y(lOCO),Z(lCCO),XP(2COO),YP(2COO),ZP( 20CO) ,XX{2OO 1Cl,ZX(2COO),AB(l0l,DUFFER(5G0),~(5} ,B{5),C(5l,~ W ORD(3) ,XXWOR0(3), 2XWORC3( 3),E( 1000), IE{ lCOOl co~~C N X,Y,Z,HA,R,NC,E REAC INPUT TAPE 5,822,LI 822 FCRt-'tiT(IlOl WRITE OUTPUT TAPE 6,824,LI 824 FOR~AT(6r FCC ,I3l WRITE OUTPUT TAPE 6,115,NC 115 FORMAT(30H NC IS NUMBER OF PTS IMAGED= ,110) C R IS THE SPECIMEN RADIUS C RA IS DISTANCE FROM CENTER OF SPECI~EN TO SCREEN READ INPUT TAPE 5,l,RA,A(ll,A(2l,B(2l,C(ll,C(2) 1 FORt'AT(3El5.8l C(3l=l.57C79c3-C(2) AB( l)=COSF(A(l)) A8(2l=COSF(A(2)) A8(3l=COSF(B(2)) A8(4l=CCSFIC(2) l ~8(5l=COSF(C(3l l AB(6)=COSF(C(lJ) 32 COSB1=-SQRTF(l.O-AB(ll*2.Cl IF(~BSF(COSBl)-.01) 771,772,772 111 ccse1=0.o 772 REAC INPUT TAPE 5,904,CISB3,COSA3 904 FOR~AT(2El0.6) 762 CO 86 I=l,NC Z(l)=Z(I)*(RA/R) Y(Il=Y(l)*(RA/Rl X( I l=X( I )(RA/R) C T~E ABGVE STATEMENT EXPANDS THE COORCS. PAGE 159 146 C POI N TS I N T H E RC TAT E O SYSTE M AR E N O W CO M PUTED 25 XP(I)=X(ll A B (l) +Y(I) -11C O S B l+l(l)*A B (6) YP(I)=X(I) ttAB (2)+Y(I)* AG (3)+Z(Il*A B (4) 86 Z P ( I ) = X ( I l COS A 3 + Y ( I l *CI SC 3 + l ( I ) ii/I. B ( 5) CO 5 c 6 1 = 1, N C 584 XX(I)=XP (1)*((4.0* R A)/(3.0 RA+YP (!))) ABC=(({4.0 *R A)/(3. 0 R A+YP(Il))**2.0)*(ZP(I)** 2.0+XP(I) *2 1.0)-(XX( 1)**2.0) IF(~ B SF(A B C)-l.OE+12)572,572,559 559 IF(ZP(Il)5 6 4,564,563 563 Z X (Il=S Q~ TF(A B C) G O T C 56 6 564 ZX( I)=-S QR TF(A B C) G O TC 566 572 ZX( I )= O .O 5 6 6 C ON T H J U E REAC I N PUT TAPE 5,92, SCALE,A r ULT,ASU B T C ~ULT IS FACT O R BY W HIC H ENE R GY IS* DETE RM I N ES NU M 0 F SIG N IFICA N C T FIGU RE S TO BE PLOTTED 92 FOR ~ AT(3E 2 0.8) C SCALE IS DIVIDI N G FACTOR GIV E S C OO R O S. I N 1NCHES TO BE PLOTTE D C ALL ~ EASU R E M E N TS W E R E I N A N GSTRO M S U N TIL THIS POINT REAC INPUT TAPE 5,707,XS,ZS 707 FCR ~ AT(2El0.6) C G 7 08 I= 1, N C XX( I)=XX( !)+XS 708 ZX(I)=ZX(I)+ZS CO Sl I=l, N C ZX( I)=ZX(l)/SCALE 91 XX(I)=XX(I)/SC A LE WRITE O U TPUT TAPE 6,62,R 62 FOR ~ AT(l9 H R=SPHERE RADIUS= ,E20.8) WRITE OUTPGT TAPE 6,101,AB(l),A B (2),COSA3,COSB1,AB(3), CISB3,AB(6), 1AB(4),A 8 (5l,RA 101 FOR~AT{20H AD(l)=COSF(A(l))= ,E20.8/20H A8(2l = COSF(A (2))= ,E20.8 l/9 H CO S A3~ ,E20.8/9H COSBl= ,E20.8/20H AB(3)=COSF(B (2))= ,E20.8 2/9H CISB3= ,E20.8/20H AB(6)=COSF(C(l))= ,E20.8/19H AB(4)=CGSF(C 31))= ,E20.8/20H AB(5)=COSF(C(3})= ,E20.8/34H RA=SPEC lfl.EN TO SCRE 4EN crsTANCE= ,E2C.8) WRITE OUTP U T TAPE 6,102,SCALE,A M ULT,ASUBT 102 FCR ~ AT(SH SCALE= ,E20.8/8H A~ULT= ,E20.8/8H ASUBT= ~E 20.8) WRITE OUTPUT TAPE 6,710,XS,ZS 710 FOR~AT(25H XS IS XX COO R O. SHIFT= ,E20.8/25H ZS IS Z X COORC. S H I lFT= ,E20.8) PAGE 160 147 CALL P~CTS{BUFFER(500),500) CALL PLOT ( 14.5,14.5,-3) C ABOVE RESETS THE ORIGIN TO THE CENTER OF THE PAPER DO 722 1-=:l,N~ IE( I )=(El I )-ASUBT)*AMULT XK=BCOir\( IE( I)) IF(AESF(ZX( I))-14.0)721,720,720 721 CALL SY MB L4(XX (I),ZX (I),.10,XK,0.0,3) GO TO 722 720 WRITE OUTPUT TAPE 6,723,ZX(I),I 723 FORtJAT(E20.8,IlO) 722 CONTINUE REAC INPUT TAPE 5,602,NL 602 FCRtJAT(4l10) C NL ANC NLL ARE NUMBER OF LINES OF BCC TG BE PRINTED(lS C NEIGrBORS) CO 829 I-:;l,NL REAC INPUT TAPE 5,603,XA,YA,X~ORO 603 FUR~AT(2El5.6,3A6) T A N D 2ND C XA A N D YA ARE COORDS OF LINE (LOWER LEFT) 829 CALL SYP:8L4(XA,YA,.25,XWOR0(3),0.0,+18) CALL PLCT {-14.5,-14.5,-3) CALL AXIS~(C.0,14.59,28.0,C.C,28.0) CALL AXIS8!14.56,0.0,28.0,90.0,28.0) 5 RETLR N END PAGE 161 148 Generation of a SingZe DisZocation Points in the perfect FCC lattice are perturbed in such a way as to form a dislocation with any~ and orien tation of the dislocation line. This is accomplished through the use of a rotatable coordinate system, the z axis being the axis along which the screw component of the dislocation lies, and the x axis being the axis along which the edge component of the Burgers vector is generated. After the points in the lattice have been perturbed, image points are selected based on the thin shell criterion. The projection and final coordinates are obtained by the same procedure as shown in Appendix I. PAGE 162 149 GENERATION OF SINGLE DISLOCATION C l~AGE INTERPRETATION DIMENSION A(2000),B(2000},C(2COO),X(l500},Y(l500},Z(l5 00) COM~CN X,Y,Z,HA,R,NC,XDS,YOS,ZDS REWIND 4 PRINT 321 321 FOR~AT(52HllOAO DATA TAPE ON A4 MARKED ON LOG SHEET H IT START) PAUSE C NBC IS TOTAL NUMBER OF RECORDS WRITTEN NC=O REAC INPUT TAPE 5,222,NBC 222 FORMAT( I 10) REA C INPUT TAPE 5,16,HA,R REAC INPUT TAPE 5,20CO,CSEA1,CSEB1,CSEC1,CSEA2,CSEB2,C SEC2,CSEA3, 1CSEB3,CSEC3,ANU,XOS,YDS,ZDS,BEDGE,BSCREW 2000 FORNAT(5El0.6) C XDS,YDS,AND ZDS ARE COORDS. OF DISLOCATION AT SURFACE 2100 FOR~AT(3El0.6} 16 FOR M AT(3El0.6l 501 R2=Ria2.0 HAB=(R**2.0l-(R-HA)**2.0 00 71 IAA=l,NBC REAC INPUT TAPE 5,501,NDD FCRMAT(6l10) REAC TAPE 4,(A(IB),B(IB),C(IBl,IB=l,~00) DO 71 l=l,NCO CALLMIXDIS{A ,CSEA1,CSEB1,CSEC1,CSEA2,CSE B2,CSEC2, 1CSEA3,CSEB3,CSEC3,ANU,I,XDS,YDS,ZDS,XDL,YDL,ZDL,U,V,XD Ll,YDLl,ZDLl ,B ,c 2,BEOGE,BSCREW) 1113 GA=A(I)**2+B(I}**2+C(I)**2 C HA IS THE S~ELL THICKNESS C R IS THE SP~ERE RADIUS FB2=R2-GA IF(F82)71,7,7 7 IF(~AB-FB2)71,10,10 10 NC=t--iC+l X{NC)=A(l) Y(NC)=B(I) Z(NC)=C(I) 71 CONTINUE 72 REWIND 4 WRITE OUTPUT TAPE 6,3110,BEOGE,BSCREW 3110 FOR~AT(9H BEDGE= ,El5.6/l0H BSCREW= ,El5.6) PRINT 532 532 FOR~AT(42HlSAVE DATA TAPE A4 FOR SANWALD HIT START) PAGE 163 PAUSE CALL PROJ CALL EXIT END C SUBRGUTINE MIXDIS 150 SUBROUTINE MIXOIS(Al,B1,Cl,CSEA11,CSEB11,CSEC11,CSEA21 ,CSEB21 1,CSEC21,CSEA31,CSEB31,CSEC31,ANU1,Il,XDS1,YDS1,ZDS1,XD L,YDL,ZDL, 2U,V,XDL1,YDL1,ZCL1,3EDGE,BSCREW) DIMENSION Al(2000l,B1(2000J,Cl(2000) Al ( 11 )=Al{ I l l-XOSl 81( ll)=Bl(Il)-YDSl C l ( I l ) = C 1 ( I l ) -Z OS 1 XOL=Al( ll)*CSEAll+Bl( 11 )*CSEBll+Cl( Il)*CSECll YDL=Al( Ill*CSEA2l+Bl(Il)*CSEB2l+Cl{ll)*CSEC21 ZDL=Al(Il)*CSEA3l+Bl(Il)*CSEB3l+Cl(ll)*CSEC31 ASB=XCL**2+YDL**2 IF(ASB-.05)2008,2009,2009 2009 V=-3EDGE* ( ( ( 1.-2 .*ANUl) *LOGF { XDL**2+YDLu)) + (( XDL**2) /(XOLH,2+ 1YDL*)))/(25.132741*(1.-ANU1)) IF(XOL)2018,2003,2003 2003 U=( ( ATANF(YDL/XDL)+(XDL*YDL)/(2.(l.-ANUl)*(XDL**2+YO L*2)))/ 16.283152lU!:OGE GO TO 2004 2018 IF(YCL)2021,2002,2002 2002 U=( (ATANF(YDL/XOL)+3.1415926+(XDL*YOL)/{2.*(l.-ANU1)* (XDL**2+ 1YCL*2)))/6.283152)*BEDGE GO TO 2004 2021 U=( ( ATANF(YOL/XOL)-3.1415926+(XDL*YDL)/(2.(l.-ANU1) (XDL**2+ 1YCL)))/6.283152)EOGE 2004 IF(YCL)llll,1112,1112 1112 ZDL=ZDL+(ATANF(-XOL/YOLl/6.283152)*BSCREW GO TO 1113 1111 ZDL=ZOL+((3.l415927+ATANF(-XDL/YOL))/6.283152)BSCREW 1113 XOLl=XDL+U YDLl=YOL+V ZOLl=ZOL 2008 Al( Ill=XDLl*CSEAll+YOLl*CSEA2l+ZOLl*CSEA31 Bl(Ill=XDLlCSEBll+YDLl*CSE82l+ZDLl*CSE83l Cl(ll)=XOLl*CSECll+YOLlCSEC2l+ZOLl*CSEC3l Al( Il)=Al( Ill+XDSl BU ll)=Bl( Il)+YOSl Cl( Il)=Cl( Il>+ZOSl IF(ll-100)2005,2006,2005 2006 WRITE OUTPUT TAPE 6,2007 2007 FORMAT(28H SUB MIXDIS HAS BEEN CALLED) 2005 RETLRN END SUBROUTINE PROJ PAGE 164 :15;1, C PROJECTION C FIRST CALC. DIR. COS. C POSITION PTS. IN THE ROTATED SYSTEM OIMENSIONX{ 1500l,Y(l500l,Z(l500),XP(2000),YP(2COO),ZP( 20CO) ,XX(2CO 10),ZX(2000),AB(l0),BUFFER(500),A(5),B(5l,C{5),X W ORD(3) ,XXWOR0(3) COMrGN X,Y,Z,HA,R,NC,XDS,YDS,ZDS WRITE OUTPUT TAPE 6,115,NC 115 FOR~AT(30H NC IS NUMBER OF PTS IMAGED= ,110) C R IS THE SPECIMEN RADIUS C RA IS DISTANCE FROM CENTER OF SPECIMEN TO SCREEN REAC INPUT TAPE 5,1,RA,AB{ll,COSB1,AB{6),AB(2),AB(3),A B(4),COSA3, 1CISB3,A B (5) 1 FOR~ 1 AT(5El0.6) 762 CO 86 I=l,NC Z(I}=Z(I)*{RA/R) Y(I)=Y(I)*(RA/R) X(l)=X(Il (RA/R) C THE ABOVE STATE M ENT EXPANDS T ~ E COOROS. C POINTS IN ThE ROTATED SYS~EM ARE NOW COMPUTED 25 XP(I)=X(I)* AB(ll +Y(I)*COSBl+Z(Il*AB(6) YP( I)=X(Il*AB(2l+Y(I l*AB(3l+Z(Il*AB(4) 86 ZP( I )=X( I )*COSA3+Y( I )*CIS03+Z( I l*AB(5) UO 566 I= 1,NC 584 XX(I)=XP (1)*((4.0*RA)/(3.0*RA+YP (I))) A8C=(((4.0-11-RA)/(3.0*RA+YP(I}))**2.0)*(ZP(I)** 2.C+XP(I)-11-*2 l.O)-(XX(I)**2.0) IF(ABSF(ABC)-7.0E+ll)572,572,559 559 IF(ZP(l))564,564,563 563 ZX(I)=SQRTF(ABC) GO TO 566 564 ZX( I)=-SQRTF(ABC) GO TO 566 572 ZX(Il=O.O 566 CONTINUE 8722 REAC INPUT TAPE 5 1 92, SCALE,816 92 FOR~AT(2E20.8) C B16=+ SUBROUTINE DISMAR CALLED,OIS~AR NOT CALLED C SCAlE IS DIVIDING FACTOR GIVES COOROS. IN INCHES TO BE PLOTTED C All ~EASUREMENTS WERE IN ANGS1ROMS UNTIL THIS POINT READ INPUT TAPE 5,707,XS,ZS 707 FORMAT(2El0.6) CO 708 I=l,NC XX( I )=XX( I )+XS 708 ZX(Il=ZX(ll+ZS 00 91 I=l,NC ZX(I)=ZX(I)/SCALE 91 XX(I)=XX(l)/SCALE 8724 WRITE OUTPUT TAPE 6,62,HA,R PAGE 165 152 62 FORMAT(22H HA=SHELL THICKNESS= ,E2C.8/19H R=SPHERE RADIUS= ,E2 10. 8) WRITE OUTPUT TAPE 6,101,AB(ll,AB(2),COSA3,COSB1,AB13l, CISD3,A8(6), 1AB(4),A[3(5) ,RA 101 FORrAT(20H AB(l}=COSF(A(l))= ,E20.8/20H AB(2}=COSF{A (2))= ,E20.8 l/9H COSA3= ,E20.8/9H COSBl= ,E20.8/20H AB(3)=COSF(B (2))= ,E20.8 2/9H CISB3= ,E20.8/20H AB(6)=COSF{C(l) )=, ,E20.8/19H A8(4)=COSF(C 31))= ,E20.8/20H AB{5)=COSF(C(3) )= ,E20.8/34H RA=SPEC IMEN TO SCRE 4EN CISTA N CE= ,E20.8) WRITE OUTPUT TAPE 6,102,SCALE 102 FOR r AT(9H SCALE= ,E20.8) WRITE CUTPUT TAPE 6,710,XS,ZS 710 FOR M AT(23H XS IS XX COORO. SHIFT,E15.6/23H ZS IS ZX COORO. SHIFT 1,E15.6) CALL PLOTS(BUFFER(5C0),500} IF(El6)1954,1954,1955 1955 1954 C CALL DISMAR(XOS,YOS,ZDS ,R,RA,AB,COSB1,COSA3,CISB3,SC ALE,XS,ZS) CALL PLCT {14.5,14.5,-3) ABOVE R E SETS THE ORIGIN TO THE CENTER OF THE PAPER XK=520000000000 B C C 721 720 723 722 DO 722 I = l,NC IF(ABSF(lX(I))-14.0)721,720,720 CALL SYMBL4(XX (I},ZX {I),.14,XK,O.O,l) GO TO 722 WRITE OUTPUT TAPE 6,723,ZX(l),I FOR~AT(E20.8,IlC) CONTI N UE ABOVE PUTS SMALL SQUARES AT LOCATION OF PTS., NOW AXIS MLST BE SHIFTED ANO DRAWN, TO ACCOLNT FOR LOCATION OF SYMBOLS AT LOWER REAC INPUT TAPE 5,602,NL 602 FORPAT{2110) C NL ANO NLL ARE NUMBER OF LINES OF BCD TO BE PRINTEO(lS C NEIGI--BORS) DO 829 I=l,NL READ INPUT TAPE 5,603,XA,YA,XWORD 603 FORMAT(2El5.6,3A6) T AND 2ND C XA AND YA ARE COORDS OF LINE (LOWER LEFT) 829 CALL SY~BL4(XA,YA,.25,XWOR0(3),0.0,+18) CALL PLOT (-14.5,-14.5,-3) CALL AXISB(C.0,14.59,28.0,0.0,28.0) CALL AXISB( 14.56,0.0,28.C,90.0,28.0l 5 RETURN ENC PAGE 166 153 SUBROUTINE CISMAR(XO,YD,ZD,Rl,RAl ,AB1,COSB11,COSA31, CISB31, lSCALEl,XSl,ZSl) OIMENSION ABl PAGE 167 154 Generation of Two Dislocations This program has the same flexibility as that described for the single dislocation, in that each dislo cation can be made to have any Burgers vector or orien tation of the dislocation line. This is accomplished through the use of two rotatable coordinate systems, the generation being basically the same as that described for the generation of a single dislocation. Here again image points are selected based on the shell criterion, and the final projection is accomplished through the use of the equations in Appendix I. PAGE 168 155 GENERATION OF TWO DISLOCATIO N S C IMAGE INTERPRETATIO N C FCC LATTICE IS GENERATED DIMENSIO N A(2COO),B(2000)iCC2COO),X(l500),Y(l500),Z(l5 00) COM~ON X,Y, Z,HA,R,NC,XDSl,YDSl,ZDSl,XDS2,YOS2~zos2 REr J I N D 4 PRII\T 321 321 FORMAT(52HlLOAD DATA TAPE ON A4 MARKED ON lOG SHEET H IT START) PAUSE C NBC IS TOTAL NU M BER OF RECORDS WRITTEN NC=O REAC INPUT TAPE 5,222,NBC 222 FORl"AT(IlO} 2000 C 2100 16 501 REAC INPUT TAPE 5,16,HA,R READ INPUT TAPE 5,2000,CSEA1,CSEB1,CSEC1,CSEA2;CSEB2,C SEC2,CSEA3, 1CSEE3,CSEC3,ANU,CS2Al,CS2B1,CS2Cl,CS2A2,CS2B2,CS2C2,CS 2A3,CS2B3, 2CS2C3,XOS1,YOS1,ZDS1,XDS2,YDS2,ZDS2,BEDGE1,BSCR W 1,BEDG E2 ,BSCR\. i 2 FOR~ AT( 5El0.6) XOS,YDS,A N D ZDS ARE COORDS. OF DISLOCATION AT SURFACE FORf-/AT(3El0.6) FORl"AT( 3E10.6) R2=R**2.0 HAB=(R**2.0)-(R-HA)**2.0 00 71 IAA=l,NSC READ INPUT TAPE 5,501,NDD FORt'AT(6IlOl READ TAPE 4,(A(IBl,B(lBl,C(IB),IB=l,NDO) DO 71 1=1,NCD CALL MIXOIS(A ,CSEA1,CSEB1,CSEC1,CSEA2,CSE B2,CSEC2, 1CSEA3,CSEB3,CSEC3,ANU,I,XOS1,YDS1,ZDS1,XOS2,YOS2,ZDS2, U,V,XDLl, 2YDL1,ZDL1,CS2Al,CS2B1,CS2Cl,CS2A2,CS2B2,CS2C2,CS2A3,CS 2B3,CS2C3, ,B ,c 3BEDGEl,BSCRWl,BEDGE2,BSCRW2) 1113 GA=ACI>**2+8(1)**2+C(l)**2 C HA IS THE SHELL THICKNESS C R IS THE SPHERE RADIUS FB2=R2-GA IF(FB2)71,7,7 7 IF(rAB-FB2)71,10,10 10 NC=NC+l ,((NC)=A( I) Y(NC)=B(I) Z(NC)=C(I} PAGE 169 71 CONTir--.JUE 72 REWIND 4 156 WRITE OUTPUT TAPE 6,3110,BEDGE1,BSCRW1,BEDGE2,0SCRW2 3110 FORMAT(9H BEOGE1=,El5.6/10H BSCRW2= ,E15.6/9H BEOGE 2=,El5.6/ llOH BSCRW2= ,E15.6) PRINT 532 532 FORMAT{42HlSAVE DATA TAPE A4 FOR SANWALD PAUSE CALL PROJ CALL EXIT ENO HIT START) SUBROUTINE MIXDIS(Al,Bl,Cl,CSEA11,CSEB11,CSEC11,CSEA21 ,CSEB21 1,CSEC21,CSEA31,CSEB31,CSEC31,ANU1,Il,XDS1,YDS1,ZOS1,XD S2,YDS2,ZDS2 2,U,V,XDL1,YDL1,ZDL1,CS2Al,CS2B1,CS2Cl,CS2A2,CS2B2,CS2C 2,CS2A3, 3CS2B3,CS2C3,BEDGE1,BSCRW1,BEDGE2,BSCRW2) DIMENSION Al(2000},81(2000),Cl(2000) Al ( I 1 )=Al{ I 1 )-XOSl Bl( IU::=:Bl( I 1 l-:-YDSl Cl( Il )=Cl( 11)-ZDSl XDL=Al(ll)*CSEAll+Bl(Il)*CSEBll+Cl(ll)*CSECll YOL=Al{ ll)*CSEA21+8l(ll)*CSE82l+Cl(ll)*CSEC21 ZDL=Al(Il)*CSEA3l+Bl{Il)*CSEB3l+Cl(ll)*CSEC31 ASB=XCL**2+YDL**2 IF(ASB-.05)2008,2009,2009 2009 V=-GEDGEl*( ((l.-2.*ANUl)*LOGF(XDL**2+YDL**2) )+((XDL**2 )/{XDL**2+ 1YDL**2)))/(25.132741*(1.-ANU1)) IF(XOL)2018,2003,2003 2003 U=( ( ATANF(YOL/XOL)+(XDL*YDL)/(2.*(l.-ANUl)*(XDL**2+YD l**2)))/ 16.283152)*BEDGE1 GO TO 2004 2018 IF(YCL)202l,2002,2002 2002 U=l (ATANF(YDL/XOL)+3.1415926+(XDLYOL)/(2.*(l.-ANU1)* (XDL+ 1YDL**2)) )/6.283152)*BEDGE1 GO TO 2004 2021 U=(( ATANF(YDL/XDL)-3.1415926+(XDL*YDL)/(2.(l.-ANU1)* (XOL*2+ 1YCL)))/6.283l52)*BEDGE1 2004 IF(YCLlllll,1112,1112 1112 ZOL=ZDL+(ATANF(-XDL/YOL)/6.283152)*BSCRW1 GO TO 1113 1111 ZDL=ZDL+((3.1415927+ATANF(-XDL/YOL))/6.283152)BSCRWl 1113 XDLl=XDL+U YDLl=YDL+V ZDLl=ZOL 2008 Al(Il)=XDLl*CSEAll+YDLl*CSEA2l+ZOLl*CSEA31 Bl(Il)=XDLlCSEBll+YDLl*CSEB2l+ZDLl*CSEB31 Cl(Il)=XDLl*CSECll+YDLl*CSEC2l+ZOLlCSEC31 PAGE 170 157 Al ( I 1 }=:Al( 11 )+XDSl 81( Il)==Sl( I l)+YDSl Cl( Il>=Cl( IU+ZDSl Al( Ill=Al( lll-XDS2 Bl( Il )=Bl( I 1 )-YCS2 Cl( I l)=Cl( I l l-ZDS2 XDL=Al(Il)*CS2Al+Sl(Il}*CS2Bl+Cl(Il}*CS2Cl YDL=Al(ll)*CS2A2+Bl{ll)*CS2B2+Cl(Il)*CS2C2 ZOL=Al( ll)*CS2A3+Bl( Il)*CS2B3+Cl(Il)*CS2C3 ASB=XDL**2+YDL**2 IF(ASB-.05}20081,20091,2C091 20091 V=-BEDGE2*{((1.-2.*ANUll*LOGF(XDL**2+YDL**2})+{(XDL,2 )/(XDL**2+ lYDL**2}))/(25.132741*(1.-ANUl)l IF(XCL)20181,20031,20031 20031 U=( ( ATANF(YDL/XDL)+(XDL*YDL)/(2.*(l.-ANUl)*(XDL**2+YD L**2)))/ 16.283152l*BEDGE2 GO TO 20041 20181 IF(YDL)20211,20021,20021 20021 U=( (ATANF(YDL/XDL)+3.1415926+(XDL*YCL)/(2.*(l.-ANU1)* ( XDL*-i;-2+ 1YDL**2))}/6.283152)*BEDGE2 GO TO 20041 20211 U=( ( ATANF(YOL/XDL)-3.1415926+(XDL*YDL}/(2.*(l.-ANU1)+ (XDL**2+ 1YCL~)))/6.283152)*BEDGE2 20041 IF(YDL)lllll,11121,11121 11121 ZDL=ZOL+(ATANF(-XDL/YDL)/6.283152)*BSCRW2 GO TC 11131 11111 ZOL=ZCL+((3.1415927+ATANF(-XOL/YDL))/6.283152)*BSCRW2 11131 XDL l=XOL+U YDLl=YOL+V ZDLl=ZDL 20081 Al( Il)=XDLl*CS2Al+YDLl*CS2A2+ZDLl*CS2A3 Bl(Il)=XOLl*CS2Bl+YDLl*CS2B2+ZDLlCS2B3 Cl(Ill=XDLl*CS2Cl+YDLl*CS2C2+ZOLl*CS2C3 Al ( I 1 )=Al( I 1 l+XDS2 Bl( I ll=Bl( lll+YCS2 Cl( Ill =Cl( I 1) +ZDS2 IF(Il-100}20051,20061,20051 20061 WRITE OUTPUT TAPE 6,20071 20071 FOR~AT(28H SUB MIXDIS HAS BEEN CALLED) 20051 RETURN END SUBROUTINE PROJ C PROJECTION C FIRST CALC. DIR. COS. C POSITION PTS. IN THE ROTATED SYSTEM DIMENSION X ( 150 0 ) Y ( 150 0 l Z { 1 5 CO) X P ( 2 0 0 0} Y P ( 2 CO C) Z P ( 2000) ,XX(200 l0l,ZX(2000),AB(l0l,BUFFER(500l,A(5},B(5l,C(5},XWORD(3) ,XXWORD(3) COM~CN X,Y~Z,HA,R,NC,XDSl,YDSl,ZDSl,XOS2,YDS2,ZDS2 PAGE 171 158 WRITE OUTPUT TAPE 6,115,NC 115 FOR~AT(30H NC IS NUMBER OF PTS IMAGED= ,110) C R IS THE SPECIMEN RADIUS C RA IS DISTANCE FROM CENTER OF SPECIMEN TO SCREEN REAC INPUT TAPE 5,1,RA,AB(l),COSB1,AB(6),AB{2),AB(3),A 8(4) ,COSA3, 1CISB3,A B (5) l FORMAT(5 E l0.6) 762 DO 86 I=l,NC Z( I )=Z( I )*(RA/Rl Y(I)=Y(I)*(RA/R) X( I )=X( I ) (RA/R) C THE AGOVE STATEME N T EXPANDS THE COOROS. C POI N TS IN T H E ROTATED SYSTEM ARE NOW COMPUTED C C C 25 XP(I)=X(I)* AB(l) +Y(I)*COSBl+Z(I)*AB(6) YP( I)=X(I)*AB(2)+Y(I}*AB(3)+Z(l)*AB(4) 86 ZP(I)-=X(Il*COSA3+Y(I)*CISB3+Z(l)*AB(5) DO 566 l=l,NC 584 XX(I)=XP (1)*((4.0*RA)/(3.0*RA+YP (I))) ABC=(((4.0*RA)/(3.0RA+YP(l)))**2.0)*(ZP(I)** 2.0+XP(I)**2 559 563 564 572 566 8722 92 707 708 91 8 7 2 1 + 62 l.O)-{XX{I)**2.0) IF(ABSF(A B C)-l.OE+06)572,572,559 IF(ZP{I))564,564,563 ZX( I)=S Q RTF(ABC) GO TO 566 ZX( I l=-SQRTF(ABCl GO TO 566 ZX( I)=O.O CONTINUc REAC INPUT TAPE 5,92, SCALE,816 FORMAT(2E20.8l 816=+ SUBROUTINE OISMAR CALLED,DIS~AR NOT CALLED SCALE IS DIVIDI N G FACTOR GIVES COORDS. IN INCHES TO BE PLOTTED ALL MEASURE~ENTS WERE IN ANGSTROMS UNTIL THIS POINT REAC INPUT TAPE 5,707,XS,ZS FORr-'AT(2El0.6) CO 708 I=d, NC XX( I )=XX( I )+XS ZX(Il=ZX(I}+ZS DO 'H I=l,NC ZX( I)=ZX( !)/SCALE XX(l)=XX(l)/SCALE ~RITE OUTPUT TAPE 6,62,HA,R FOR~AT(22H HA=SHELL THICKNESS= ,E20.8/19H R=SPHERE RADIUS= ,E2 10.8) WRIT~ OUTPUT TAPE 6,101,AB(ll,AB(2l,COSA3,COSB1,AB{3), CISB3,AB(6), 1Aff(4),AB(5) ,RA .101 FORMAT(20H AB(ll=COSF(A(l)l= ,E20.8/20H AB(2)=COSF(A (2))= ,E20.8 PAGE 172 159 l/9H COSA3= ,E20.8/9H COSBl= ,E20.8/20H AB(3)=COSFlB {2))= ,E20.8 2/9H CISB3= ,E20.8/20H AB{6)=COSF(C(l))= ,E20.8/19H AB(4)=COSF(C 31) )= E20.8/20H AB( 5l=COSF (C(3 ) )= ,E20.8/34H RA=SPEC IMEN TO SCRE 4EN CISTANCE= ,EZ0.8) WRITE OUTPUT TAPE 6,102,SCALE 102 FORMAT(9H SCALE= ,E20.8) WRITE OUTPUT TAPE 6,710,XS,ZS 710 FOR~AT(23H XS IS XX COORD. SHIFT,El5.6/23H ZS IS ZX COORD. SHIFT 1,E15.6) CALL PLOTS(BUFFER(500),500) IF(Bl6)1954,1954,1955 1955 CALL OISMAR(XDS1,YDS1,ZDS1,XDS2,YOS2,ZDS2,R,RA~AB,COSB l,COSA 3, 1CISB3,SCALE,XS,ZS) 1954 CALL PLOT (14.5,14.5,-3) C ABOVE RESETS THE ORIGIN TO THE CENTER OF THE PAPER B XK=520000COOCOO CO 722 I=l,NC IF(ABSF(ZX( I) )-14.0)721,720,720 721 CALL SYMBL4(XX (I),ZX (I),.14,XK,0.0,1) GO TO 722 720 WRITE OUTPUT TAPE 6,723,ZX(Il,I 723 FOR M AT(E20.8,110) 722 CONTI N UE C ABOVE PUTS SMALL SQUARES AT LOCATION OF PTS., NOW AXIS MUST BE C SHIFTED AND DRAWN, TO ACCOUNT FOR LOCATION OF SYMBOLS AT LOriER READ INPUT TAPE 5,602,NL 602 FOR.MAT(2110) C NL ANO NLL ARE NUMBER OF LINES OF BCD TC BE PRINTED(lS C NEIGHBORS) 00 829 I=:l,NL REAC INPUT TAPE 5,603,XA,YA,XWORD 603 FORMAT(2El5.6,3A6) T AND 2ND C XA ANO YA ARE COORDS OF LINE (LOWER LEFT) 829 CALL SY~BL4(XA~YA,.25,XHOR0(3),0.0,+18) CALL PLOT (-14.5,-14.5,-3) CALL AXISB(C.0,14.59,28.0,0.0,28.0) CALL AXISB(l4.56,0.0,2B.0,90.0,28.0) 5 RETURN END SUBROUTINE DISMAR(XCS1,YOS1,ZOS1,XDS2,YDS2,ZDS2,Rl,RA1 ,ABl,COSBll, 1COSA31,CISB31,SCALE1,XS1,ZS1) DIMENSION ABl(lO) Xl=XCSl(RAl/Rl) Yl=YCSl(RAl/Rl) Zl=ZCSl*(RAl/Rl) PAGE 173 160 X2=Xl*A B l(ll+Yl*COS B ll+ZlA B 1(6l Y2=Xl*A B 1(2)+YlA B 1(3)+Zl A B 1(4) Z2=Xl*COSA3l+Yl CISB3l+Zl A B 1(5) X3=X2*({4. R Al)/(3.*RAl+Y2) l ABCl=( ((4.* R ~l)/(3. R A l+Y2l)**2.) {Z2*2.+X2**2.)-{X3* .. 2.) IF(A B SF(A B C1)-l.E+06)5721,5721,5591 5591 IF(Z2)5641 ~ 5 6 41,5631 5631 Z3=S QR TF(A B C1) GO TO 56 6 1 5641 Z3 = -S QR TF(A B C1) GO TO 5 66 1 5721 Z3= 0 .0 5661 Z3=(Z3+ZS1)/SCALE1 X3={X 3 +XS1)/SCALE1 CALL PLOT(l4. 5 ,14.5,-3l B XR=54 0 00000 C OOO CALL SY M BL4(X3,Z3,.25,XR,O.O,l) CALL PL O T(-14.5,-14.5,-3) Xl=XCS2( R A1/ R l) Yl=YDS2 ( R A1/ R l) Zl=Z C S2( R A1/ R l) X2=Xl*A D l(l)+YlCOS B ll+Zl*A B 1(6) Y2=Xl*A B 1(2)+YlA B 1(3l+Zl A B 1(4) Z2=XlCOSA3l+YlCIS B 3l+Zl*AB1(5) X3~X2((4.RAl)/(3.RAl+Y2)) ABC!. -= ( ( (4.RAl)/(3.RAl+Y2) )**2. l{Z2* 2.+X2*. )-(X3* +2.) IF(A B SF(ABC1)-l.E+06)57211,57211,559ll 55911 IF(Z2)56411,5 6 411,563ll 56311 Z3=S CR TF{ABC1) GO TC 56611 56411 Z3~-S Q RTF(A B Cll GO TO 56611 57211 Z3=C.O 56611 Z3=(Z3+ZS1)/SCALE1 X3=(X3+XS1)/SCALE1 CALL PLOT(l4.5,14.5,-3) CALL SY MS L4(X3,Z3,.25,XR,0.0,1) CALL PLOT(-14.5,-14.5,-3) RETURN ENO PAGE 174 BIBLIOGRAPHY 1. E.W. Muller, Z. F. Physik 131, 136 (1951). 2. A. J. W. Moore, Int. J. Phys. Chem. Solids 23, 907 (1962). 3. E.W. Muller, Adv. in Electronics and Electron Physics 13, 83 (1960). 4. F. Kirchner, Naturwissenshaften 41, 136 (1954). 5. E.W. Muller, 5th Field Emission Symposium Chicago, (1958) 6. E.W. Muller, Phys. Rev. 102, 618 (1956). 7. T. T. Tsong and E.W. Muller, J. Chem. Phys. 41, 3279 (1964) 8. E. W. Muller, J. Appl. Phys. 28, 1 (1957). 9. D. G. Brandon, Battelle Memorial Jnstitute Geneva, (1965). 10. S. Ranganathan and A. J. W. Moore, private communi cation. 11. D. G. Brandon, J. Sci. Inst. 41, 373 (1964). 12. R. W. Newman, R. C. Sanwald and J. J. Hren, to be published, J. Sci. Inst. 13. M. Drechsler and P. Wolf, Fourth International Confer ence on Electron Microscopy Berlin, (1958). 14. E.W. Muller, Z. F. Physik 156, 399 (1959). 15. E.W. Muller, Imperfection in Crystals, A. I. M. E. Symposium (edited by J.B. Newkirk and J. H. Wernick) Interscience, New York (1962). 16. J. J. Hren, Acta Met. 13, 479 (1965). 17. S. Ranganathan, K. M. Bowkett, J. J. Hren and B. Ralph, Phil. Mag. 12, 841 (1965). 161 PAGE 175 162 BlBLlOGRAPHY--Continued 18. M. Drechsler, E. Pankow and R. Vanselow, Z. F. Phys. Chem. 4, 17 (1955). 19. E. W. Muller, J. Appl. Phys. 30, 1843 (1959). 20. D. G. Brandon and M. Wald, Phil. Mag. 6, 1035 (1961). 21. E.W. Muller, J. Phys. Soc. Japan 18, 1 (1963). 22. D. W. Pashley, Rep. Prog. Phys. 28, 291 (1965). 23. S. Ranganathan, J. Appl . Phys. 31, 4346 (1960). 24. S. Ranganathan and A. J. Melmed, Phil. Mag. 14, 1309 (1966). 25. H.F. Ryan and J. Suiter, J. Less-Common Metals 9, 258 (1965). 26. C.H. Sisam, Analytic Geometry Henry Holt and Co., New York (1936). 27. M. Drechsler and H. Liepack, Colloques Internationaux Du Centre National De La Rechesche Scientifique No 152 (1965). 28. E.W. Muller, Surface Science 2, 485 (1964). 29. A. J. W. Moore and J. F. Nicholas, J. Phys. Chem. Solids 20, 222 (1961). 30. L.A. Girfalco and V. G. Weizer, Phys. Rev. 114, 687 (1959). 31. W. T. Read, Dislocations in Crystals McGraw Hill, New York (1953). 32. Technical Documentary Report No. WADD TR6l-?2 Wright Patterson Air Force Base (1962). 33. F. C. Frank, Phil. Mag. 42, 809 (1951). 34. A.H. Cottrell, Dislocations and Plastic Flow in Crystals Oxford University Press, London (1953). PAGE 176 163 BIBLIOGRAPHY--Continued 35. J. S. Hirschorn, J. Less-Common Metals 5, 493 (1963). 36. E.W. Muller, Third European Conf erence on Electron Microscopy Prague (1963). 37. A. J. Melmed, Surface Science 5, 359 (1966). PAGE 177 BIOGRAPHICAL SKETCH Roger Carl Sanwald was born March 1, 1941, at Newark, New Jersey. In June, 1959, he was graduated from Chatham High School. In June, 1963, he received the degree of Bachelor of Engineering from Stevens Institute of Technology in Hoboken, New Jersey. In September, 1963, he enrolled in the Graduate School of the University of Florida. He had a College of Engineering Fellowship the first year, while enrolled in the Department of Metallurgy and Materials Engineering. From September, 1964, he has held a NASA traineeship while he has pursued his work toward the degree of Doctor of Philosophy. Roger Carl Sanwald is married to the former Judith Amanda Craig. He is a member of the American Society for Metals, the American Institute of Metallurgical and Petroleum Engineers, the Society of the Sigma Xi, and Alpha Sigma Mu. 164 PAGE 178 This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Engineering and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 12, 1967 Dean, Graduate School Supervisory Committee: Chai~ J. /1 ~ ~-cY ~ (f:or C. V. Smith) |