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By John H. Hollomon

In a recent paper,' a very early suggestion by Ludwik2 concerning the nature of the mechanical behavior of metals has becn reexamined and extended. In essence it was suggested that there exists, at least under certain conditions, a mechanical equation of state; that the stress required for flow depends upon the instantaneous values of the strain, strain rate, and temperature, and not upon their past values. It is proposed that under these circumstances thc stress does not depend upon past history of these variables, but is like the temperature of a gas that depends only upon the instantaneous values of the pressure and volume. This notion has not been applied to present-day studies of metals, though it is especially applicable to the study of creep. It is the purpose of this paper to analyze the significance and possible range of applicability of this idea and to determine how it influences the accepted ideas concerning creep. Stress-strain Relations It is wcll known that the stress-strain curves of metals are influenced by both train rate and temperature. For a fixed strain rate and temperature there is associated with each metal a stress-strain curve. If either the strain rate or temperature is changed a new stress-strain curve will be characteristic (Fig. I). A much more genera1 relation between the effects of these variables is possible, however. Consider the stress-strain curves of Fig. I determined at

Jan 1, 1947

His face wreathed in smiles, J. Richard Lucas, Chairman of the Keynote Session, conceded that "the papers appeared to have been of some interest." This was a modest description of the opening day of technical sessions at the SME Fall Meeting and Exhibition, October 13 to 15, in Tampa, Fla. The auditorium had capacity for 800 spectators and there was hardly a vacant seat visible. The topic, Resource Development of the Fertilizer Minerals, was covered in four authoritative papers. H. S. Ten Eyck set the stage. Resources are always viewed in relation to future requirements, and these depend largely on population. If the population trend continues, the world's 3.3 billion people will increase to 6.7 billion people by the year 2000. About 70% of this increased population will be in developing areas which are not capable of supplying sufficient food and fiber for their needs. Fertilizer is a means of bridging the crop shortage. Projections of fertilizer requirements by 1970 are from 53 million to 67.5 million long tons of plant nutrient which compares with 1964 consumption of 37.7 million tons. Looking ahead to 1980, the growth continues with an estimated demand of 80 million to 113 million tons combined of N P and K.

Jan 12, 1966

By Oswald J. Heinrich

During the last twenty years much has been done to investigate and define the Mesozoic formatibn of the United States along the Atlantic States, as well as in' the Territories. The investigations of Professors Hitchcock, Emmons, Hayden, H. D. Rogers, and those now in progress in connection with the geologicd surveys of Penn-

Jan 1, 1879

By P. D. P. Hamilton

In any mining district where benefication of the ores is practiced, one can usually observe several types of flow sheets even though the various producers are milling ore from the same geological formations. It is obviously impossible to describe all the mills in the Porcupine district. However, the following description of some of them is of particular interest in that it shows where one mill has paid particular attention to corduroy blanket practice; one to gravity concentration of sulfides for intensive cyanidation of the values therein contained; one to flotation of values and cyanidation of this product only; and others where small variations of standard all-slime cyanidation practice prevail. In all cases it will be seen that cyanidation is the basic principle. Discussions involving the difference in the flow sheets and the reasons for them will always be of value to those interested in the art. The author of this article, however, believes that such discussions should originate from the operators of the mills under discussion. For this reason no comparisons or discussion will be included in the description of the mills mentioned. Should any reader of this article obtain information therefrom which will be of value to him in helping to solve his own particular problems, the author will consider that the article has satisfactorily served its purpose. All values mentioned throughout this article are based on the value of gold at $20.67 per ounce. Coniaurum Mines, Limited The ore of the Coniaurium mines consists of a mineralized schist and quartz containing about 2.5 per cent sulfides. Owing to the low grade of ore developed in the mine, extreme care has had to be exercised throughout the entire mining and milling operation in order that an operating profit might be obtained. The management, however, has been able to keep costs down, with the result that at present the outlook for the property is considerably brighter than it has been in the past. This is partly due to the fact that it is now known how to mine and treat

Jan 1, 1935

By W. S. Landis

In brief, this is the story of an attempt to Americanize a process originally developed in Europe. The story will be recited in two sections, the first dealing with the process as developed by the European engineers; the second containing the improvements and modifications of American origin, together with some details of the operating results of the unit as finally assembled. Developments in other directions in this same field have not, up to the present, pointed out an economic future for the process, therefore our record here is largely a report of development progress. Whether some particularly favorable location will be found at some future time for a commercial installation cannot as yet be foretold. The project is now at a standstill. The peculiar tools of the electrometallurgist should be of great interest to the zinc smelter. The low thermal efficiency of the zinc retort, its comparatively small-scale unit and its intermittent cycle as it stood in 1925, should make this art a particularly fertile field for the electric furnace. Yet in spite of years of work recorded in volumes of literature, and an expenditure of large sums of money, there is not a single electrothermic zinc smelter operating anywhere in the world. In this industry the electric furnace plays only a secondary part today. Historical The first serious proposal to use the electric furnace for the smelting of zinc ores was made by the Cowles Brothers in 1885, but their experimental work did not advance the art to any extent. The next investigator of this field was Dr. Gustav DeLaval, a Swedish engineer well known to the industrial world through his development of the centrifugal and the steam turbine. The DeLaval cream separator is a familiar household appliance in most parts of the world. Although Dr. DeLaval commenced his experiments in the field of electric zinc smelting in 1893, it was not until 1898 that he had a practicable furnace, even then operating but intermittently. His work was done at Trollhättan, Sweden. He continued to improve this early fur-

Jan 1, 1937

By F. C. Schora, H. P. Meissner

Iron ore from Cerro Bolivar, Segre', and Sierre Grande was reduced in fluid beds at about SOOT, using gas analyzing 20.5 pct CO, 41 pct Hz, and 38.5 pct N2. Except in the early stages of reduction, the rate of oxygen loss from the bed was directly proportional to the oxygen concentration in the bed, independent of gas composition as long as the gas was reducing, and little affected by temperature. The percentage reduction of the bed solids at any time was independent of particle size. I HE reduction of spheres or cubes of dense iron ore in a slowly moving stream of either pure H, or pure CO has been found to take place at distinct interfaces between well defined layers of iron and the iron oxides.'y4 These interfaces generally penetrate the ore lump in topochemical fashion, remaining parallel to the original contours of the specimen. Thus in the earlier stages of reducing hematite, the specimen's core is Fe,O, surrounded by successive shells of the lower oxides, and finally by an exterior shell of iron. McKewan5 assumed that the rate limiting step occurred at the Fe-FeO interface, with reducing gas penetrating up to this interface through the external porous iron shell, and reduction of the higher oxides within the ore lump occurring by diffusion of iron through the dense FeO shell. He developed an analytical expression based on this model relating the amount of oxygen removed from

Jan 1, 1962

By John V. W. Reynders

Our entry into the World War suddenly brought home to us in a startling way the vital importance of manganese. Since the war, much has been written and said upon the subject of manganese and a great deal of time and money have been spent in attempts to find an adequate substitute for this metal in the manufacture of steel; none has been found. It is true that ways are known by which it is possible to make small amounts of good steel without manganese, but taking into consideration the diminishing quality of iron ores and fuels as well as the increasing exaction of specifications for finished products, it may safely be assumed that, high-grade manganese ore will be more and more a necessity in the production of steel as time goes on. Taking world resources as a whole it may be well to point out that the problem of obtaining needed supplies of manganese from a strictly commercial viewpoint is now less serious than it was before the war. Formerly it was feared that world resources of high-grade manganese ore might be exhausted in a relatively short time, but, largely as a result of the stress laid upon the importance of manganese during the war, the world's available supply of high-grade manganese ore has been greatly augmented by new discoveries and fuller development of previously known deposits. Sources of Production Russia alone, of countries producing the major part of the world's stee1, has within its geographical boundaries sufficient manganese to supply its needs. The domestic production of England is inadequate and English requirements for high-grade manganese are supplied principally from India. France also draws largely upon India, as it depends upon foreign sources of supply except with respect to a small tonnage of silico-manga-nese and silico-spiegel produced in the Hautes-Pyrénées. Germany and the United States likewise depend upon foreign sources of high-grade ore, though each possesses comparatively large reserves of ferruginous manganese ore and manganiferous iron ore. At present the world demand for high-grade manganese ore is met with ease by the production of Russia, India, Gold Coast and Brazil.

Jan 1, 1927

By L. A. Wright, H. L. McKinley

PINE Creek mine of Union Carbide Nuclear Co. is some 23 miles northwest of Bishop, Calif., in the Sierra Nevada Mts. Office and mill are 7800 ft above sea level, the 1500 level portal is at 9300 ft, the A level portal at 10,800 ft, and the 3050 portal at 11,500 ft. Original claims were located in 1916, and the mine worked intermittently until 1936, when it was purchased by Union Carbide Corp. Since then the property has been operated continuously, except for two brief periods when operations were suspended because of manpower shortages. Mine production through 1956 amounted to 2,792,425 dry tons of ore.

Jan 10, 1957

By F. R. Hensel

Up to the present time few attempts have been made to produce hard nonmagnetic materials by heat treatment of austenitic steels. The usual result has been to cause them to pass into the martensitic stage, thus destroying to a great extent the typical austenitic properties, since this involves a transition of a portion of the iron from a face-centered to a body-centered type of atomic arrangement. High-strength nonmagnetic materials such as might be provided by properly hardening austenite are required in the electrical industry for different machine parts. In Europe nonmagnetic retainer rings for restraining the end-turns of the rotor windings in turbo-alternators have been developed as a special application of high-strength austenite. In testing1 a 28,000-kva. three-phase Siemens turbogenerator running at 3000 r.p.m. at full load current, the losses in the short-circuit test were 450 kw. for the magnetic ring and 286 kw. for the nonmagnetic, and the difference should be attributable to the magnetic ring. The differences in temperatures during the same test are given in Table 1 and these show the advantage of a nonmagnetic over a magnetic retainer ring. Hardening in this case was obtained by cold working, such as hammering, rolling or some means for stretching the steel while at a temperature near 500" C. One means for stretching2 is to rotate the forged or rolled ring in a centrifugal machine so as to develop within it stress higher than

Jan 1, 1931

By Will H. Coghill

FLOTATION has been so closely allied with the sulfide minerals and their early and associated oily reagents that the term "oil flotation" has erroneously been applied to the entire flotation process. Today, the term "chemical flotation" might be used to advantage. Flotation literature has dealt primarily with the removal of sulfides. from non-sulfide gangues, and the separation of one sulfide from another.. Now, the application of the process to the separation of nonsulfide minerals is coming to the fore; and from the standpoint of the number of commercial minerals involved, the flotation of nonsulfides offers the larger field. Here, soap or soap-forming substances, such as the fatty acids and their derivatives, are used; hence there may well be a subdivision of the chemical flotation process called "soap flotation." Early experimenters learned that it was easy to form a froth or foam that would carry a part or all of the ore charge over the lip of the flotation machine; but this could not be regarded as flotation because of the lack of selectivity. Doubtless soap was one of the first reagents tried, and doubtless it was rejected on account of its gangue-carrying properties. Now this same gangue-carrying ability makes soap an acceptable reagent, for we are dealing with that, which among sulfides, is considered "gangue." As in the flotation of sulfides, so in the flotation of nonsulfides, the selective flocculation of the mineral and the dispersion of the gangue are the first objectives. In fact, these two phenomena epitomize flotation. They depend upon the unlike surface properties of the respective minerals; and, as long as the components of an ore have unlike surface properties, a separation is possible, at least in theory. Often in nonsulfide flotation it makes little difference which mineral floats and which is depressed; generally only a separation is sought.

Jan 1, 1932

By Carle Hayward

THIS work was first undertaken in the metallurgical laboratory of the Massachusetts Institute of Technology in 1907 by L. A. Dickinson, E. Phelps, and V. S. Rood, under the author's direction. They attempted to determine the diagram for the system Cu2S-NiS, but several inexplicable results were obtained. The following year, P. H. Mayer with Mr. Dickinson carried, on some experiments with nickel sulphide and found that the molten material slowly lost sulphur until it reached a composition corresponding to the compound Ni3S2, established by Bornemann.1 At this point the composition remained stable. The freezing points of the mixtures gradually fell off with the loss of sulphur. Since the above facts seemed to explain the erratic results obtained in the first experiments, Mr. Mayer with F. Jaeger attempted in 1910 to establish a correct equilibrium diagram for the Cu2S-Ni3S2 series. They prepared pure Cu2S and Ni3S2 and determined the cooling curves for various mixtures. The fusions were made in a platinum resistance furnace and the temperatures were measured with a platinum-iridium thermocouple connected with a Sullivan galvanometer. The curve was established for mixtures containing more than 40 per cent. Cu2S, but they were unable to determine certainly the direction of the curve beyond that point. Nothing further was attempted with this series until the present year, when the work was carried to a successful conclusion with the assistance of A. Butts, Jr., and E. H. Weil. The results of this work are given in the following pages.

Jan 1, 1914

Jan 1, 1970

By A. L. Holley

THE application of scientific methods to the investigation of natural laws and to the conduct of the useful arts which are founded upon them, is year by year mitigating the asperity and enlarging the outcome of human endeavor. More notably, perhaps, are these the facts in that system of productive and constructive arts of, which engineering is the general name. In metallurgical engineering especially, within the period of our own recollection, how rapid has been the rate and how wide the scope of progress: the scientific discovery and mining of metalliferous veins; the economical separation and reduction of ores of every grade; the production and regulation of high temperatures; the varied improvements in the manufacture of iron, in saved heat and work, in uniformity and range of products ; and, most important of all, the creation and the utilization, to be counted by the million tons a year, of the cheap constructive steels. Wonderful as this range and degree of development may appear to the public eye, the close and thoughtful observer must, nevertheless, conclude that neither the profession nor the craft of engineering may congratulate themselves too complaisantly, but that they should rather acknowledge to each other the embarrassing incompleteness of the union between engineering science and art. There is a small school of philosophers whom we may designate as original investigators, men who come close to nature, who search into first principles, and who follow that scientific and therefore fruitful method by which the relations of matter and force are discovered, classified, and brought within the reach of practice. These wonderful - men do not indeed create the laws of nature, as they sometimes almost seem to, but they go up into the trembling mountain and the thick darkness and bring down the tables upon which they are written. * President's Opening Address. The discussion following this paper, and also the Joint Discussion of the Institute and the American Society of Civil Engineers, on the subject of Technical Education., are published in a separate volume.

Jan 1, 1876

By A. S. Eve

In the spring of 1930, the question was raised as to the possibility of estimating the depth to which the pyrrhotite-nickel deposit at the Falconbridge mine extended in the earth. This body is 7500 ft. long, it dips at 80° to the north, and it is fairly straight with slight sinuosities. Dr. F. W. Lee suggested that it might be possible to solve the problem by measuring the vertical magnetic intensity, with a good magnetometer, both near the surface of the earth and also at a level a few feet higher in the air. The writer regarded this suggestion with some scepticism, but it was decided to put the scheme to a practical test. In July, 1930, Dr. Lee, Dr. Keys and the writer, with Mr. Joyce and Mr. Belloc as assistants, made a firm, light wooden platform (Fig. 1) measuring 4 by 4 ft. on the top, which was 10 ft. above ground. This platform could be moved easily a few yards by four men, and carried across country in a wagon or by motor truck. Some iron nails were used in its construction, but we were not able to detect their influence on the Askania magnetometer. Nevertheless, in future work, the use of iron should be completely avoided. Readings were taken along a traverse at right angles to the orebody, at several stations, both on the ground and on the platform. The excess readings, over the dike, as compared with those about 800 ft. away, are shown in Table 1. The largest reading, 6.8, is open to suspicion and should be rejected. The depth of the top of the orebody is known by electrical survey, and by actual drilling, to be about 112 to 123 ft., and the covering is sandy glacial drift. Similar readings were taken over a long straight magnetic diabase dike. These also are shown in Table 1. The magnetic vertical intensity is consistently less at the higher level, which is 10 ft. above the lower level.

Jan 1, 1932

By J. Allen Veatch

THERE exists in., the territory embraced between the summit of the Sierra Nevada and the coast a great dike and vein system that appears never to have been recognized in its entirety, nad many facts concerning it have never been published. This is important, not only in affording numerous clues bearing on the origin, nature, and age of the mercury deposits, and eliminating many controverted questions, but it. incidentally affords numerous facts bearing on the genesis of gold, copper, chromite, magnesite, dolomite, a type of calcite deposit, and numerous associated minerals. The mineral springs of the State almost without exception emanate from these dikes. A great range of the hydrocarbons also exists throughout the entire system of these dikes. Seepages from their, walls have frequently been mistaken as emanating from the inclosing sandstone and shale, resulting in heavy expenditures in the futile development of these Cretaceous beds, which contain no petroleum within the territory. The theory of mountain making by lateral or tangential pressure is inconsistent with some of the following recorded facts. The limitations of this paper do not admit of a description of each of the hundred or more mercury deposits embraced in the area, and therefore only typical examples will be described. The accompanying map, Fig. 1, is not submitted as complete, or as entirely accurate, but it is as nearly so as the notes at hand will admit. The field is too extensive to be completely covered by a single individual. The heavy continuous lines represent sections actually surveyed! The sections and extensions- in. dotted line are assumed, and are either unsurveyed, or -buried under younger beds. There are numerous branches connecting the great parallels that the small-scale map does not admit of plotting, even if the survey had been complete. Likewise, the conclusions in some instances may be faulty, but it is to be hoped that as a whole they may serve as a foundation on which a more complete structure may be built in the future.

Jan 2, 1914

By E. C. Harder

IN view of the gradually decreasing known reserves of high-grade manganese ore and the rapidly increasing consumption of iron-manganese alloys in the steel industry, it is well to turn our attention to the development of low-grade manganiferous ore deposits and to the utilization of these ores in steel manufacture. The dependence of this country on imported manganese ore and ferromanganese has been forcibly brought to the attention of steel makers during the last few years by the cessation of manganese-ore imports from Russia, the restriction of manganese-ore imports from India and of ferromanganese imports from Europe. Should the importation of manganese ore from Brazil cease, the steel industry of this country would face a serious situation. The present practice of steel making requires ferromanganese in large quantities and ferro-manganese of the grade used at present can be produced only from high-grade manganese ores such as the United States appears to possess only in insignificant amounts. There are in the United States, however, large reserves of manganiferous iron ore which could probably be used extensively in the steel industry if certain changes in the practice of steel manufacture were made. Thus high-grade manganese ore could be conserved. Among the most important deposits of manganiferous iron ore in this country are those of the Cuyuna iron-ore district of Minnesota. It is probable that in case of restriction of foreign supplies, the users of manganese in the United States would have to rely to a large extent on the Cuyuna district to make up the shortage.

Jan 9, 1917

By Lawrence Wright

THE occurrence of gold, gold-silver, silver-lead-zinc ores in the post-Cambrian sediments in the Black Hills of South Dakota, and their genetic relationship to the Tertiary intrusives, is well known and accepted, but the occurrence of gold in the pre-Cambrian rocks at the Homestake mine, which underlie this Tertiary gold mineralization, has given rise to differ-ences of opinion as to the period of major gold introduction into the older rocks. Fig. 1 illustrates. the distribution of mineralization along the axis of the Tertiary uplift in both pre-Cambrian and post-Cambrian rocks. The following possibilities have been presented: 1. The Homestake gold deposit was formed in pre-Cambrian time and the deposits in the overlying Cambrian sediments formed during Ter-tiary time. 2. The Homestake lode existed at the beginning of Cambrian time, but was enriched by the passage of Tertiary auriferous solutions. 3. The Homestake lode was represented by pre-Cambrian quartz-chlorite sulfide mineralization, very low in gold content and the gold superimposed on the older nearly barren mineralization during the Ter-tiary metallogenic epoch. 4. The sulfides and sulfarsenides as well as gold were deposited in both pre-Cambrian and post-Cambrian rocks during the Tertiary period. Those favoring the first proposition feel that the question is pretty well settled, while the proponents of the second theory are more flexible in their view, feeling that the amount of gold introduced in either period is still an open question. The third and fourth theories, with some modifi-cations, are supported by the proponents of the Tertiary theory, who feel that the evidence strongly favors that period. The following pages are intended to present the evidence and to give the reaction of various observers with my own comments and conclu-sions; also, the accumulated facts of observation gathered over a period of close association with the area under discussion, beginning in 1919.

Jan 1, 1936

By R. B. Mears, E. H. Dix, R. H. Brown

In the Institute of Metals Division Lecture of 1940, E. H. Dix, J~.(I)~ discussed the acceleration of the rate of corrosion by high constant stresses. Experimental data obtained on this subject at the Aluminum Research Laboratories and a theory compatible with these facts were included in that lecture. The present paper contains data and ideas subsequently developed at the Aluminum Research Laboratories. Dix pointed out that in order for acceleration of corrosion by constant stress to occur: (1) There must exist in the alloy a susceptibility to selective corrosion along more or less continuous paths, as for in- stance at grain boundaries, and (2) There must exist a condition of high stress acting in a direction tending to pull the metal apart along these continuous paths. Acceleration of corrosion rate by stress is not confined to any one alloy system. Table I lists some of the metals and alloys and some of the corroding environments in which acceleration of corrosion by stress was sufficient to cause cracking (stress- corrosion cracking). It will be noted that the variety of alloys in which stress-corrosion cracking has occurred is rather large. Also, it will be observed that

Jan 1, 1945

By L. T. Larson, M. L. Picklesimer

The relationship between the apparent angle of rotation of monochromatic plane polarized light and the tilt of the basal pole from the surface normal has been experimentally determined for zirconium over the wavelength range of 500 to 655 mp. This relationship allows the determination of the spatial orientation of the basal pole of an individual grain in a polycvystal-ling zivrconium specimen to within ±3 deg by three simple tneasurements with a polarized-light metallurgical microscope. The method of measurement is discussed in detail. THE optical anisotropy of materials having noncubic crystal structures has long been used to reveal features by polarized-light microscopy. Petrographers have used measurements of certain optical properties to identify and classify transparent or translucent minerals. More recent work (i.e., Cameron1) has extended such measurements to opaque minerals in reflected light. Few attempts have been made to make similar measurements on noncubic metals. Couling and pearsall2 have reported that a sensitive tint plate can be used in a polarized-light metallurgical microscope to determine the position of the basal-plane trace in a grain of polycrystalline magnesium. Reed-Hill3 has reported that the same technique can be used for zirconium. We have found that the precision of measurement can be increased to about ±0.5 deg by using a Nakamura plate4,5 to determine the exact extinction position after the sensitive tint plate has been used to locate approximately the basal-plane trace. This report describes a method for measurement of another optical property, the apparent angle of rotation. This measurement permits determination of the angle between the basal pole of a grain of a hcp metal and the normal to the surface of the specimen. When the two measurements are combined, the orientation of the basal pole in space can be determined from three simple measurements on a single surface. One to two hundred such determinations will permit plotting of a basal-pole figure for the polycrystalline material with reasonable accuracy. When normally incident, monochromatic, plane-polarized light is reflected from the surface of an optically anisotropic material, the light may be converted to elliptically polarized light, the plane of vibration may be rotated, or both may occur. The el- lipticity, the angle of rotation, and the reflectivity can be related to the indices of refraction and the absorption coefficients of the material.6,7 Ellipticity values can be determined with an elliptical compensator, but not with the ease and precision desirable for the present purposes. Measurement of the angle of rotation requires only the determination of the angle from the crossed position (90 deg to the polarizer) that the analyzer must be rotated to obtain extinction when the trace of the optical axis in the surface is at 45 deg to the vibration direction of the polarizer. The angle of rotation of the analyzer is approximately 6/5 that of the true angle of rotation of the light as reflected from the specimen because there is a small amount of additional rotation produced during the passage of the reflected light through the mirror of the microscope. Since we are presently interested only in determining the tilt of the basal pole, the angle of rotation of the analyzer (the apparent angle of rotation of the light, i.e., uncorrected) can be used. Precision of the measurement can be increased substantially by the use of a Nakamura plate4,5 in determining the extinction position. In an optically uniaxial material (hcp or tetragonal crystal structure) the angle of rotation depends only on the optical properties of the material and the orientation of the optical axis of the grain relative to the plane of incidence of the plane-polarized light.7,8 Thus, in a metal such as zirconium, the apparent angle of rotation at the 45-deg position in any given wavelength of light is a direct measure of the tilt of the basal pole from the normal to the surface. If the optical properties vary with wavelength, the apparent angle of rotation for any given tilt of the basal pole will vary. None of the required information exists in the literature for zirconium nor for any other non-cubic metal. MEASUREMENTS ON SINGLE-CRYSTAL ZIRCONIUM A single-crystal sphere of zirconium 9/16 in. in diam was spark-cut from a single-crystal rod grown from iodide bar by an electron-beam zone-melting process.9 The damaged surface was removed by chemical polishing in a 45/45/10 mixture (by vol) of water, concentrated HNO3, and HF (48 pct) and then electropolishing at 50 v in a bath1' of methyl alcohol and perchloric acid (95/5 by vol) at -70-C. The single-crystal sphere was mounted in a five-axis goniometer stage having a removable eucentric X-ray diffraction goniometer head for the two inner orientation axes. The basal pole of the single-crysta sphere was aligned parallel to a third axis of the goniometer stage by using the sensitive tint method to determine the basal-plane trace at several rotational positions of the sphere. The alignment was then checked by removing the sphere and eucentric gonio-

Jan 1, 1967

By R. L. Fullman

IN the investigation of metallurgical transformations and the relationships between microstructure and properties of metals, it frequently is desirable to obtain a measurement of the relative amounts of the various phases present and of the mean size of particles into which each phase is dispersed. The relative amounts of the phases can be measured by the classical methods of area, lineal, and point analysis,1-5 in accordance with the principle that the volume fraction of a phase, the fraction of a polished cross section occupied by the phase, the fraction of a random line occupied by the phase, and the fraction of randomly arrayed points occupied by the phase are all equal. The validity of this relationship depends only on the attainment of a truly random sample of area, length, or points, and not on the size, shape, or distribution of the particles constituting the phase. Smith and Guttman8 have derived a relationship between the interface area per unit volume S, and the measurable quantities L., the interface length per unit area on a cross section, and NL, the number of interfaces per unit length intersected by a random line. Their equation, Sv = — L8 = 2NL is also valid regardless of the distribution of particle sizes and shapes. In contrast to the situation concerning measurement of relative fractions of phases and of interface area, the measurement of particle sizes in opaque samples has not been subjected to a complete analysis. It has been common to measure some lineal or area dimension of particles on a polished cross section and to use the mean value as a qualitative measure of particle size. In the present paper, quantitative relationships are established among the various mean dimensions on a polished cross section and the actual dimensions of the particles present. Particles of Uniform Size Spheres: If a metal sample contains particles of a phase a dispersed in the form of spheres of uniform size, a polished cross section through the sample will reveal circular areas of phase a with radii from 0 to ?, the radius of the spheres. Consider a cube of unit dimensions to be cut from the sample. If a cross section parallel to one of the cube faces is examined, the average number of particles per unit area (N,) equals the number of particles per unit volume (Nv) times the probability p1 that the plane would intersect a single sphere positioned at random within the unit cube. Since, of the various possible positions for the cross-sectional plane over the unit length from top to bottom of the cube, only those positions existing over the length 2r would lead to the plane intersecting the sphere, the probability of intersecting a single sphere is just 2r. N8= Nvp1 = Nd-2r [1] Applying the equality of area and volume fractions, the relationship is found between sphere size and average area s of uniform spheres intersected by a random cross section, 4 - f = NV V = Nr . — pra = N s = Nd . 2rs S = —pr2 [2] A similar analysis reveals the average traverse length across spheres of uniform size when random lines are passed through the sample. If a randomly oriented unit cube is cut from the sample and a randomly positioned line is passed through the cube parallel to a cube edge, the number of spheres intersected by the line (Nl) equals the number of spheres per unit volume times the probability p1 of the line hitting a single randomly placed sphere in the cube. Since possible positions of the line occupy unit area, and possible positions for which it will pass through the sphere occupy an area of pr2, the probability of the line hitting a randomly placed single sphere is pr2. NL = Nv p1 = Nvpr2 [3] Combining this relationship with the equality of volume and lineal fraction, the desired relationship is obtained between radius and mean lineal traverse length -i, for spheres of uniform size. 4 - - 3 l=4/3r [4] Circular Plates: Consider a sample containing particles of a phase a in the form of circular plates of uniform radius r and thickness t, where r >> t. If the plates are randomly oriented, as in a sufficiently large sample of a fine grained polycrystalline material, area and lineal analysis may be carried out with parallel cross-sectional planes and lineal traverses. If the plates are not randomly oriented, it is necessary to randomize the orientation of the cross-sectional planes and traverse directions. Let a unit cube be cut from the sample, and a cross-section plane be passed through the cube parallel to one of the cube faces. The number of plates cut by the cross-sectional plane per unit area is equal to the number of plates per unit volume times the probability of a plate intersecting a single randomly positioned and randomly oriented plate in the cube. If J is the component of the plate diameter in the direction normal to the cross-sectional planes, the probability of a plane cutting a single randomly oriented plate is equal to J, the mean value of J for all possible orientations of the plate. Let 4 be the dihedral angle between a plate and the cross-sectional plane, and let p?, d? be the probability that a plate makes an angle between 4 and ? + d? with the cross-sectional plane. Then for ran-

Jan 1, 1954