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By R. H. Hiltz, N. J. Grant

Solid-solution titanium-cerium alloys, cold-rolled to 0.010 in. thickness, and annealed, were internally oxidized to produce a fine dispersion of CeO2 in the titanium matrix. The oxidized alloys were cut into sheet stress-rupture specimens and tested for rupture life at 426°C, 538°C, and 648°C. There was noted an improvement in strength over that of pure titanium, the internally-oxidized alloys showing curves with a much flatter slope on the log stress—log rupture life plot, characteristic of the stable metal-metal oxide class of alloy. DURING the past 3 to 4-year period, considerable interest has been aroused by the potential offered by alloys based on the ultra fine dispersion of a hard insoluble phase in pure metals or solid-solution alloys for use at high temperatures. The impetus to the development and study of such alloys was provided by Irmann.1,2 through the success of the SAP alloy (pure aluminum containing about 12-14 volume pct of A12O3). Subsequent studies have thrown light on the true potential of this class of alloys in the remarkable stability of the structure after long-time exposure at high stresses and high temperatures.a-5 While success has been demonstrated in the special case of the A1-Al2O3 alloys, and more recently

Jan 1, 1959

By L. Adler

The author recently proposed a technique for handling the effect of planes of weakness on failure in geologic material.1 The technique employed an "extended" Mohr's rupture envelope, on which was superimposed a generalized three dimensional stress representation. Subsequently, the method was applied to so-called two-dimensional stress states" which were in reality degenerate three-dimensional ones. Since a two-dimensional stress solution usually refers to a plane stress condition, the terminology was confusing. The distinction between a plane stress state and a degenerate three-dimensional one is shown in Fig. 1. The solution of problems involving a generalized three-dimensional stress state is greatly aided by continual reference to the basic principle that no stress components can act on a plane, oriented as is the weakness plane, which lies outside of the weakness plane's rupture enVelope.2 This is shown in Fig. 2; and if it is also remembered that the radii of the Mohr's Circles give the orientation of such a plane, solutions are easier to follow, and perform. Sample Problem 1: Site Location (generalized three-dimensional case).' Given: 1) Both rupture envelope for the solid and plane of weakness. 2) Existing principal stresses and orientations in space (?12, ?13) obtained from a stress analysis of the proposed structure. Find: Permissible orientations of the weakness plane in order to select a suitable site.

Jan 1, 1963

By W. Hume-Rothery

IN recent interesting papers,1,2 Bardos et al. have described ternary Laves phases of the type A2(BsSi) and A4(B5Si3) where A = titanium vanadium, niobium, and so forth, and B = manganese, iron, cobalt, or nickel; other more complex formulae have also been established. In these phases, silicon plays the part of the smaller B atom of a Laves phase AB2, and the interatomic distances show that the effec- tive silicon radii lie between 1.16 and 1.21Å, and are thus very much smaller than the metallic radius of silicon for CN = 12. It does not seem to have been pointed out that these small atomic radii are almost the sape as the normal covalent radius of silicon (1.174Å). If, therefore, silicon acts as an acceptor of electrons as suggested by Bardos et al.,1 it appears to do so, not by the building up of a negatively charged silicon ion which would be large, but by the formation of what are very nearly or may be actually normal covalent bonds. Since a covalent bond contains two electrons, its formation would reduce the number of electrons available for bonding in other parts of the structure, and in this way the effect of silicon in stabilizing the o phase at high electron concentrations might be understood.

Jan 1, 1965

By C. G. Dunn, F. W. Daniels, M. J. Bolton

Recent observations on grain boundary movements in sihcon iron have indicated the possibility of studying grain growth phenomena in two-grain specimens in which several variables affecting growth are readily controlled. These variables are (1) the energy per unit area (?) in the grain boundary, (2) the orientation relationship of the two grains, and (3) the radius of curvature (7) of the moving boundary. Ideally, one grain A should be a narrow rectangular parallelepiped adjacent to a second grain ß as shown two dimensionally in Fig 1. Such a two-grain group becomes unstable when one end of grain A is shaped according to the dotted line. Since the width w of grain A remains constant during growth of grain B, T should also remain constant. The other vari- ables mentioned above can be controlled in the preparation of each two-grain specimen.l,2,3 For a parLicular orientation of each grain, is not necessarily fixed; ? may also depend on the orientation of the grain boundary, but this may be controlled to some degree. The energy relationships for the gowth of grain ß may be derived from the more simple arrangement of two grains A' and B' shown schematically in Fig 2. Here B' largely surrounds A'. Assume these grains are oriented with a common plane parallel with the flat

Jan 1, 1950

By E. A. Holbrook

The art of ore dressing is as old as civilization itself. Jason's search for the golden fleece was perhaps only the use of sheepskins for catching gold from gold-bearing sands washed over them. From Hoover's translation of Agricola we gather that in the middle ages ore dressing had advanced through the stage of hand sorting to water concentration by jigging. German methods and apparatus continued in the lead until the closing years of the past century; indeed, it is little more than a generation ago that many ore-dressing plants in this country were designed and operated by German-trained ore-dressing specialists. Meanwhile the development of the ore deposits in this country during the nineteenth century brought forth new inventions and methods. Blake's invention of the rock breaker in 1858 and the coming of the Wilfley table about 1896 are random examples of the improvements caused by our expanding needs. Yet ore dressing remained an art developed and practiced more or less by cut and try methods. The publication of the volumes on Ore Dressing by Dr. Robert H. Richards in 1903 marks the beginning of American ore-dressing practice as a science. Previously we had as guides only a few incomplete books based on foreign practice, or scattered references and articles in our own Transactions. Dr. Richards set down all the developments that had gone before, and outlined the ways for a scientific approach to the subject. He gave us a starting point. Since then ore-dressing practice and technique in this country have set the pace for the world. Coal preparation has been called "ore dressing upside down,'' because in ore dressing we generally save a small percentage of the heavy valuable mineral and discard the lighter gangue, whereas in coal preparation we save the larger quantity of light mineral and discard the smaller percentage of heavy waste. By comparison with ore dressing the history of coal preparation is brief, extending backwards hardly 100 years, for although coal had been dug and used locally for centuries it became an important article of commerce only with the coming of the industrial revolution 150 years ago. In new mines it was to he expected that the thickest and purest seams should be worked first. So long as these lasted there was little thought of coal preparation other than hand picking of the larger impurities and simple sizing to supply the real or fancied need

Jan 1, 1934

By H. M. Morse

There was little change in the oil and gas industry in Mississippi during the year 1938. No new fields were discovered and the production from the Amory field was nil. In the Jackson gas field six wells were drilled and production was obtained in each instance, but none of these wells increased the area from which the gas is produced. Outside the Jackson gas field nine wells were drilled, of which details are given in the accompanying table. The seeming lack of activity in Mississippi belied what was actually happening, for a great many geophysical crews of the major companies were active in all parts of the state. There were approximately 65 structures mapped and for the most part these have been leased. It is believed that 1939 will see an increase in wildcatting over a large part of the state.

Jan 1, 1939

Delay in classifying the war minerals claims geographically and by minerals' has resulted because a part of the claims is in the hands of the commission which is holding hearings at points easy of access from the mining districts. It was necessary for the commission to take many claims for use at these hearings. Twenty days will be allowed claimants to file typewritten or printed briefs based on the record made by the commission's findings. At the end of twenty days, the case will be finally submitted to the Secretary of the Interior. No new facts will be considered, those which have been submitted to the commission only being given consideration. The Attorney General has issued a decision whereby only such claimants as were directly requested by the government to start operations will receive consideration under the War Minerals Act.

Jan 9, 1919

By H. M. Morse

There was little change in the oil and gas industry in Mississippi during the year 1938. No new fields were discovered and the production from the Amory field was nil. In the Jackson gas field six wells were drilled and production was obtained in each instance, but none of these wells increased the area from which the gas is produced. Outside the Jackson gas field nine wells were drilled, of which details are given in the accompanying table. The seeming lack of activity in Mississippi belied what was actually happening, for a great many geophysical crews of the major companies were active in all parts of the state. There were approximately 65 structures mapped and for the most part these have been leased. It is believed that 1939 will see an increase in wildcatting over a large part of the state.

Jan 1, 1939

By E. G. Gaylord

A BRIEF review of the oil conditions in California for a few years prior to 1923 is desirable for a clear comprehension of the development during that year and the results of the development. California's oil production during the five years preceding 1922 had been running from 100,000,000 to 115,000,000 bbl. annually. Crude oil stocks in storage had been declining steadily from 1915 to 1919, when there was a slight increase. They continued to decline through practically all of 1920, when they were drawn on to the extent of 11,350,000 bbl., leaving 20,930,000 bbl. on hand. In 1921, with production approximately 10,000,000 bbl. greater than the previous year, crude-oil stocks increased until, at the end of the year, there were 33,289,000 bbl. on hand. It would seem, therefore, that at the end of 1921 the so-called Pacific Coast province could absorb between 100,000,000 and 105,000,000 bbl. of oil per year, any in excess of this amount going into storage, not being, able to compete in other markets with oil outside the state. During 1922, daily production increased from 316,000 bbl. at the beginning of the year, to 500,000, at the end. This increase was the result entirely of production from the new fields, in the Los Angeles basin. In fact, considerable oil from the older fields of the state was shut-in beginning with March, 1922. As of 1922, the past development of the oil industry of California had been, in a general way, quite regular. As the need grew for more oil, pioneer drilling was increased, resulting in the discovery of new fields here and there. The good prices for crude oil that prevailed during the later part of the War and into 1921 had stimulated an extensive wildcatting campaign by California's larger oil corporations, as well as by smaller independent oil companies and by individuals. Many wildcat wells in California have required, two or three years for their completion, consequently the intensive wildcatting, which might be said to have started in California in 1917 and continued into 1922, found its greatest results with, the discovery of three very productive fields which, for all practical purposes, might as well have come in simultaneously.

Jan 3, 1924

By R. F. Lee, G. Purcell

Tests carried out in the Department of Mining and Metallurgy at the New Mexico Institute of Mining and Technology indicate that high density rigid foams may, with the addition of suitable fillers, have structural properties comparable with concrete. Since polyurethanes have other useful properties that are unique to the foams, they have great potential value for both mining and civil engineering applications. Polyurethane, a generic term for a wide variety of closely related foam compounds, suggests rather misleadingly that foams are simple polymers of the urethane monomer ethyl carbamate (H,NCOOC,H,) . Polyurethane foams certainly do contain urethane groups, but these need not necessarily preponderate. Other reactive groups may be, and usually are, present so that the final polyurethane product will most likely contain, in addition to urethane groups, aliphatic and aromatic hydrocarbon, ester, ether, amide and urea groups.

Jan 9, 1963

By Kenneth N. Weaver

Portland cement can be made from relatively abundant industrial minerals and rocks, and this may explain why cement producers placed little emphasis on geology during the early days of the industry. After World War II, however, the industry began to recognize the need for geological exploration, and some of the larger companies began to build a permanent geological staff. This development had its roots in the following factors: Rapid expansion of the industry; closing of many cement plants due to inadequate raw materials; necessity for upgrading the quality of portland cement; and the increasingly large capital investment required to build new cement plants, which in turn, required a higher degree of accuracy in reserve calculations. Whereas the geologist's function during the active expansion of capacity was primarily in exploration, one of his major functions today is the development of existing reserves.

Jan 1, 1965

By Robert W. Coughlin

Pulverized coal has been introduced at the anode where it is oxidized during the electrowinning of metals from their aqueous solutions This substitutes an electrochemical reaction between coal and water for the usual oxygen-evolving half-cell reaction at the anode, with the result that overall cell potential is significantly lowered and the consumption of electrical energy is correspondingly reduced. Experimental results are reported for electrowinning copper as an example of this novel modification of the electrowinning process in this example, the use of coal lowered the consumption of electricity by two-thirds of the requirement of the conventional process When the potential electrical energy of the coal consumed is considered the new, coal-assisted process consumes about 45% less energy, and the energy costs (combined costs of coal and electricity) are reduced by about 50%.

Jan 1, 1981

By A. S. Richardson

PROGRESS in health and safety in the mining field has been greatly affected by war conditions. Some of the instruments commonly used in ventilation and dust prevention work have been practically unobtainable. Much equipment, such as fans, has been purchasable only when it was possible to make positive statements to the effect that it was absolutely necessary for the production of definite quantities of materials essential to the war. Even then, long delays were usually experienced in obtaining such equipment from the manufacturers. These circumstances have, of course, not been conducive to the development of new and better methods, though attention has been directed by governmental agencies towards the recognition of existing hazards to which little attention had previously been paid.

Jan 1, 1945

By C. D. Michaelson

Bridging the gap between have and have-not nations is one of the necessities of the present era. The responsibility for accomplishing this must be assumed by the affluent industrial societies of the world, both through their governments and through private enterprise. Despite a discouraging number of instances where companies established in underdeveloped countries have been expropriated, U.S. industry must not abandon such ventures. Instead, it should tailor them to the needs and aspirations of the host nations. One way in which industry can minimize the danger of expropnation is to enter into partnership with the government of the host nation. Kennecott Copper has recently done this in Chile, becoming 49% owner of a company formed to expand and operate the El Teniente mine. So far, the partnership has produced marked benefits not only for Chile, but also for Kennecott. The memory of D.C. Jackling serves chiefly to remind us how much we owe to the pioneers of our industry. Jackling was one of the giants. Orphaned at the age of two, he made his way through sheer persistence and plain hard work. His brilliance gave rise to a vision, and his tenacity made his vision a reality. Foreseeing that economies of scale would make the handling of low-grade ore profitable, he introduced the revolutionary concept of open-pit copper mining. During World War I, the defense effort of the United States was irnrneasurably strengthened by the tremendous supplies of copper from the western mines that Jackling had made possible. Again in World War 11, the demands for copper were supplied, and the nation once more had profound reason to be grateful to this rugged, stubborn man who had long ago ventured into the mountains and unlocked their wealth. We are now embarked upon another great era of change, and we must devise new concepts to meet it. The task is not only to create new techniques of mining, but to forge better ties between men and institutions which will let us work in harmony with the techniques we have. We might recall that while Jackling was a man of towering pride and formidable will, he was also adept at getting others to share his vision and join his purpose. Today our reach for cooperative endeavor must go far beyond our own shores as we seek to develop the buried riches of distant lands. And we would do well to remember that Jackling's remarkable insight would have come to nothing had not British investors been willing to provide much of the initial capital. As was true of so many early enterprises in America, European capital was absolutely essential. CORPORATIONS ABROAD - AN EXPANDING ROLE Today the situation is quite different. The United States is now a capital-surpIus nation and looks abroad for opportunities to employ its capital profitably. On the other hand, we have become dependent on foreign sources for supplies of vital raw materials, while we seek

Jan 1, 1970

By M. M. Fidlar, Ralph E. Esarey

A marked decrease in drilling during 1937 in Indiana seemed to have no undesirable effect upon the oil and gas industry. Instead, production figures for oil showed a small increase over those for the preceding year, although not so much gas was marketed as in 1936. Drilling activity was centered in proven areas, and the proportion of dry holes to the number of wells drilled decreased somewhat. During 1937, including old wells drilled to deeper horizons, 134 wells were completed in Indiana, of which 39 were oil wells, 39 gas wells, and 56 dry holes. Nearly 75 per cent of the holes were drilled in proven areas. Of the 36 wells drilled in wildcat areas, only four found commercial production. Wildcat wells were especially numerous in Vigo, Pike, Perry and Spencer Counties. Exploratory drilling activity in Vigo County resulted in part from the drilling of the discovery well in the new Prairie Creek field. In Pike County, wildcat wells were drilled in the search for new supplies of gas to meet the demand in the southwestern portion of Indiana. Drilling in both proven and wildcat areas in Perry and Spencer Counties continued to be heavy, because of the presence of shallow Missis-sippian sands and the resulting low cost of drilling and equipping. At the end of the year, 42 of the wells started in 1937 were not completed. Only a few locations were drilled in the old Trenton field, but the wells, many of them 40 years old, continued to furnish a small supply of gas for domestic consumption. Oil production in the Jay County portion of the Trenton area was nearly twice that of 1936, owing in part to a vacuum system installed on several leases. The Harrison County (Laconia) gas field is being developed northward to offset water encroaching from the south. This water in the black shale threatens to put an end to the steady production that this field has furnished for many years. During 1937, the Oaktown gas field again furnished more gas than any other field, but production in that area showed a marked decrease over the marketed gas for 1936. During the early months of the year, seven wells were completed in the Oaktown field to bring the total number of producing wells to over 40. At present, all the operators in the area are

Jan 1, 1938

By E. Miller, J. M. Eldridge, K. L. Komarek

Aclivilies of magnesium in liquid Alg-Si alloys have been delermined between 5 and 60 at. pcl Si, close to the melling point of Mg2Si, by an improved isopieslic melhod. Silicon specinrens, held in alumina crucibles and graplrile conlainevs of special design, were healed in a letrlpevalure gvadient and equilibrated with mag-nesilcrrl rapor in a closed lilanium system. The ther-madynamic Junctions were calculated and compared with the thermodyuamic properties of the other three mg- Gvoup IVB systems. Lattice paramelers of three Mg2X compounds were measured. The bonding in the Mg2X compounds is largely covalent with small and uarying amounts of metallic and ionic conlvibutions. The Mg-Si phase diagram1 has one congruent melting compound, Mg2Si, of essentially stoichiometric composition, two eutectics, and very limited terminal solid solubilities. Little information is available on the thermodynamic properties of this system. The free energy of formation of Mg2Si has been determined by the Knudsen cell technique2 in the range 572" to 680oC, by the transportation method3 between 858" and 950oC, and by the electromotive-force method4 in the range 400o to 600°C. Kubaschewski and villa5 and caulfield6 have measured the heat of formation of Mg2Si. An electromotive-force study of magnesium-rich liquid alloys was recently published by Sryvalin el al.7 The present investigation was undertaken to complete a general survey of the thermodynamic properties of the homologous series of Mg-Group IVB systems, i.e., Mg-Pb,a9,Mg-Sn,10,11 mg-Ge,12and Mg-Si. An isopiestic technique, previously developed for similar measurements on liquid Mg-sn11 and Mg-Ge alloys,12 was modified for the Mg-Si system. Specimens of the nonvolatile component, silicon, were contained in dense alumina crucibles placed inside covered graphite crucibles which were heated in a temperature gradient in an evacuated and sealed titanium reaction tube and equilibrated with magnesium vapor of known vapor pressure. The alumina crucibles prevented contact between the highly corrosive liquid Mg-Si alloys and graphite. The graphite cruci- bles effectively preserved the high-temperature equilibrium composition of the liquid alloys containing highly volatile magnesium on termination of the experiments during the quench to room temperature. EXPERIMENTAL PROCEDURE Silicon of semiconductor-grade purity (E. I. du Pont de Nemours and Co., Brevard, N.C.) and 99.99+ pct Mg (Dominion Magnesium Ltd., Toronto, Canada) were used. Graphite crucibles with press-fitted lids were machined from high-density (1.92 g per cu cm) rods (Basic Carbon Corp., Sanborn, N.Y.) which had a maximum ash content of less than 0.04 pct. The alumina crucibles had a purity of 99.7+ pct (Triangle RR grade, Morganite, Inc., Long Island City, N.Y.). In preliminary runs the liquid alloys were contained in graphite crucibles following the exact procedure developed for the Mg-Ge system.'2 These runs failed due to appreciable reaction between the molten Mg-Si alloys and graphite, and the results have been discarded. The procedure was then modified and the Mg-Si alloys were subsequently held in alumina crucibles. For most of the runs alumina crucibles of known weight and approximately 6.3 mm ID, 12.5 mm height, 1.0 mm wall thickness were loaded with weighed amounts of silicon and encapsuled in tightly covered weighed graphite crucibles 5/16 in. ID, 2 in. helght, 3/32 in. wall thickness). The graphite crucibles were machined from rods which were 85 pct of the theoretical density. These crucibles were therefore sufficiently porous so as to permit magnesium vapor to effuse through the silicon under the experimental conditions of approximately 970O to 1220°C and 1 day equilibration time. However, negligible magnesium was lost from the crucible during the quench due to the slow effusion rate through the pores of the graphite. The inner alumina crucible prevented the liquid alloys from contacting the graphite, and the very tightly fitting graphite crucible lids served to retain any magnesium vaporizing from the alloys inside the crucibles during the quenching step.12 The loaded silicon-alumina-graphite cells were positioned, one above another, on a 16-in.-long titanium thermocouple well and tied securely to the titanium tube with thin molybdenum wires held in grooves around the circumference of the graphite crucibles. A thin (0.005-in.) molybdenum strip prevented contact between the graphite crucibles and the titanium. This assembly was lowered into a titanium reaction tube (la in. ID, 16 in. long, $ in. wall thickness) closed on one end which contained a 11/2-in.-long cylinder of magnesium at the bottom. The inner titanium thermocouple well was positioned eccentrically in the large tube because of the eccentric mounting of the cells on the well. Appropriate modifications of the titanium cap"'12 were made to join the inner and outer titanium

Jan 1, 1968

By J. L. Wilson, D. L. Emmons

The Tucker Hill perlite deposit is located in the northwestern portion of the Devils Garden lava field in Lake County, south-central Oregon. The perlite occurs in the chill margin of a late Miocene composite rhyolite lava dome that exhibits distinct cooling zonation. The chill margin is composed of an outer glass envelope, which contains commercial-grade granular and vesicular perlite, and an inner glass envelope, which contains onionskin perlite of local commercial quality. Glassy rhyolite and rhyolite compose the lava dome core. The entire dome complex encompasses at least 5 km2 (2 sq miles). Expansion tests in commercial-scale furnaces indicate that the perlites are suitable for a wide range of applications and compare favorably with commercial perlite ores. The large tonnage deposit appears competitive for markets in the midwestern and upper northeastern states, and the Pacific Northwest of the US.

Jan 1, 1986

By Knut L. Sandvik, Aage Solbakken

it is an established fact that the collecting properties of xanthates and anionic collectors are dependent on the hydrocarbon chain length. This is ascribed to the greater adsorption densities caused by increasing the xanthate chain length. As batch flotation tests showed similar trends for amines, an investigation of the adsorbing properties of amines was undertaken. Because earlier investigations showed adsorption densities for dodecylamine salts on quartz varying as much as 20 times under apparently similar conditions,1-4 it was decided to make a survey of the changes in adsorption density due to the drying conditions of the quartz. EXPERIMENTAL PROCEDURE The quartz, handpicked in the pegmatite quarries of Iveland, Setesdal, had the following composition: SiO2 99.81, A12O3 0.01, Fe2O3 0.05 and loss by ig-

Jan 1, 1969

By L. Moussoulos

Pyrite concentrates produced at the Olympias mines, Northern Greece, show an unusually high gold and silver content (16-20 and 25-40 ppm respectively), associated with the presence of arsenopyrite; their high As content (5-8%) has hindered their utilization so far. In the present paper a simple and environmentally safe method that permits their complete utilization for the production of SO2 gas, As203, and the recovery of Au and Ag is described. According to this method, the pyrites are subjected to a single-stage oxidizing roasting in a fluid bed reactor. The precious metals, and approximately 45% of the contained As, remain with the cinders, while the rest of the As is removed with the SO2 - bearing gases and can be recovered by suitable treatment. The gases, after further cleaning, can be utilized for the production of sulfuric acid.

Jan 1, 1984

By M. H. Richman, E. S. Meieran

An analytical method is presented which enables the electron diffraction patterns from a twinned region to be indexed. The method has only been applied to (112) twins in bcc lattices, but can easily be adapted to other types of twinned lattices. Calculations of all twin spots, including those superimposed on matrix spots, is made possible by this method. SINCE the advent of transmission electron microscopy, the analysis of selected-area electron diffraction patterns has become very important in the interpretation of electron micrographs. Frequently, as in the case of some martensites, twinned areas are observed and give rise to complex diffraction patterns, which so far can be analyzed only for the simplest orientations of the thin foil specimen with respect to the electron beam. In order to simplify the analysis of twin patterns of more complex orientations, an analytical method was devised which is capable of predicting the positions of all reflections from both matrix and twin regions for all possible foil orientations. Although the method has only been applied to the case of bcc metals twinned about a (112) plane, it is equally applicable to other Bravais lattices. This method has been successfully applied to selected-area electron diffraction patterns of Fe-Ni and Fe-Ni-C martensites and drawn tungsten wire. PROCEDURE Imagine the reciprocal lattice corresponding to a bcc material, with each point hkl representing the crystallographic plane (hkl). Since the extinction conditions require that for this lattice, only reflections where h + k + 1 = even integer are allowed, the three dimensional reciprocal lattice is actually fcc. Since in real space twinned regions and matrix regions are mirrored across a (112) plane, the twinned reciprocal lattice is derived from the matrix reciprocal lattice by a rotation of 180 deg about a (112) direction. While it is possible to calculate how a point hkl will shift under a twinning transformation to a new point htktlt using only the reciprocal lattice based on the orthonormal vectors [100], [010], and [001], the calculations are greatly simplified if a new orthonormal coordinate system containing the twin axis is introduced. The twinning operation then becomes a simple rotation of 180 deg about a principal coordinate axis (in this case the twinning axis), and a retransformation back to the original reciprocal space will give the position htktIt of the twinned reflection.

Jan 1, 1963