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Jan 1, 1957

Jan 1, 1955

Jan 1, 1969

By Joseph B. Story, John T. Clarke

A modification, of the Kelvin bridge using an inductor was used to measure the conductivities of molten sodium chloride, calcium chloride, and mixtures thereof. A capillary-type four-lead fused quartz dipping cell was constructed. The effect of a small amount of potassium chloride on the conductivity of the melt was determined. IN the electrowinning of sodium and chlorine from sodium chloride in Downs cells, calcium chloride is added to the sodium chloride in an approximately equimolar amount to permit cell operation at a lower temperature. This lower temperature improves current efficiency and ease of operation, but decreases voltage efficiency by decreasing the electrical conductivity of the melt. A knowledge of the electrical conductivity of the Downs cell melt is necessary to the understanding of Downs cell operation, for the IR voltage drop across the melt is responsible for a significant part of the electrical power consumed by the cell. The work of previous investigators on fused sodium chloride-calcium chloride mixtures was reviewed from the standpoint of consistency with the results of others on the conductivities of the pure constituents. The electrical conductivity of pure fused sodium chloride has been established with accuracy by Van Artsdalen and Yaffe,1 by Edwards, Taylor, Russell, and Maranville,2 and by Lee and Pearson.3 It has also been studied by a number of other investigators.4-13 The best values for calcium chloride are believed to be those of Lee and Pearson,3 since the sodium chloride and potassium chloride conductivity values given by them check the best literature values. A number of other investigators have also reported values for calcium chloride.'-"."' Electrical conductivity data for fused sodium chloride-calcium chloride mixtures have been reported by Sandonnini,12 Vereshchetina and Luz-hnaya,18 Ryschkewitsch,11 Barzakovskii,5 and Alaby-shev and Kulakovskaya.15 Sandonnini reports no data at temperatures lower than 850°C. The conductivity values reported by Vereshchetina and Luz-hnaya, by Ryschkewitsch, and by Barzakovskii for pure sodium chloride and for pure calcium chlor- ide vary somewhat from the literature values believed to be best and the sodium chloride-calcium mixture data of these authors and of Alabyshev and Kulakovskaya (who give no data for the pure compounds) show significant variations. In view of these facts, it seemed that an investigation of the electrical conductivities of fused sodium chloride-calcium mixtures would be worthwhile. Apparatus Bridge System—The bridge used,in this investigation is diagrammed in Fig. 1. In general, the elaborations of this bridge over the simple Wheatstone bridge were for the purposes of a) eliminating lead resistances or making them accurately measurable, and b) eliminating major capacitative and inductive effects or balancing them against each other. Lead resistances in the lower arms of the bridge were kept as low as possible by making the leads short and of low resistance wire. This could not be done in the cell arm of the bridge because of the remoteness of the cell from the other bridge components and because of the high-resistance nickel or chrome1 wires used in the cell. Since maximum sensitivity is attained when the resistances of the four arms of the bridge are equal, 50-O resistors were used in the lower arms of the bridge when fused salt conductivities were being measured, and 1000-O resistors were used when the cell was being calibrated with normal aqueous KCl solution. The cell resistance was of the order of 50 ohms with fused salts and of the order of 1000 ohms with normal aqueous KC1 solution. The resistors used were non-inductive and accurate to 50.05 pct (General Radio Co. Types 500-C and 500-H). The lead resistance between point A in the cell and point E at the measuring resistor was eliminated by using the kelvin bridge principle; that is, the voltape drop across the lead from point A to point E was divided between the cell arm of the bridge and the measuring arm of the bridge in proporation to the resistaxes in the lower arms of the bridge.

Jan 1, 1958

By Davis M. Lapham

Ore bodies at Cornwall, Pennsylvania, have been mined since 1742 principally for iron from magnetite, but also for copper (in chalcopyrite), silver (in chalcopyrite), gold (in chalcopyrite), cobalt (in pyrite), sulfur, and crushed limestone aggregate. Metallization occurs as a limestone replacement above the essentially conformable, south dipping portion of a discordant saucershaped diabase sheet. Pre-Triassic and Triassic faulting have emplaced host Cambrian limestone and two Ordovician(?) hornfels units, now for the most part resting above diabase. Thermal metamorphism of carbonate-bearing country rocks largely to calc-silicates, is ubiquitous. Metasomatism, particularly of potassium, is unique to ore zones. Associated magnetite- actinolite mineralization replaces, in a preferential sequence, all earlier metamorphic and metasomatic minerals and rarely diabase chilled margin. Mineralogical (diopside, actinolite, mica, sulfides, oxides) and chemical (Fe, Cu, S, Co) zoning of the eastern ore body is focused down dip, near its western edge, along a major vertical fault through diabase. This zoning is spatially related to magnetite veins in diabase, to an ore body within diabase, and to a shear zone that probably extends through the sheet.

Jan 1, 1968

Jan 1, 1970

By Eugene McAuliffe

AS this is written, the probability A is that the bituminous coal out- put for 1936 will approximate 420,000,000 tons (of 2000 lb.) with an average working time for all mines of 205 days. The results for 1935 approximated 369,224,000 tons, with the mines working about 185 days. The increase in production of 14 per cent is heartening. In anthracite, it is probable that the preliminary figures covering output for 1935, 51,- 003,000 tons, will be exceeded this year although the percentage of in- crease will almost certainly be small. The substantial tonnage of anthracite illegally mined, estimated to be 3,600,- 000 tons in 1935, with a similar large Lonnage in 1936, is in addition to the figure given above; it represents an open and flagrant disregard of law and property rights. It is hard to understand why the officials of the State of Pennsylvania do not enforce the laws against theft and brigandage, more so when the legitimate employees of the established producing proper-

Jan 1, 1937

By Gerald J. Anderson

The Marquette District of Central Northern Michigan is the oldest of the Lake Superior iron districts with a mining history dating from 1852 up to the present. The total production of all types of ore through 1965 was 338,120, 146 long tons. Emphasis up to the 1950's was on mining the direct shipping ores; however, demands for higher-grade and better-structured ores in the last decade has resulted in a complete shift in emphasis to beneficiating the lowgrade Negaunee Iron-formation to produce high-grade iron ore pellets. The Precambrian sediments of the Marquette Range have been correlated with the Animikie Series of the Thunder Bay District, Ontario, and are contained in a westward plunging synclinorium that is approximately 3 3 miles in length and 3 to 6 miles in width. -The rock unit of major economic importance is the Negaunee Iron-formation that reaches a thickness of 2500 feet near the City of Negaunee. The primary iron content of the Negaunee Iron-formation averages 26 per cent, whereas the altered iron-formation approximates 31 per cent. The Negaunee Iron-formation has been divided into three principal types: diagenetic, oxidized, and metamorphic. The ores produced to date fall into four categories: high-grade soft ores, high-grade hard ores, siliceous ores, and concentrates and agglomerates (pellets). The soft ores generally are found in synclines and in fault-trough structures, bounded by mafic dikes along at least one side, and are mainly found in the basal portion of the Negaunee. In contrast, the hard ores occur in the uppermost portion of the Negaunee and are controlled by structural features such as folds, faults, and dikes. Iron-formation that has undergone diagenetic and metamorphic alteration makes up the main beneficiating ore types. Studies on the Negaunee Iron-formation have played an important role in the formulation of many of the theories on the origin of the Precambrian iron-formations.

Jan 1, 1968

Jan 1, 1882

By E. H. Kelton, R. D. Grissinger

In designing structural members of steel and some other materials the design engineer has available recognized values of elastic modulus and safe working stress that may be substituted in well-known engineering formulas to give him desired cross-sectional areas and deflections for a particular part. Zinc, on the other hand, has no generally recognized modulus. It flows under continuously applied loads. Therefore, the designer of structural members must use a range of working stresses depending on the continuous load and the amount of permissible distortion. The only method of obtaining these values is by creep testing. Creep testing of zinc has been carried on for some 20 years by this laboratory, covering a wide range of zinc and zinc alloys. Types of test included tension, compression, and simple and cantilever beams at temperatures ranging from o to I50°C. Specimen conditions ranged from as-cast or as-rolled to accelerated aged in steam or dry air. Effects of plating were investigated also. It is not the intention of this paper to expound all these factors but rather to present the final data on well-known zinc die-casting alloys and to suggest means of applying these data to engineering design. Early Work In the early work on creep testing of zinc, the direct-loading tension method was used. However, direct loading of die castings was unsatisfactory because of the high loads required. For instance, a standard die-cast tension bar having a cross-sectional area of about 0.049 sq. in. would require a direct load of about 1000 lb. for a unit stress of 20,000 lb. per sq. in. Also, tests were made at nominal room temperature, but normal variations caused fluctuations greater than those due to experimental error or other causes, so that control of this variable within ±1/2°C. seemed essential. These two factors led to the development of an apparatus described by J. Ruzicka,' in which the specimens were loaded by means of a lever and were held at constant temperature by means of an oil bath. While Ruzicka's apparatus yielded excellent check results, it seemed desirable to have some setup by which literally hundreds of die-cast specimens could be tested at the same time. This would permit the conducting of a number of low-load tests designed to run 10 years or longer and still leave capacity for current high-load work. Beam testing in a constant-temperature room was the answer. Test Method Preliminary work indicated that die-cast beams of various lengths and cross scctions yielded the same result within reasonable limits regardless of whether they were tested as simple beams, uniformly loaded beams, or cantilevers. Accordingly, the simplest type was adopted. This was a,

Jan 1, 1945

By E. S. Moore

Although nickel was in use in alloys long before the Christian era, the metal was not discovered until 1751, when Cronstedt recognized it in niccolite from Sweden. The Chinese apparently used a nickel alloy, known as "pahfong" thousands of years ago and Persian coins dating farther back than 200 B.C. consist of a copper-nickel alloy similar to some of those now in use in coins. The pure metal was first prepared about 1775 and the metal was refined for commercial purposes some years later. The nickel industry did not attain much prominence until the ores in New Caledonia were discovered. Exports of ore from that country began in 1875 and since that date there has been an ever-increasing world output of nickel and a great expansion in the uses of the metal except during short periods of commercial and industrial depression. Previous to 1875, for a number of years, Norway was the chief producer, although the metal was mined also in Sweden, Finland, Italy, Germany, Austria, France, China, the United States and other countries. With the discovery of nickel deposits in the Sudbury district, Ontario, a new factor was introduced into the nickel industry and Sudbury has since become the overwhelmingly dominant feature in the world's nickel markets. Although nickel was first discovered in Canada as early as 1848 near the north shore of Lake Huron and nickel-copper sulfides were found in 1856 by surveyors working close to the present location of the important rnine at Creighton, in the Sudbury field, it was not until 1883 that any importance was attached to this field. In that year copper sulfide attracted attention where the main line of the Canadian Pacific railroad was being projected across Canada to the Pacific coast. Mining began near this discovery in 1889 but in the meantime operations had begun in 1886 in other parts of the field and by 1890 about 100,000 tons of ore had been raised. In the Sudbury field, as in Norway and at the old Lancaster Gap mine in Pennsylvania, the ore was first mined for its copper content and difficully in the refining of the copper led to the discovery of the important proportions of nickel in the ore. No. 3 property of the International Nickel Co., now known as the Frood, the greatest deposit in the

Jan 1, 1932

By AIME AIME

AN IMPORTANT innovation in Alabama in the semi-longwall type of coal mining as applied to low-dipping thin seams has been introduced by the Galloway Coal Co., mining the Mary Lee high-ash seam which averages only 26 in. thick. The property is their Mill Creek mine, near Carbon Hill, Ala., in the Warrior coal field. The results, although not final, indicate that roof conditions can be successfully handled by this method without loss of walls, with a rate of recovery much above that possible with the room-and-pillar method, and at the low production cost necessary for this grade of coal. Double cross-headings are turned off a main haulageway on 60-ft. centers and driven 40 ft. wide with thin chain pillars between. These double entries are turned at 525-ft. intervals, leaving a solid block of coal 440 ft. across; that is, from the outside of one pair of headings to the inside of the next pair is 440 ft. The bottom, which consists of hard fireclay, is taken in these entries 8 ft. wide and 4 ft. deep for head room.

Jan 1, 1934

By O. W. Shimer, D. C. Helms, C. E. Brown

THE early history and progress of anthracite stripping, from the first known operation at Summit Hill in 1821 through 1917, was covered in 1917 in a paper by J. B. Warriner,1 then chief engineer, now president of the Lehigh Navigation Coal Co., and stripping methods between 1917 and 1931 were the subject of a paper in 1931 by H. H. Otto,2 of the Hudson Coal Company. The present paper is confined to methods used today on pitching beds in the western middle and southern anthracite fields. The anthracite industry, not unlike other large industries, had been affected for more than a decade by the adverse economic conditions of the country. High development and operating costs and the competition of other fuels caused some collieries to be closed temporarily, while others were permanently abandoned. In order to obtain a larger tonnage at a lower development cost, a larger stripping program was put into effect by many companies. In 1932, stripping operations accounted for 334 million tons, or 7 per cent of the total anthracite production of the year, and in the next 10 years this increased until more than double that amount, or over 14 per cent of the total production, was from stripping operations. These figures represent the total production of the entire anthracite region; unfortunately, the figures have not been separated from the different fields. There are some operations in the southern and middle western fields where the tonnage obtained from stripping amounts to more than 30 per cent of the total colliery production. Stripping operations on pitching beds present difficult problems. Owing to the various geological formations, a different situation is encountered in nearly every stripping pit; and because of the proximity of an overlying bed, it is very often necessary to load the entire overburden into trucks and haul it out of the pit to the dump area, although back filling and casting is done whenever possible. Beds in the areas described in this paper pitch from 20° to 90° and some are on the invert in, many sharp folds and faults. PLANNING The more successful strippings have had adequate engineering analysis. Some work is being done without much preliminary planning, but many strippings would have failed completely if close supervision by the engineering department had not been maintained. Locations of stripping areas are determined as follows: 1. By constructing cross sections over the areas and drilling sufficient diamond-drill or well drill holes to study the strata and to determine the location, thickness and pitch of the beds.

Jan 1, 1944

By A. H. Tousley

Current problems in the cement industry are discussed and suggestions for solving them by modernization are made. Cement facility modernization is discussed in considerable detail with examples illustrating before and after conditions. Itemized are the improvements realized in better working conditions, better efficiency, higher production, lower costs, and better community relations. These improvements are related to the overall profitability of the particular plant or facility.

Jan 1, 1972

By F. Weinberg

It has been suggested, for some time, that the behavior of metals under high-temperature creep conditions is strongly influenced by the behavior of the grain boundaries present in the material. This has led to a number of investigations1-6 in which attempts were made to measure the properties of the grain boundaries themselves, both to determine their contribution to the creep process and to obtain further knowledge of their intrinsic structure and properties. The present investigation was undertaken to extend the observed data under more closely controlled experimental conditions and to attempt to clarify sonie of the discrepancies in the literature. A brief report of the preliminary results has been published.7 EXPERIMENTAL PROCEDURE For the present investigation, controlled orientation specimens were used having plane boundaries which terminated at the specimen surface. Trierys-tal specimens were used instead of the bicrystal type of Puttick and King,' in order to simplify the gripping technique required to apply a shear stress along the boundary. The aluminum used was obtained from the Aluminum Francaise and was of 99.994 pct purity, containing 0.002 pct each of Cu, Fe, and Si. The aluminum tricrystals were grown horizontally from the melt, in graphite boats, and in an argon atmosphere, following the technique developed by Chalmers.' The orientation of the tricrystals was controlled by using seed crystals separated by mica inserts. Thermal conditions were adjusted to produce tricrystals having boundaries which were macroscopically straight, parallel to the growth direction, and perpendicular to the top surface. Following growth, the tricrystals were carefully cut with a jeweller's saw to the test specimen shapes shown in Fig. l(a) and l(b). In Fig. 1(a), the boundaries subjected to shear stresses are indicated AB and CD: Points A and C are the ends of the mica inserts separating the seed crystals; points B and D are at lateral cuts made in the specimen. It was found that the slots left by the mica inserts at A and C had to be enlarged slightly, by cutting, to ensure that the boundary terminated freely at these points. A shear stress was applied to the boundarks by gripping the upper two legs in file-faced Inconel grips and suspending a weight from the lower part of the specimen held in similar grips, as shown schematically in the diagram. The test specimens shown in Fig. 1(b) were cut from long

Jan 1, 1959

By O. W. Shimer, D. C. Helms, C. E. Brown

The early history and progress of anthracite stripping, from the first known operation at Summit Hill in 1821 through 1917, was covered in 1917 in a paper by J. B. Warriner,1 then chief engineer, now president of the Lehigh Navigation Coal Co., and stripping methods between 1917 and 1931 were the subject of a paper in 1931 by H. H. Otto,2 of the Hudson Coal Company. . The present paper is confined to methods used today on pitching beds in the western middle and southern anthracite fields. The anthracite industry, not unlike other large industries, had been affected for more than a decade by the adverse economic conditions of the country. High development and operating costs and the competition of other fuels caused some collieries to be closed temporarily, while others were permanently abandoned. In order to obtain a larger tonnage at a lower development cost, a larger stripping program was put into effect by many companies. In 1932, stripping operations accounted for 3½ million tons, or 7 per cent of the total anthracite production of the year, and in the next 10 years this increased until more than double that amount, or over 14 per cent of the total production, was from stripping operations. These figures represent the total production of the entire anthracite region; unfortunately, the figures have not been separated from the different fields. There are some operations in the southern and middle western fields where the tonnage obtained from stripping amounts to more than 30 per cent of the total colliery production. Stripping operations on pitching beds present difficult problems. Owing to the various geological formations, a different situation is encountered in nearly every stripping pit; and because of the proximity of an overlying bed, it is very often necessary to load the entire overburden into trucks and haul it out of the pit to the dump area, although back filling and casting is done whenever possible. Beds in the areas described in this paper pitch from 20" to 90 and some are on the invert in many sharp folds and faults. Planning The more successful strippings have had adequate engineering analysis. Some work is being done without much preliminary planning, but many strippings would have failed completely if close supervision by the engineering department had not been . maintained. Locations of stripping areas are determined as follows: I. By constructing cross sections over the areas and drilling sufficient diamond-drill or well drill holes to study the strata and to determine the location, thickness

Jan 1, 1944

By O. W. Shimer, D. C. Helms, C. E. Brown

The early history and progress of anthracite stripping, from the first known operation at Summit Hill in 1821 through 1917, was covered in 1917 in a paper by J. B. Warriner,1 then chief engineer, now president of the Lehigh Navigation Coal Co., and stripping methods between 1917 and 1931 were the subject of a paper in 1931 by H. H. Otto,2 of the Hudson Coal Company. The present paper is confined to methods used today on pitching beds in the western middle and southern anthracite fields. The anthracite industry, not unlike other large industries, had been affected for more than a decade by the adverse economic conditions of the country. High development and operating costs and the competition of other fuels caused some collieries to be closed temporarily, while others were permanently abandoned. In order to obtain a larger tonnage at a lower development cost, a larger stripping program was put into effect by many companies. In 1932, stripping operations accounted for 3½ million tons, or 7 per cent of the total anthracite production of the year, and in the next 10 years this increased until more than double that amount, or over 14 per cent of the total production, was from stripping operations. These figures represent the total production of the entire anthracite region; unfortunately, the figures have not been separated from the different fields. There are some operations in the southern and middle western fields where the tonnage obtained from stripping amounts to more than 30 per cent of the total colliery production. Stripping operations on pitching beds present difficult problems. Owing to the various geological formations, a different situation is encountered in nearly every stripping pit; and because of the proximity of an overlying bed, it is very often necessary to load the entire overburden into trucks and haul it out of the pit to the dump area, although back filling and casting is done whenever possible. Beds in the areas described in this paper pitch from 20" to 90° and some are on the invert in many sharp folds and faults. Planning The more successful strippings have had adequate engineering analysis. Some work is being done without much preliminary planning, but many strippings would have failed completely if close supervision by the engineering department had not been maintained. Locations of stripping areas are determined as follows: I. By constructing cross sections over the areas and drilling sufficient diamond-drill or well drill holes to study the strata and to determine the location, thickness and pitch of the beds.

Jan 1, 1946

By KENNETH CROPPER

USUALLY we forget about the things which move along smoothly. There are no causes for worry when there are no troubles. But when troubles arise we must put forth some thought and effort to alleviate them. No one "worries about his automobile until it begins to knock, develop flat tires, or refuses to start. However, when these ailments become apparent we get wide awake on the spot, and do our best to repair the car. So we find ourselves now. The old bus seems to hit all the ditches. There is rough ground on all sides and none of us is able to do anything toward improvement. A state of pessimism stares us in the face, and there is sufficient cause for it. With lead at 3.75 cents, copper about 8 cents, zinc under 4 cents, -and silver about 26 cents, one has a reason for grief. When we think of the 5000 or less men employed in Utah mines and smelters in 1931, as compared with about 12,000 in 1929, the calamity seems even worse. In view of the present economic crisis we cannot even think of new men getting a job.

Jan 1, 1931

By E. L. Tucker

PREVIOUS investigations of E. L. Tucker and R. E. Head' related in particular to the effect of cyanogen compounds on galena, sphalerite, and pyrite, and their behavior in the presence of such com- pounds. The present contribution deals with the float- ability of the copper minerals and of pyrite. All experiments were performed on minus 200-mesh material, the method of preparation being the same as that described in the previous article. Flotation tests were made in a small mechanically agitated machine of 50-g. capacity designed in this laboratory by Gates and Jacobsen." General Naval Stores pine oil No. 5 and potassium xanthate were used in constant amount in all of the experiments; these agents were selected because of their present wide use in the flotation of copper minerals. Lime additions were varied to secure four 'final hydrogen ion concentrations, namely, neutral, 0.0005 per cent CaO, 0.003 per cent CaO, and 0.02 per cent CaO. The variation in lime addition was introduced to include the range of alkalinities employed in actual flotation practice.

Jan 1, 1926

By H. E. Cook

The Ising model for the internal energy of a binary alloy has been used to obtain a general equation for the critical resolved shear stress of partially ordered crystals. The equation expresses the stress in terms of the Warren "alphas " and can be used to estimate the variation in strength with order without the assumption, present in the original formulation of this problem in terms of domain size, that order is complete within each domain and that the domains are of ungorm size and shape. In addition, it is the general equation, according to the Ising model, for strengthening by short-range order. Two applications of the equation are considered: One is an estimate of the variation in strength of CkAu with long-range order. The other is an estimate of the variation in strength of FeCo with quench temperature. Reasonable agreement is found with the variations reported in the literature. When the internal energy of an alloy crystal depends upon the distribution of solute, the strength of the crystal will also depend upon it because a portion of the applied stress for plastic deformation will be: where V is the volume of the crystal and E(E) is the energy change associated with the solute redistribution caused by the plastic strain, E. We expect T to equal zero for a crystal having a random arrangement of solute because the arrangement would remain random after plastic deformation. Likewise, we expect it to equal zero when the crystal is perfectly ordered because the motion of paired dislocations found in such crystals does not disrupt order. However, when short-range order exists or when long-range order is incomplete, plastic deformation will decrease the amount of order and additional work, proportional to the ordering energy, will be expended. Fisher' estimated T for crystals having short-range order by assuming an interaction energy between neighboring atoms and estimating the change in the number of unlike neighbors as a dislocation moved through the crystal. (His analysis was limited and several workers2"6 have since given more complete ones.) Fisher minimized the importance of a strengthening mechanism of this type for paired dislocations in a structure having long-range order. ~ottrell,' however, pointed out that T could be appreciable for ordered crystals having antiphase domains. He attributed the strengthening to the increase in surface energy of the domains as they were cut by paired dislocations. Ardley,' in his test of Cottrell's theory, found that r for Cu3Au crystals obeyed the equation: for 1 > t where 1 is the domain size, t is the domain wall thickness, and y is the surface energy of an antiphase boundary. His experiments represent the classic confirmation of the strengthening mechanism proposed by Cottrell. However, the assumptions involved in using Cottrell's theory are valid only for large domain size in CU~AU,~"~ i.e., when Eq. [2] reduces to: For small domains, ~linn~ has questioned Ardley's assumption that order was complete, and, indeed, Stoloff and ~avies" fpnd it incomplete until a size of approximately lOOA was reached. Even when the order within a domain is complete, it is not obvious how one determines the appropriate value for I in a structure where domains vary in size and are irregularly shaped. The purpose of this paper is to estimate T without restrictions upon the degree of order and domain shape. Our major assumption will be the use of a generalization of the model proposed by Bethe" (the Ising model) for the internal energy. This will in fact allow us to combine the theories for strengthening by short-range order and by antiphase domains into a single, general formalism. We will use the results to estimate the variation in strength of Cu3Au crystals with long-range order8 and the variation in flow stress of FeCo crystals with quench temperature.12'13 INTERNAL ENERGY For simplicity, we restrict our considerations to those binary solid solutions which can be described as an arrangement of atoms on a Bravais lattice. An atom site will be indexed by three numbers (PI, pz, p3) determined by the vector: from the origin fixed at atom (0, 0, 0) to the atom site where a', an, and a, are the lattice translation vectors. We write: For the energy of the crystal where pi(p) is the probability (either zero or one) of finding an atom of type i (i = 1, 2) at site (p), which is shorthand for (pl, pz, p3) and pj(p + r) is that for an atom of type j (j = 1, 2) at the site (p + r), which is shorthand for (pl + rl, The coupling parameter, resents the energy associated with the pair Pi(p), Pj(p + r). The crystal is assumed large enough so that surface effects can be neglected; therefore, trans-lational and inversion symmetry require the coupling parameters to obey the relations

Jan 1, 1969