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By J. Jones-Parra, J. C. Calhoun

INTRODUCTION The purpose of this paper is to present the results obtained by solving the fractional flow' and frontal advance' equations to obtain oil recovery at water breakthrough as a function of the length of the system being flooded, the linear velocity of injection and the displacing phase viscosity. The method of solution makes use of the stabilized zone concept first presented by Terwilliger, et al3 when calculating saturation distributions in a gravity drainage system. In order to check the calculated results, the equations and method of solution were applied to the system on which Rapoport and Leas' reported experimental data. METHOD OF SOLUTION The solution of the fractional flow equation and the frontal advance equation is obtained by assuming that a water front forms to sweep the oil out ahead of it. There is ample evidence that under the proper conditions a water 'front does form. If this water front is not to dissipate, the water saturations which comprise it must have equal linear velocities. These water saturations are said to comprise the stabilized zone. Equation (2) indicates that for all these saturations and hence fw must plot as a straight line against saturation for those water saturations comprising the stabilized zone. The water saturations lying between the highest in the stabilized zone and the highest that can be achieved in the porous medium comprise the variable zone. Regardless of the numerical value of the length of the system, the variable zone must eventually span all of it, so that for long enough systems in Equation (1) will be small enough to be neglected. Hence, for the saturations in the variable zone Equation (1) can be simplified to In order to determine the composite fw function applicable to the entire range of saturations in the porous medium, it is necessary to find the water saturation which separates the stabilized zone from the variable zone. This dividing saturation must be on the curve defined by Equation (4), and a tangent curve at that point must pass through the initial water saturation to satisfy Equation (3). Hence, the valid fw curve is given by a straight line starting at the initial water saturation, tangent to the curve of Equation (4): and along the curve of Equation (4) from the point of tangency on. The point of tangency determines the water saturation dividing the zones. The length and shape of the stabilized zone can be found by solving Equation (1) for Integrating this expression numerically between any two saturations in the stabilized zone gives the distance separating these two saturations. By repeated integration, the saturation distribution of the stabilized zone can be determined. The total length of the stabilized zone is given by The integral appearing in Equation (5) is a function of the viscosity ratio, the capillary properties of the rockfluid system, the relative permeabilities, and the initial water saturation. Considering these variables fixed and designating the value of the integral by writing caw for and v/60 for q/A; Equation (5) becomes The saturation distribution of the stabilized zone can be integrated over its length and divided by the length to obtain the average water saturation The volume of water in the stabilized zone is then given by The average water saturation in the variable zone can be found by extending the straight line segment of (the segment joining Sw and Sw) until it intersects fw = 1.00. The saturation at the intersection is. This method of finding is nothing more than the graphical solution of the equation The volume of water in the variable zone at breakthrough is then given by Assuming no initial water saturation, the recovery at breakthrough expressed as percent pore volume is given by

Jan 1, 1953

By R. E. Grace

WHEN an Fe-C alloy, partially reacted to pearl-ite, is quenched rapidly enough to suddenly stop the growth process, it may be expected that any carbon concentration gradients will freeze in situ in the newly formed martensite. A short tempering treatment should precipitate finely dispersed iron carbide particles in approximately the same distribution in which carbon existed during the growth process. Examination with the electron microscope of areas of tempered martensite adjacent to the pearlite-martensite interface should reveal the existence of preferred areas of precipitation. Fig. 1 is an example of an electron micrograph used in this study. Small sections of a commercial steel (0.9 pct C, 1.6 pct Mn, 0.25 pct Si) were austenitized at 945°C in air for 45 min, furnace-cooled to 697°C, iso-thermally reacted to pearlite for 30 min, and quenched into iced brine. Two samples were tempered 5 and 20 min, respectively, in air at 498°C and were subsequently quenched in water. The samples were mechanically polished and etched in 1 pct picral. Collodion replicas were made by the conventional technique; these were shadowed with chromium at about 650' from the horizontal, and electron micrographs were taken at a linear magnification of 14,200X. Prints were made at a magnification of 44,100X. Lineal analysis of the prints was made as a function of the interlamellar spacing and volume percent of iron carbide. Straight lines were drawn into tempered martensite between adjacent lamellae at a distance S/4 or S/6 apart, where S is defined as the interlamellar spacing measured at the pearlite-martensite interface. The volume percent of iron carbide was measured along each of these traces with a "gage length" of S/10. The results of analyses of two samples are shown in Table I. The area included between the S/2 and S distance from the interface could be considered homogeneous within the sensitivity of the measuring technique. This value was used as a proportionality factor to evaluate an arbitrary carbon content of homogeneous austenite, and also as a parameter by which error in measurements could be evaluated. Results of the lineal analysis are shown in Figs. 2 and 3. Fig. 4 shows the statistical nature of the interlamellar spacing for all lamellae measured in this

Jan 1, 1954

By R. A. Swalin, B. W. Roberts

ONSIDERABLE controversy has centered about the existence of an order-disorder transformation in alloys in the composition vicinity of 75 atomic pct Ni and 25 atomic pct Cr. All the evidence to date for such a transformation is of a rather indirect nature, since direct X-ray confirmation is difficult to obtain as a result of the small difference in X-ray scattering factors between nickel and chromium. As a result, superlattice lines would be very faint if present. Several investigators1-5 have observed anomalies in the electrical resistivity vs temperature curves of alloys in the vicinity of 25 atomic pct Cr. The observed change is characterized by a rather broad resistivity peak at about 540°C. A discontinuous drop in resistivity is normally associated with ordering. However, Taylor and Hin-ton4 have rationalized that the discrepancy could conceivably be caused by changes in the Brillouin zones upon ordering. The same investigators measured the specific heat as a function of temperature and have also observed anomalies in the vicinity of 540°C. Nordheim and Grant5 have proposed, because of the absence of superlattice lines and the absence of a discontinuity in the resistivity vs temperature curve, that long-range order is improbable and that the results could be better explained by short-range ordering. To date, however, the question has not been resolved satisfactorily. Neutron diffraction techniques provide a method for partially resolving the question since nickel and chromium have considerably different atomic scattering factors in the case of neutrons and, therefore, superlattice peaks, if ordering occurs, should be readily discernable. The results of a neutron diffraction investigation* coupled with a dilatometric in- vestigation are reported here. An alloy containing 28.0 atomic pct Cr was prepared from carbonyl nickel and electrolytic chromium in the form of a 1-in. diam x 10-in. ingot. Melting was done in an alumina crucible and casting in a graphite mold. The ingot was homogenized at 1250°C for seven days, after which a 5-in. long piece was cut into shavings in order to obtain a suitable neutron diffraction sample. Hardness samples and samples suitable for dilatometric analysis (% in. diam x 2 in.) were prepared from the remainlng piece. The neutron diffraction sample, along with a hardness sample, was annealed at 501°C for seven days, at which time the temperature was lowered to 479°C and held at this temperature for twenty-four more days. The hardness was observed to increase from 133 to 147 Vhn during the course of the annealing treatment, in qualitative agreement with Nordheim and Grant." Neutron diffraction traces taken at room temperature and at 42 °K are shown in Fig. 1. The insert shows the trace taken at room temperature with short counting times and thus only fair statistical accuracy. No evidence for superstructure reflections of Cu3Au-type ordering is observed. The statistical accuracy of the 4.2°K trace was improved by using longer counting periods. Several weak intensities are observed at small angles, including an aluminum reflection from the cryostat assembly and a very

Jan 1, 1958

By C. F. Knutson, Martin Felsenthal, J. D. Lung, J. D. Sudbury

This paper has been written to summarize the laboratory and field studies leading to injection of Pacific Ocean water into the Third Grubb formation. Laboratory tests for studying the permeabilities of cores to Pacific Ocean water are described. Resu1ts of these tests showed that the water could be successfully injected if it is chertzically stable, sterile, and properly filtered. Clay swelling was shown to be a comparatively minor factor in permeability reduction. Studies leading to development of an economical system for properly conditioning the ocean water are discussed. It was necessary to control pH, calcium carbonate deposition, bacterial growth, iron solubility. oxygen, and suspended .solids. A relatively simple system combining the use of sulifur dioxide, a quaternary rlmmonium chloride, and filtration was shown to adeqrrately control all these factors. This combination has not previozrsly been used as a water treating technique. Evaluation tests of the water conditioning system prior to injection are described, and results of such tests are presented. Control measures needed to insure a continusously adequate water treatment are reviewed. INTRODUCTION The utilization of sea water for waterflooding has only recently been attempted. Knowledge of treating requirements has been limited, and in most cases no water treating has been done. Prior to the initiation of the Third Grubb zone pilot water flood at Ventura, Calif.,1,2 extensive laboratory research was done in an effort to define the treatment necessary to provide a suitable injection fluid. Close coordination and cooperation between field and research personnel were found to be the foremost requisites in developing the most economical and practical method for treating ocean water. LABORATORY STUDIES Plugging Tendencies of Untreated Ocean Water A program of laboratory studies was initiated for the purpose of evaluating whether ocean water could be injected into the formation for prolonged periods of time without causing plugging of the pore spaces of the formation. Because of the fairly low permeability of the formation (averaging about 40 md), it was realized from the outset that a water of exceptionally high quality was needed. Ocean water used in the laboratory studies was carefully filtered through ultrafine fritted glass filters, having nominal maximum pore sizes of 1.4 microns. Permeability tests were then performed on representative core material from the formation. It soon became apparent in these tests that filtration alone was not a sufficient treatment for the ocean water. Prolonged injection of the filtered water caused rapid plugging of the core samples. The permeability decline of a typical sample is illustrated by Curve A in Fig. 1. The several factors which can contribute to this plugging tendency are briefly reviewed below: Interaction with Formation Clays An evaluation of clay swelling effects on liquid permeability was made using core samples from the Grubb formation. An oil was chosen for determining the reference liquid permeability because it would not tend to react with any clays that might be present in the formation. The samples were then extracted, dried, and resat-urated with either treated ocean water, formation water, or fresh water. Permeabilities to the various waters were then determined. Results of this test series are summarized in Fig. 7. The difference between the oil permeability and water

Jan 1, 1957

By C. E. Westley, R. R. Beebe

Recent works by Plaksh et al.,1,2 Glembotsky,3 and others have renewed interest in the effects of oxidation on the surface activity of the sulfides. Since considerable supporting data are available on galena-potassium ethyl xanthate, this system is being studied at South Dakota School of Mines and Technology by contact angle measurement, using the technique of McGlashan and McLeod.4 The contact-angle method is well adapted to investigation of the influence of small amounts of oxygen on the establishment of a stable contact angle on galena in potassium ethyl xanthate solution. Samples of massive galena containing less than 0.001 pct Cu, 0.001 pct Ag, and 0.01 pct Zn are mounted in castable plastic and ground and polished manually in a glove box containing the desired proportions of oxygen and purified nitrogen. Oxygen and nitrogen contents of the test atmosphere are determined with an Orsatt analyzer. In this way a reproducible surface is created in contact with whatever oxygen concentration is desired. Water and all reagents are prepared and stored in the test atmosphere. An absorbent is used to control carbon dioxide that might be present. Final polishing is done with levigated alumina on a glass plate. The prepared sample is placed in the test solution and transferred to the contact angle apparatus in a sealed box, and the test is conducted while both sample and solution are maintained under a slight positive pressure of the test atmosphere. Table I gives the results of contact angle measurements made at various 02-Nz concentrations.5 Each value of contact angle is the average of at least six individual measurements made with different bubbles at different locations on the sample surface. A period of 20 to 25 min has been found by preliminary tests to be sufficient to permit develop- ment of maximum angle, and all measurements are made after at least this much time has elapsed. Exposure of prepared samples to pure water under the test atmosphere before exposure to the xanthate results in a small but measureable and consistent reduction in contact angle. This is particularly true in the oxygen-deficient atmospheres; for example, at 2.2 pct O2 such water contact reduces the average angle about 8" from that obtained by immediate testing in xanthate. The general trend of increasing contact angle with increasing oxygen concentration corresponds to the results of Plaksin et a12 The absence of diminish-ment of contact angle at very high oxygen concentration has been attributed to the high concentration of xanthate per unit of galena surface: At oxygen concentrations below 1 pct (the limit of accuracy of the Orsatt analyzer), no contact angle is obtained. Furthermore, no contact angle is obtained at any oxygen concentration in the absence of xanthate. This result seems to be in disagreement with those cited by Plaksin.1 Further work on the interaction of galena and potassium ethyl xanthate or a similar collector as influenced by oxygen is contemplated. The true nature of the interrelation is as yet unknown. For example, it is not known whether the governing oxygen reaction is with the mineral, with the reagent, or with both.

Jan 1, 1961

By C. G. Dunn

The measurement of interfacial teusions or grain boundary energies in metals is an important metallurgical problem, especially since it bears on growth and nucleation processes. Statistical methods of measuring equilibrium grain boundary angles, which relate interfacial tensions, have been used by some investigators1'2 but more direct methods would be preferred. A direct way, based on a simple extension of a successful method of growing single crystals of silicon ferrite to predetermined orientations in flat specimens,3 is to produce three-grain specimens in such a way that the equilibrium common grain boundary will be perpendicular to the surface of the specimen. The angles to be measured then appear as the grain boundary angles in the surface of the specimen. The method of producing a single crystal specimen with a predetermined orientation is basically the same as that employed by Yoshida and Fuji-wara4 in their work on aluminum wires. The specimen, prior to any single crystal transformation, must have a matrix which will support the growth of a crystal of the desired orientation. Growth may occur through exaggerated grain growth or, if the matrix is strained a critical amount. by normal recrystallization. In either case the matrix must have stability at the temperature used to effect the transformation, this stability appearing as a long induction period for exaggerated grain growth or for the formation of recrystallization nuclei. Additional control is gained by using a high temperature gradient through which the specimen is moved at a suitable rate. To obtain the desired orientation, the specimen is fed a short way into the furnace producing one or more crystals in the hot end. After the specimen is removed from the furnace, the orientation of a crystal is determined, so that this crystal may be reoriented by properly deforming a narrow piece of the matrix between the end crystal and the rest of the specimen. This part of the specimen in the case of silicon ferrite is preferably held at red heat during the reorienting process. Fig 1 shows a specimen which is ready to be transformed into a single crystal through growth of a reoriented "seed" crystal. When the method just described is applied to the production of a three-grain specimen two crystals are grown first side by side in one end, the specimen is then reversed and the third grain grown from the opposite end. A suitable group of grains each of different orientation is one where all three grains have the same crystallo-graphic plane in the plane of the specimen. Fig 2 shows a photograph of a three-grain group prepared as described. The notches are introduced in the open ends of the grain boundaries to anchor them during the final annealing operations which ultimately should bring the grain boundaries to the lowest energy configuration. Two-grain groups with different crystallographic planes in the plane of the specimen can be similarly prepared and treated. If the two grains have different surface (gas to metal) energies per unit area, the equilibrium condition leads to a curved grain boundary.* * This feature of the problem was pointed out by J. c. Fisher. References 1. Cyril Stanley Smith: Grains. Phases and Interfaces: An Interpretation of Microstructure. Trans. AIME 175, 15. Met. Tech. TP 2387, June 1948. 2. D. Harker and E. Parker: Grain Shape and Grain Growth. Trans. A.S.M. (194,5) 34, 156-195. 3. C. G. Dunn, A. Seddon, and R. Wylie: Unpublished Work, 1946-1947. 4. Takeo Fujiwara: A Method of Producing A Long Single-Crystal Wire of Aluminum with any Desired Crystallographic Axis. Jnl. of Sci. Horoshima Univ. A, (1939) 9, No. 3, 227.

Jan 1, 1950

By M. E. Fine

THE electrical conductivity (a) and the thermal conductivity (K) of metallic conductors, as demonstrated by Smith and Palmer1 and Hume-Rothery,2 are to a good approximation related by an empirical equation, K = A s T + B, in which T is the absolute temperature, and A and B are constants, the Lorenz constant and the lattice con- ductivity. Strictly speaking, the lattice conductivity is not a constant, but to some extent depends upon the electronic conductivity.3 By plotting K as a function of sT, this equation has been proved to a

Jan 1, 1951

By J. H. Mallery, J. H. Keeler

THE crystal structure of Zral, the compound richest in zirconium (91.03 pct Zr by weight) in the system Zr-Al, has been found to be ordered face-centered-cubic. This compound is of particular interest, since it is encountered as the precipitated phase in potentially useful zirconium-rich alloys. It forms peritectoidally from body-centered-cubic zirconium and Zr,Al at about 975°C. The Zr-A1 constitution diagram has been reported by McPherson and Hansen.' For the primary determination a 100 g ingot was prepared by arc-melting in a water-cooled copper crucible under purified argon. The raw materials were crystal-bar zirconium (a purity of better than 99.7 pct) and spectroscopically pure aluminum. The ingot was double melted, examined metallographi-cally, and found to be quite uniform as shown in Fig. 1. The hardness of the as-cast button was VHN 390 to 445. A 20 mil diam specimen was prepared from annealed (900°C) filings of Zr,Al. The specimen was mounted in a 5 cm radius powder camera and an 8 hr exposure was taken using copper radiation with a nickel filter. The structure of the compound Zral was found to be ordered face-centered-cubic of the Cuau type, with an a,, value of 4.37220.003A. Its calculated density is 5.976e0.013 g per cu cm. The relative intensities for the lines were calculated using the formula 1+cos2 20 I oc----------------- • p • /F7 sin2 0 cos 0 1 + COS' 20 where----------------is the Lorentz polarization fac- sin2 0 cos 0 tor, p is the multiplicity factor, and F is the structure factor. The X-ray data are given in Table I. The compound has exhibited plasticity when present as particles in a matrix of a zirconium. During hot working at about 900°C, the compound has been found to deform plastically to the extent that some of the particles exhibit a length-to-width ratio greater than 4:1 as shown in Fig. 2. In answer to the discussion of ref. 1, McPherson and Hansen pointed out that the compound may be cold rolled and reduced as much as 33 pct. Zr,Al has also been found2 under certain conditions to precipitate in lamellar form at grain boundaries of the a phase of hexagonal-close-packed zirconium. Acknowledgment This work was carried out under Contract W-31-109-Eng 52 for the United States Atomic Energy Commission.

Jan 1, 1956

By F. H. Ellinger

TI HE crystal structure of neodymium has been re-A ported by Quill1 and by Klemm and Bommer"s to be hexagonal-close-packed, Strukturbericht-type A3, with a, = 3.650.4, c, = 5.890.4, c/a = 1.614. Klemm and Bommer,% owever, pointed out that the observed intensities of the lines of the neodymium powder pattern agreed only in part with the intensities calculated for the type A3 structure, and that the neodymium powder pattern showed extra lines which could not be traced to impurities. They found that these extra lines could be accounted for by doubling the c-axis, but apparently they did not consider their evidence to be sufficiently conclusive to justify reporting the doubled c-axis. In the present work, the structure of neodymium was found to be double hexagonal-close-packed. The neodymium metal used in the present investigation was obtained through the courtesy of F. H. Spedding of the Ames Laboratory. Its major impurity was stated to be 0.03 pct Ca. The author's spectrographic analysis indicated high purity and carbon was determined to be less than 30 ppm. X-ray powder specimens were prepared from filings stress relieved in vacuum at 200°C for 50 hr. Powder photographs were taken in an 114.6 mm diam camera (Straumanis film mounting) at room temperature and in a 19 cm diam Unicam high temperature camera at elevated temperatures. Copper x-radia-tion was used for all the photographs. The powder pattern of neodymium was found to index to a hexagonal unit cell with dimensions, as determined from the high angle Ka doublets, of: a.= 3.655+0.001.4; c, = 11.796+0.004.4; and c/a = 3.227 at room temperature. The measured density of neodymium shows that this cell contains 4 atoms, the calculated density is accordingly D, = 7.02 g per cu cm. The diffraction data for the first 15 lines of the powder pattern are listed in Table I. In a double hexagonal-close-packed unit cell, the 4 atoms would be in the positions (0,0,0), (0,0,%), (1/3, 2/3, 1/4), and (2/3,1/3,%) of space group P 6,/mmc. On the basis of these positions, the relative intensities of the reflections were calculated using the expression: 1+cos2 20 I oc \Frp 1 ' sin2 0 cos 0 where F is the structure factor and p is the multi- plicity factor. Since absorption was not taken into consideration, comparison of observed and calculated intensities should be made only for neighboring reflections. Each atom has 12 neighbors, 6 in the basal plane at a distance of 3.655.4 and 3 above and 3 below at the closer distance of 3.626.4. These distances correspond to an average radius of the neodymium atom of 1.820.4. The description of this structure in terms of A, B, and C close-packed layers of atoms is ABAC ABAC. ...... The structure repeats every fourth layer and the stacking sequence of the second, third, and fourth layers corresponds to cubic-close-packing of atoms. A series of powder photographs taken at successively increasing temperatures up to 680a10"C revealed that the double hexagonal-close-packed structure is stable up to this temperature. Acknowledgments The authw wishes to thank W. H. Zachariasen and A. S. Coffinberry for their helpful comments on this work and also Marion Gibbs for measuring the diffraction films. This work was sponsored by the United States Atomic Energy Commission.

Jan 1, 1956

By C. B. Jordan

LAVES and Wallbaum' have described TiRu and TiOs as having the same structure as TiFe; they had previously described the latter compound as having the CsCl structure. On the other hand, a careful investigation of the structure of TiFe by Duwez and Taylor- ailed to reveal the ordered CsCl structure, finding instead a random arrangement of titanium and iron atoms on the points of a body-centered-cubic space lattice. Consequently, it is not clear whether Laves and Wallbaum proposed the ordered or the disordered structure for TiRu and TiOs. In order to settle this question as to the structural type and to obtain values of the lattice para- meters (thereby completely determining the structures), the present investigation was undertaken. Samples having the compositions TiRu and TiOs were weighed out accurately from iodide titanium supplied by the New Jersey Zinc Co., ruthenium powder from Goldsmith Brothers Smelting & Refining Co., and osmium powder from Wildberg Brothers Smelting & Refining Co. No analysis was made of any of these metals. According to the respective analyses of the suppliers, total impurities amount to less than 0.03 pct in the titanium and to less than 0.5 pct in the osmium. The purity of the ruthenium is not known. The alloys were prepared by arc melting in a furnace of the type described by Schramm, Gordon, and Kaufmann.9 pecimens for X-ray diffraction were obtained by crushing the ingots and passing the resulting powders through a 200 mesh screen. Diffraction patterns were made with a powder

Jan 1, 1956

By Edward I. Salkovitz

RECENTLY a set of crystallographic angles for bismuth was calculated and a standard (111) projection was constructed. Since these calculations

Jan 1, 1957

By A. Taylor, S. Leber

WHEN studies of deformation texture, lattice stress, and crystal orientation are made, it is frequently required to know the interplanar angles and subsequently to embody the results of the investigation in a stereographic projection. Tables of

Jan 1, 1955

By Edward I. Salkovitz

THE determination of the orientation of metal single crystals and the studies of plastic deformation are greatly facilitated by the use of the stereographic projection. To draw a standard projection of the crystal it is necessary to have a tabulation of the angles between the various crystal-lographic planes of the particular crystal studied. Bozorth' has published such a table for the cubic system which can be found in many reference books; such convenient tabulation for the hexagonal metals, however, does not appear to be in the literature, although scores of workers have calculated these angles. Therefore, Table I of angles for magnesium, zinc, and cadmium has been compiled. Since the c/a ratio is different for each of these metals, three separate calculations are necessary. The angles 4 between (HKIL) and (hkil) were calculated by means of the formula: Hh+Kk+½(Hk + hK)1+~3aLl cos The c/a ratios were calculated from data given in the 1948 edition of Metals Handbook. Fig. 1 is a typical (0001) standard projection of zinc. For all practical purposes, it serves for cadmium as well, as can be seen from the table. For magnesium, however, it is necessary to construct another net. R. M. Bozorth: Physical Review. (1925) 26, 390.

Jan 1, 1952

By W. J. Romanow

RECENT papers by Williams, Sherwood, and Boothby,1 Seybolt et al.2 Heikes; and Roberts4 attest to the increasing importance of the hexagonal ferromagnetic compound MnBi. Since the magnetic properties of MnBi crystals are highly anisotropic, it is useful and necessary to have appropriate aids for determining their orientations—a (00.1) standard stereographic projection and a corresponding table of crystallographic angles for this structure. The tables of crystallographic angles for the hexagonal system available in the literature5 are computed for c/a ratios greater than that of MnBi and cannot satisfactorily be used to find the crystallographic angles of MnBi. A table of angles between the crystallographic planes (HK.L) and (hk.1) was calculated for MnBi from the formula0 F=are cos Hh+Kk+ 1/2(Hk+hK) + 3/4 (a/c)2Ll/v(H2 + K2 + HK + 3/4 (a/c)2L2) (h2+ k2+hk + 3/4 (a/c)2l2) on an IBM Type 650 Magnetic Drum Data Processing machine. The lattice constants used are listed in Table I and are those determined by Willis and Rooksby7 at 20°C. More recently published values obtained by both X-ray and neutron diffraction"' show no difference from these significant to the calculation. From the crystallographic angles between the basal plane (00.1) and the general plane (hk.1) in Table I, an (00.1) standard stereographic projection was made for MnBi which is shown in Fig. 1. Acknowledgments The author wishes to thank R. G. Treuting and J. N. Hobstetter for their technical assistance, and Miss D. P. Booth for programing the IBM machine for the calculations. He appreciates the opportunity to read the paper of B. W. Roberts of General Electric Co. prior to its publication. References 1H. J. Williams, R. C. Sherwood, and 0. L. Boothby: Bulletin Amer. Physical Soc., 1956 vol. 1, p. 132. 2 A. U. Seybolt, H. Hangen, B. W. Roberts, and P. Yurcisin: AIME Trans., 1956, vol. 206, p. 605: Journal oF Metals, May 1956. 3R. R. Heikes: Physical Review, 1955, vol. 99, p. 446. .B. W. Roberts: A Neutron Diffraction Study of the Structures of Manganese B.W.Roberts:ANeutronDiffractionBismuthide, Physical. Review, 1956, vol. 104, p. 607. 5 A. Taylor and S. Leber: AIME Trans., 1954, vol. 200, p. 190; JouRNal of Metals February 1954. « C. S. Barrett: Structure of Metals, 2nd Ed., p. 635. McGraw-Hill Book Co. New York and London, 1952. 7B. T. M. Willis and H. P. Rooksby: Proceedings Physical Soc., London, 1954, vol. 67B, p. 290.

Jan 1, 1958

By R. B. Russell

SINCE Cahn's¹ paper on plastic deformation in orthorhombic (a) uranium an increasing amount of work is to be expected on orientation of single crystals and preferred orientation in this metal. It would seem desirable to have a table of angles between crystallographic planes (hkl) and (HKL). Table I shows these angles as distances from (loo), (0l0), and (00l), and in Fig. 1 as a standard (001) stereographic projection. Only the locations of poles whose diffracting planes produce Cu Ka reflections up to 0 = 70.5" (i.e. (154) ) are given." The relative reflection of these planes are represented by circle diameters. It is hoped that the projection may also be used in making inverse pole figures, originally suggested by Harris.' Values of 4 were calculated from the expression hH kK 1L cos = hH/a²+kK/b²-lL/c² where the lattice parameters a, b, c, and the spac-ings d were taken from the work of Jacob and Warren." (They report a = 2.852 kX (2.8578 A), b = 5.865 kX (5.8768 A), c = 4.945 kX (4.9550 A).) The parameters reported by them may be subject to revision,4,5 accordingly the interplanar angles are listed to only the nearest tenth of a degree. References ¹R. W. Cahn: Acta Metallurgica (1953) 1, p. 49. ¹G. B. Harris: Philosophical Magazine (1952) 43, p. 113. 3C. W. Jacob and B. E. Warren: Journal ACS (1937) 59, p. 2588. 4J. S. Lukesh: Acta Crystallographica (1949) 2, p. 420. ³ J. Thewlis: Nature (1951) 168, p. 198.

Jan 1, 1954

By J. F. Nicholas

THE angles between the crystallographic planes in cubic metals were originally given by Bozorth and have been republished many times. Recently, Salkovitz2 tabulated the angles for the hexagonal metals (magnesium, zinc, and cadmium). he most commonly used metal which does not fail into either of these groups is tin, which has a tetragonal structure with an axial ratio of c/a = 0.54558 (99.99 pct purity, 25°C).² ccordingly, Table I has been prepared giving the possible values for the angle, 4, between planes of the types {HKL) and {hkl). The formula used was: Hh + Kk + LI — cos? = —--------------—-----------— A [(H2 + K2 + L2-a2c2(h2 + k2 + r;)] All angles are given correct to the nearest minute as this is sufficient for all practical purposes. Every angle depends in a different way on variations in the value of c/a, but even in the worst cases a variation of about 0.0003 in this ratio is needed to vary an angle by l', which corresponds to a temperature change of 20 °C or can be caused by impurities.

Jan 1, 1952

By C. J. McHargue

THE angles between the crystallographic planes in cubic crystals have been given by Bozorth,¹ in magnesium, zinc, and cadmium by Salkovitz,² and in tin by Nicholas. The determination of the orientation of single crystals and the study of plastic deformation require the knowledge of these angles and are often facilitated by use of a standard projection. Tables of these angles and standard projections have not been published for titanium or zirconium which have c/a ratios considerably less than those hexagonal metals for which this data is available. Accordingly, Table I has been prepared giving these angles. Fig. 1 presents a standard (0001) projection of titanium. The angle between (h1k1i1l1) and (h2k2i2l2) was calculated by means of the formula: h1h2 + K1K2 + 1/2 (h1k2 + h2k1 + cos 4 = [(hi2 + K + h1k1 + 3/4 l12 (h2² + C" a² 3/4 — l1 l2 C² k2²+ h2k2 + 3/4 —l2²) ] 1/2 c² The c/a ratio for titanium is taken from the data

Jan 1, 1953

By P. Pietrokowsky

RECENTLY a comprehensive investigation of the various sigma binary alloys was published by Duwez and Baen.1 It was shown that the following conditions are necessary for the formation of the sigma phase: (1) the difference in atomic diameters of the two alloying metals must not be more than about 8 pct; (2) of the two elements which form sigma, one must exhibit a body-centered cubic structure while the second metal must be face-centered cubic, at least in one of its allotropic forms. In addition, a tetragonal unit cell was proposed and powder patterns of six sigma phases were indexed for d spacings greater than 1.640 kx. Since the indexing, based on tetragonal symmetry, led to only fair agreement, the possibility of sigma being orthorhombic was investigated. Various methods for indexing orthorhombic powder patterns are de- scribed in the literature.' The method used in this investigation was a graphical examination of the differences in values of sin' 9 for the adjacent lines of the powder diffraction pattern. This method led to a complete indexing of the Cr-Co (60-40) pattern. Because of the strong similarity between the various sigma alloys, it was then possible to assign indexes to the patterns of Fe-Cr (50-50), Fe-Cr-Mo (30-30-40), Fe-V (50-50), Co-V (50-50), and Ni-V (50-50). The values of sin' 9 and indexes of the six sigma alloys, together with visually estimated intensities, are presented in table 1. In general, the similitude between the six patterns is quite good. For spacings greater than 1.75 kx a few differences are noticeable. In the Ni-V pattern, no (320) exists and the two strong lines (030) and (031) are separated by only two lines of lower intensity, unlike any of the other sigma patterns. However, very little is known about the Ni-V phase diagram and it is possible that the equal atomic percentage composition in this system lies in a two-phase region. At lower spacings there is a close analogy between the patterns. The planes

Jan 1, 1951

By Martin F. Littmann

FOR magnetic devices employing reactors with high rates of flux change very thin magnetic tapes have been employed. One of the more interesting of

Jan 1, 1957

By T. P. Meloy

In a recent paper, Gaudin and Meloy1 derived a size distribution function for single, impact fracture. Mular2 extended the use of this size distribution to describe the output of carefully controlled grinding experiments. As it stands, the function is difficult to compute. Consequently, a series of tables of the function are presented below: where M(x) is the cumulative fraction finer than size x when x is particle size. Weight percents are tabulated for the single parameter, r, in three different size ranges: from 0 to 1, from 1 to 10 and from 10 to 100. Tables I, II and III are cumulative percent finer and Tables IV, V and VI are for weight percent passing screen size x, retained on the screen below. A number of conditions from the table are plotted in Figs. 1 and 2. Fig. 1 is a plot of the cumulative percent finer vs. size, and Fig. 2 is the weight per-

Jan 1, 1963