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By E. F. Adkins, R. W. Douglas

The variation of the spectral emissivity of osmium and ruthenium with temperature can be expressed by the following relations: DURING a study of the sintering characteristics of osmium and ruthenium, discrepancies in the previously accepted melting points of the metals were noted. The melting temperatures had not been measured at the time of the work, but were estimated in the literature as 2700°C for osmium and 2450°C for ruthenium.' A recent investigation by Baird2 has shown the melting temperatures of osmium and ruthenium are 3000" and 2250° C, respectively. In the present work, osmium bars sintered at temperatures as high as 2900° C showed no evidence of melting, whereas ruthenium showed definite evidence of melting at temperatures as low as 2250°C. In view of the apparent discrepancies, the melting temperatures of both metals were determined. The spectral emissivities of both metals were determined as an aid to studies of the sintering characteristics of these metals. MATERIALS AND PROCEDURES The materials used in these studies were metal powders supplied by Englehard Industries, Inc., through the courtesy of The International Nickel Co. Analytical data from three laboratories, although not in complete agreement as to the purity of the material or the major impurity elements, indicated that the maximum impurity levels were 0.5 and 1.0 wt pct in osmium and ruthenium, respectively. The metal powders were pressed to 1/4 by 1/4 by 6-in. bars at 20Tsi using carbon tetrachloride as a lubricant. The pressed bars were presintered 1 hr at 1200°C after heating 1 hr at 400°C. The presintered densities of the several bars of the metals ranged from 65 to 80 pct of ideal density. Specimens were self-resistance heated in a vacuum sintering bell. Temperatures were measured with an L&N, disappearing filament optical pyrometer. True temperatures were read at the base of an 0.30-in.-diam, 0.150-in. deep hole drilled into the center of a presintered bar. Apparent temperatures were read from the surface of the bar adjacent to the hole. MELTING POINT DETERMINATION The melting temperature of the metals was determined as the temperature at which a slight additional increase in current through the specimen resulted in a reduced temperature at the base of the hole. The apparent reduction in temperature is a result of the liquid metal filling the hole, thus altering the black-body conditions. The melting points of osmium and ruthenium, based on three deter-

Jan 1, 1962

By Elmars Ence, Harold Margolin

The partial isothermal section of the Ti-Cu-Zr system at 750°C has been studied. The crystal structure of Ti2Cu has been detevnlined as tetragonal and when expressed as face-centered tetragonal, a = 4.164, c = 3.611 A, and c/a = 0.867. TITANIUM systems in which ß-titanium decomposes eutectoidally into a and a compound rather readily are known as active eutectoid systems. Recently in active eutectoid alloys based on the Ti-Cu system, and containing additionally A1 and Zr, it has been found that zirconium may produce both moderate weakening and marked strengthening at elevated temperatures.' In order to obtain some insight into the behavior of zirconium, an attempt was made to determine whether compounds of the type Tix Zr and TixZryCu exist. Only results of the Ti-Zr-Cu studies are reported here since data on Ti-Zr alloys will be presented in a separate publication. EXPERIMENTAL PROCEDURE Alloys were nonconsumably arc melted in argon atmosphere as 15-g buttons, prepared from materials of the following purities: Bureau of Mines electrolytic Ti — Bhn 63 OFHC copper Reactor Grade Zirconium Bhn .C 150 Prior to alloy preparation the electrolytic titanium was premelted to avoid excessive spattering of the charges during melting. The weight losses during melting did not exceed 1 pct of the total weight of the charge. The alloys prepared are listed in Table I together with analyzed compositions. in cases where they were determined. Specimens for heat treatment were wrapped in molybdenum sheet, annealed in argon-filled quartz capsules, and quenched by breaking the capsules under iced brine. About 1 to 2 sec are required to remove the capsule from the furnace, drop it into the iced brine and break it. Because of the insulation effect of the quartz, this delay in quenching should not produce a large drop in temperature. In many instances, no difference or little difference has been detected between the same specimen quartz capsule quenched or directly quenched in vacuum by an impinging stream of water.' Ti-Cu-Zr alloys were homogenized for 48 hr at 825°C prior to annealing at 750°C. Annealing for 5 days at 750°C produced the same results as 11 days at temperature and it was therefore assumed that 5 days was sufficient to establish equilibrium. Titanium-copper alloys were homogenized for 40 hr at 900°C prior to heat treatment for 6 days at 750°C. Specimens for metallographic examination were electropolished and etched with the following solution:

Jan 1, 1962

By R. E. Reed-Hill

The lattice reorientations in (1011) and (1013) twins of pure magnesium have been investigated using polarized light. Both forms (Ire subject 20 almost complete second-order twinning on the (1012) plane of the first-order twin. The observed deviations of composition planes by 6 and 3 deg, respectively, from coherent twinning planes are caused by second-order twinning. The experimental data indicate that (1011) and (1013) are reciprocal twins with a twinning shear 0.136. In a previous paper1 it was shown that the primary fracture mechanism in single crystals of magnesium strained in tension parallel to the basal plane is parting or fracture inside twins. In the temperature interval 25oto286oC, the observed twinning form, dominant in fracture, was a very small twin which formed in bands. Because these twins were very small and appeared in extensively deformed regions they did not lend themselves to X-ray analysis. Lacking Cou-ling and Pearsall&apos;s2 recently reported polarized light technique for magnesium, the twinning plane indices (K1) were originally determined by two surface composition plane measurements. This plane fell in a major zone (axis < 1120>) at an angle of 55.75 deg to the parent crystal basal plane. Because its habit deviated by 6 deg from{l0ll), 61.9 deg, the experimental data per se were not consistent with simple (1011) twinning and, for expedience, the twin was designated with habit plane indices (30341, 54.5 deg, representing the lowest set of indices corresponding to measured angles. Couling, Pashak, and sturkey3 have also found bands of reoriented lattice with a similar habit in polycrystalline specimens of magnesium and certain dilute magnesium alloys. These bands are significant because their formation in large numbers during plastic flow permit specific magnesium alloys to undergo very extensive deformation by cold rolling. Using polarized light2 they showed that the basal plane in the reoriented material was nearly parallel to the band habit and, as a result, proposed a mechanism for band formation involving a double twinning process in which (1011) twins form and then retwin according to (1012). Because of the obvious similarity between the twin bands observed in single crystals and in polycrystalline specimens, it was decided to investigate if the two phenomena were not identical and, if possible, to verify the retwinning hypothesis. The present paper is a report of this investigation and shows that the twins with the (3034) habit correspond closely to retwinned (l011) twins. Evi- dence is also given for an analogous retwinning process in (1013) twins. EXPERIMENTAL PROCEDURE Rectangular single-crystal specimen deformed in tension (stress axis in the basal plane parallel to [1010], as previously described,4 were polished electrolytically by Jacquet&apos;s method5 and etched in acetic picral.2 All twins studied in this investigation belong to a major zone, whose axis, [1210], was perpendicular to both stress axis and two sides of each specimen [(1210) surfaces]. These latter contain the plane of shear of the twins and measurements of the basal plane trace of twins on these surfaces determines the lattice orientations of the twins. The position of the basal plane in each case was determined with a polarizing microscope using a modification of the technique of Couling and Pearsall.2 For each twin the four positions of maximum extinction were measured, the data averaged, and then the pair of extinctions corresponding to the basal plane was determined by rotating the microscope stage and observing the position at which the color changed from orange to blue with a gypsum red, first-order plate inserted between the crossed Nicol prisms. EXPERIMENTAL RESULTS (1013) Twins—In general, especially in speci-mens deformed at 150°C, (1073) twins appear in greater numbers and larger sizes than (1011) twins, and the lattice reorientation inside these twins is therefore easier to study and measure. Fig. 1 shows an area of an unetched and unpolished (1210) crystal surface containing a number of (1073) twins sloping upward to the right at about 29 deg with the basal plane of the original crystal (horizontal direction). Slip line segments may be seen in both the parent crystal (running horizontally) and in the twins (sloping downward to the right at approximately -22 deg with the horizontal). As may be seen in Fig. 2, the slip lines in the twin are not at the proper angle (64 deg from basal plane of the

Jan 1, 1961

By R. W. Armstrong, D. H. Feisel

Self-diffusion in pure crystalline solids has been described through extension of the Cohen and Tum-hull volume -fluctuation model originally proposed for diffusion in simple liquids. It is shown, for a number of crystalline solids, that the self-diffusion coefficient, D, obeys the relationship where a is the lattice spacing, v the Debye frequency, Z&apos; the lattice-volume fluctuation per atom required for a diffusion jump, and (?V/Vo) the relative increase in the lattice volume resulting from thermal expansion between O°K and the temperature under consideration. The calculated value of Z&apos; is dependent upon the difiusion mechanism and the lattice geometry. ThE need for an increased understanding of mass-transport processes in crystalline solids has resulted in much research to determine rates of self-diffusion.&apos; For several metals (e.g., copper, silver, and iron), the experimental values of the self-diffusion coefficient, D, have been determined over a large interval of temperature and the accuracy of D has been investigated through repeated experiments. Also, a number of empirical and theoretical diffusion studies have established a rational basis for understanding the D values for the different elements.2"6 The empirical diffusion analysis by Sherby and simnad6 supplied the main stimulus for the present study. Sherby and Simnad, reviewing the self-diffusion data for a substantial number of pure crystalline solids, suggested that sufficient data were available to establish with some certainty the apparent intrinsic material properties which influence D. Without explicit regard for any mechanism of diffusion, these investigators obtained an empirical relationship for D which correlated the self- diffusion coefficient for each element with the melting temperature, crystal structure, and valence of the particular element. Their analysis is only unsatisfactory in that the effect on D of crystal structure is not very clear and the valence to be assigned to an element is not always straightforward. The present study is an attempt, from a very different viewpoint from that of Sherby and Simnad, to establish a basis for understanding the self-diffusion coefficients for crystalline solids. This analysis was undertaken for the purpose of developing a general expression for D which would explicitly depend on the mechanism of diffusion. The main result of the study is that diffusion in crystalline solids may be described in terms of a volume-fluctuation model which was originally proposed by Cohen and Turnbull7 to describe diffusion in simple liquids. The model appears to be useful for predicting self-diffusion coefficients in crystals. The following sections describe the volume-fluctuation model for diffusion, the application of this model to crystals, and the resultant agreement with experimental data. 1) THE VOLUME-FLUCTUATION MODEL FOR DIFFUSION Cohen and Turnbull7&apos;8 developed their model for liquid self-diffusivity for the purpose of describing certain properties of liquids, glasses, and the liquid-glass transition. A hard-sphere model for the liquid structure is used. In this model, the size of a spherical molecule or atom is fixed for all temperatures by the interatomic spacing at the absolute zero of temperature. At any temperature above O°K, the atom is viewed as being contained within a larger spherical cage defined by the interatomic spacing of the atoms at that temperature. A random distribution of varying cage sizes is assumed to exist in the liquid. An individual diffusion jump occurs when the free volume of an atom (the volume of the cage containing the atom minus the volume of the atom) is large enough for the atom to jump out of its cage. With this physical picture of diffusion in liquids, Cohen and Turnbull derive the following equation for self-diffusivity: D=Doe-yv*/vf [1] where Do is the product of three factors which are the diffusion-path geometry, the diffusion-jump distance, and the velocity of the diffusing atom, v*

Jan 1, 1964

By M. Herman, G. J. London

A series of dilute Fe-Co alloy specimens were prepared from zone-melted iron. The iron used was given ten floating-zone passes in purified hydrogen. Alloying was accomplished by plating a gradient of metallic cobalt onto a 1/4-in.-diam bar of material from a pure plating bath. The deposit was radioac-tively tagged by a small addition of co80 to the plating bath. Alloy homogenization was accomplished by passing an additional floating-zone pass through the plated section of the bar. The final concentration gradient achieved was from residual to about 1000 ppm (0.1 pct) added Co. After homogenization, the bar was swaged to 0.057 in. wire and cut to tensile specimen blanks 1-1/16 in. long. Each specimen was counted in a well-type scintillation counter. The concentration of cobalt was computed by combining the normalized specimen counting rate with the known weight (measured to the nearest 0.01 mg) of cobalt added during plating. Fig. 1 shows the cobalt gradient obtained, exclusive of residual cobalt which was estimated by neutron-activation analysis to be approximately 50 ppm. Results of recrystallization for 1 hr at 750°, 700°, 650°, and 600°C with specimens containing 40, 170, 470, and 870 ppm added Co are shown in Figs. 2, 3, 4, and 5. Recrystallization was accomplished in a dynamic vacuum of less than 1 x 10 torr. The specimens were mounted and polished parallel to the swaging direction. The recrystallization re- sults show a radical coarsening of structure with increasing cobalt concentration. The coarsest structure obtained was at the 470-ppm level (at all recrystallization temperatures) with a slight grain refinement at the 870-ppm level shown. A more normal solid-solution effect is seen in Fig. 5 (60OoC, 1 hr) where the only specimen completely recrystallized is the one containing 47 ppm Co. Since the possibility existed that some atmosphere-contamination phenomenon might be responsible for the observed results the recrystallization series was repeated using sealed quartz capsules. These were filled with purified argon which was gettered during heat treatment on metallic titanium included in the capsules. The results obtained were essentially the same as shown. Another identical alloy was made using a 5-mm-diam bar of Westinghouse Puron. Specimen preparation and recrystallization treatment were also as described. The results obtained (see Table I) were somewhat erratic. Some coarsening in structure does appear at the highest cobalt concentrations. There is no apparent relationship between the coarse, columnar, directionally solidified structure of the as-melted bars and any of the recrystallized grain structures observed, all of which are much finer than the former structure. A recrystallization process can be most simply described as one of homogeneous nucleation of recrystallized grains in a cold-worked matrix followed by more or less uniform growth of the strain free grain into the cold-worked material.&apos; An impurity element may influence one or both of these processes. Aust and utter&apos; have shown that impurities strongly influence grain boundary mobility in lead. Moreover, definite recrystallization textures are produced by small concentrations of tin,3 while additions of silver or gold4 produce large boundary in-

Jan 1, 1964

By J. E. Pavlick, A. G. Guy

The total creep of tin alloys containing antimony in solid solution was observed to decrease with increase in antimony content. However, near the solubility limit an anomalous maximum in deformation was. found, in agreement with similiar observations by Bochvar. IT has long been recognized that alloys may deform more rapidly under stress if there is a tendency for a phase change or other structural change to occur simultaneously. However, the phenomena involved are complex and in some instances the opposite effect, strengthening, may result.&apos; This area of research has not attracted much interest in the western world, and although Russian investigators have studied it for the past 15 years their work has not received due attention here. Recently diametrically opposing views on one aspect of this problem have been espoused by two Russian investigators, Kornilov and Bochvar. The question is this: in certain binary alloys tested in creep at the temperature where the solute is at the limit of solid solubility, does the incipient formation of the second phase strengthen or weaken the alloy? Korni-lov&apos;s arguments in favor of strengthening were presented at the 1954 Symposium at the National Physical Laboratory, =Creep and Fracture of Metals at High Temperatures.The most recent statement of Boch-var&apos;s theory of weakening is in Oding&apos;s new book on creep and rupture.3 Although Bochvar agrees that strengthening occurs in some alloy systems, he argues that in certain alloys an anomalous weakening is observed. However, the extensive work of Korni-lov&apos;s group on creep properties of alloy systems has not revealed any anomaly of this type, but instead has tended to refute Bochvar&apos;s theory. This clear-cut controversy offered a good opportunity for a simple experiment to resolve the problem. The experiment and its results supporting Bochvar are the subject of this paper. However, it will be helpful first to sketch the present status of the larger question of the effect of a phase change or other structural change on plastic deformation. PREVIOUS WORK One of the first researches in this area was by Pfei1,37 who investigated the creep resistance of an 80 Ni-20 Cr alloy containing varying amounts of titanium. He found that the resistance to creep increased rapidly as the solid solubility limit in the alloy at 775&apos;c was approached and slightly exceeded. With further addition of titanium the creep resistance decreased, presumably because of precipitation effects. Bochvar&apos;s first work in this field was a study in 1945 of "superplasticity" in zinc alloys containing 15 to 25 pct Al.4 He found that alloys quenched from about 400 had anomalously low hardness when tested below this temperature. The following year, in a paper on the dependence of mechanical properties on the composition and structure of alloys,5 he extended some observations of Kurnakov to predict that at high temperatures of testing the hardness values for solid-solution alloys might be lower than those for the component metals. He also pointed out the influence of changes in solidus temperature in affecting curves of hardness vs composition, and he

Jan 1, 1962

By H. I-Lieh Huang, O. D. Sherby, J. E. Dorn

RECENT investigations1-4 have suggested that the total plastic strain, C, for high temperature creep under a given stress can be correlated by means of a temperature-compensated time, te- where t is the duration of test, e is the base of natural logarithms, aH is the activation energy for creep, R is the gas constant, and T is the absolute temperature. A typical set of data on which this deduction was based is given in Fig. la. Inasmuch as the e-in t curves for the various temperatures are identical except for their displacement along the In t axis where f and are appropriate functions. If the increase in creep rate with an increase in temperature is exclusively ascribable to a single process involv- ing thermal activation, (T) should equal RT and therefore it = constant [2] where ?= te Assuming the validity of this relationship, ? = ?2 at the same strain, or where the subscripts refer to two alternate test conditions.

Jan 1, 1957

By J. E. McDonald, J. G. Eberhart

A model is discussed from which the work of adhcslon .tor liquid transition metals on aluminum oxide surfaces can he calculated, A close-packed (00011 oxygen surface on A12O3 is assumed with two different types of surface sites: one type involving metal-oxygen bonds and the other man der Waals into actions. The work of adhesion is thus expressed as the sum of these two bonsding free energies. Calculated works of adhesion for nickel, titanium, chromium, and zirconium on sapphire agree well with the experimentally determined quantities. The model is extentled to the calculation of the work of adhesion and shear stress required to remove a thin metal film from a sapphire substrate and is in good agreement with experimental values. The obsewed dependence of the work of adhesion on the free energy of oxide formation of the metal is shown to also provide an interpretation of the tittle dependence of thin-film adhesion. THIS paper presents a model for the type of bonding which occurs across a metal-A12O3 interface. The model is used to explain the results of two types of experiments in which such an interface exists: 1) the adhesion of thin metal films on Al2O3 substrates and 2) the wetting of A12O3 by liquid metal drops. The adhesion of thin films to various substrates has been the subject of a variety of investigations.&apos;-&apos; Benjamin and weaver3 and Bowie,6 using the scratch test developed by Heavens,10 studied the adhesion of metallic films to glass substrates. Their observations for noble-metal film adhesion agree well with an adhesion model involving a van der Waals type of bonding between the film and the substrate. For films of metals whose free energy of oxide formation. ?F°f, has a negative value. Benjamin and weaver3 and Bowie6 found a time-dependent adhesion with an initial value that can be interpreted in terms of van der Waals interactions but a larger terminal value which was related to ?F°f, Karnow-sky and Estill7 deposited films on sapphire at elevated temperatures and noticed no time dependence of film adhesion but a similar correlation with ?F°f. Because of the kinetic problems associated with thin-film adhesion it is desirable to examine adhesion in an equilibrium system. The wetting behavior of liquid-metal drops on Al2O3 provides such a system. Systems of this metal-ceramic type have been studied extensively.11 Humenik and Kingeryl2 have measured the wetting of A12O3 (and other substrates) by several metals and have pointed out that the wetting ability of these metals increases with increasing values of -?F°f. It is thus seen that thin-film adhesion and metal wetting on A12O3 are both related to the tendency of the metal to react with the surface oxide ions of the Al2O3 substrate and, because of this, both phenomena should be explainable by an appropriate model for the metal-Al2O3 interfacial bonding. In the sections that follow, wetting and adhesion data on A2O3 are reviewed and a model is presented by which these phenomena can be interpreted. ANALYSIS OF WETTING EXPERIMENTS In an equilibrium system involving a liquid-metal drop on a solid Al2O3 substrate, the work of adhesion, WAD, is defined by the Dupre, equation as WAD = ?s + ?L -?sL [1] where ?s and ?L are the surface free energies of the solid substrate and the liquid drop, respectively, and ?sL is the interfacial free energy. The work of adhesion is the work required to separate a unit area of the solid-liquid interface into two surfaces. The work of adhesion is usually determined from a sessile-drop experiment in which yL and the contact angle, ?, are measured. The Young-Dupr6 equation is then used to calculate WAD Wad = ?L (1+ cos ?) [2] The literature of this subject has been examined and Table I shows work of adhesion data for various liquid metals as measured on A12O3 substrates. The standard free energy of oxide formation of the metal at the temperature of the wetting experiment, ?F°f . is also tabulated in kcal per g-atom of oxygen. The data is grouped according to the gaseous atmos-

Jan 1, 1965

By G. R. Speich

The age-hardening of Fe-18 to 21 pct Ni marten-sites containing small amounts of titanium, aluminum, copper, or molybdenum has been studied by hardness measurements, transmission electron microscopy, extraction replicas, and X-ray and electron diffraction. Electyon-probe microanaly-sis of the extracted precipitates was also performed. The effects of temperature and composition on the aging kinetics have been studied in detail for Fe-20 pctNi-Ti martensites. All the martensites hardened on aging at 400" to 500 "C with a single aging peak. In the Fe-20 pct Ni-Ti martensites hardening was associated with the formation of a fine dispersion of NisTi. No precipitation was observed in an Fe-21 pct Ni-2.6 pct A1 alloy, although the hardness increased from 327 to 545 Dph during aging at 500°C. Copper was precgitated from an Fe-20 pct Ni-10.7pct Cu alloy, at least during the later stages of age-hardening. Hardening in an Fe-18 pctNi-5pct Mo alloy was associated with the precipitation of an intemzetallic compound tentatively identified as NisMo. The formation of austenite and the recovery of the defect structure of the martensite occur simultaneously with the normal precipitation processes &ring aging of these alloys. THE age-hardening of substitutional iron-base martensites is an important subject for study since age-hardening is one of the principal means of strengthening the substitutional iron-base martensites in precipitation-hardening stainless steels1 and maraging steels.a~4 In contrast to Fe-C martensites, these martensites are relatively soft and ductile in the as-quenched condition, due to the low solid-solution hardening effect of the substitutional elements as compared to carbon. Also, these martensites are hardened by aging at low temperatures in contrast to Fe-C martensites which generally soften on tempering. Titanium and aluminum, singly or in combination, as well as copper and molybdenum, have been the principal alloying additions to stainless steels to obtain an age-hardening reaction.&apos; Ti-A1 or Co-Mo-Ti combinations have been used in the case of maraging steels.3"4 The present work has been confined to a study of the age-hardening characteristics of an Fe-18 to 21 pct Ni martensite to which small amounts of titanium, aluminum, copper, and molybdenum were added in order to study the effects of each alloying element separately. The age-hardening characteristics of these alloys were studied by hardness measurements, transmission electron microscopy, extraction replicas, and X-ray and electron diffraction. Electron-probe micro-analysis of the extracted precipitates was also performed. The types and dispersion of precipitates, the formation of austenite, and the recovery of the defect structure of the martensite all were studied. EXPERIMENTAL PROCEDURE Alloys. The compositions of the alloys studied are listed in Table I. The level of 18 to 21 pct Ni was chosen to assure complete transformation to martensite on quenching and is the same nickel level used in maraging steels. The concentrations of titanium, aluminum, copper, and molybdenum were chosen after a study of the and Fe-Ni-MO&apos; ternary diagrams indicated how much of each alloying element could be dissolved in austenite. Alloys with 2 and 4 pct Cu were also studied, but these did not show precipitation hardening at the nickel levels studied and will not be discussed further. The alloys were prepared from electrolytic iron and high-purity* nickel, titanium, aluminum, copper, and molybdenum by vacuum melting of 20-lb heats. These were vacuum-cast into two 8-lb ingots. The ingots were hot-rolled at 1100°C to 1/2-in. plate and subsequently cut into 3/8-in. cubes. The cubes were sealed in evacuated quartz capsules and solution-treated for 16 hr at 1100°C, except for the Fe-20 pct Ni-10.7 pct Cu alloy and Fe-21 pct Ni-2.6 pct A1 alloys which were solution-treated for 16 hr at 1300" and at 1250"C, respectively. Metallographic examina-

Jan 1, 1963

By D. H. Polonis, Gary A. Dreyer

This investigation involved a study of the reactions occurring during aging of supersaturated a phase in a CIL-Si alloy. The aging processes at temperatures below 552°C were studied by means of metallography, microhardness, and electrical resistivity measurements. During the initial stage of aging in the temperature range 275o to 525oC an anisotropic phase, which is visible in the form of striations in the micro-structure formed prior to the appearance of the -phase. This transition phase has been designated k&apos; due to its structural similarity to the k phase. The rate of K&apos; formation during aging increased with aging temperature. At 450o and 525°C the reaction appeared to reach completion before significant amounts of precipitation were observed. The overall precipitation reaction in this system involved four stages: formation of formation of at a grain boundaries, growth of particles in 7-egions, and coagulation of into large particles. The relative role of each of these processes is dependent on the temperature of aging. The formation of phase from supersaturated a during aging increases the electrical resistivity and slightly increases the hardness of a 4.9 pct silicon alloy. The formation of during aging decreases the electrical resistivity and increases the hardness until coalescence of particles begins during the final stages of aging. The coalescence of results in a decrease of both electvical resistivity and hardness. THE available literature does not record a detailed analysis of the aging processes in Cu-Si alloys. Such an analysis has been difficult due to both the limited data available and misleading microstructural observations which have been reported. Previous work suggests that a rather complicated series of processes may be involved in the aging of supersaturated a. The present investigation is concerned with an analysis of the sequence of these aging processes in a 4.9 pct Si alloy. The currently accepted version of the Cu-Si equilibrium diagram, Fig. 1, was published by C. S. Smith in 1940.&apos; This diagram shows a eutectoid at 5.2 pct Si with the high temperature phase, having its composition range entirely to the right of this value. By quenching alloys in the composition range 4.6 to 5.2 pct Si from the region, Smith was able to retain the phase. However, this phase exhibited a striated microstructure which was anisotropic to polarized light. These striations were suggested by Smith to be duplex twins. Using an oscillating crystal X-ray method Barrett2 studied alloys containing 4.0 to 5.4 pct Si quenched from the and regions. The striations observed in quenched specimens were attributed by Barrett to quenching stresses. He found that striations could also be formed by deformation after quenching and identified them as stacking faults in alloys containing 4.0 to 5.0 pct Si. Barrett related the striations in alloys containing 5.0 to 5.4 pct Si both to stacking faults and to a metastable phase which he called The striated structure reported by smith&apos; was also obtained by Hoffmann, et al. 3 By deeply etching their specimens the anisotropic phase on the surface was removed, leaving a structure which was microscopically similar to pure copper. By means of X-ray analysis before and after deformation at room temperature it was concluded that the anisotropic phase was metastable This conclusion has been questioned by Barrett who attributes the discrepancy to superposition of k and lines on the powder patterns of Hoffmann, et alm3 The decomposition of supersaturated according to smith&apos; and Anderson,4 is very sluggish. However, Nony and Dreyer,5 and Hoffmann, et al.&apos; found that deformation increases the rate of decomposition. X-ray studies by Hoffmann, et al on

Jan 1, 1962

By J. H. Westbrook, R. W. Guard

The influence of a number of alloying additions on the structure and hardness of Ni3Al (?&apos;) has been studied. Three general effects have been observed.. solid-solution hardening, strain aging, and defect hardening arising from deviations from stoichiometry. The solid-solution hardening phenomena are complicated relative to binary terminal solid solutions due to the possibility of virtually independent substitution for the component elements and to the concomitant effects of strain aging. In many superior nickel-base high-temperature alloys the elevated temperature strength is attributable to the presence of a dispersion of Ni3A1 (?&apos;) in a solid-solution matrix. Examples of such alloys are Inconel X-550, Inco 713, Waspalloy, and Nimonic 80. It is, therefore, appropriate to seek a further understanding of the properties and alloying behavior of this compound. Ni3Al, the compound most rich in nickel in the nickel-aluminum binary system, was first detected by litakal but not established with certainty until the much later studies of Bradley and Taylor&apos; and Alexander and Vaughan.3 More recently the mechanism of formation was clarified by Floyd4 who showed that Ni3Al is the product of a peritectic reaction at 1395°C between the melt and the compound NiAl in agreement with the original suggestion of Alexander and Vaughan.3 Ni3Al has the ordered face-centered-cubic or Cu3Au structure in which the nickel atoms occupy the face centers and the aluminum atoms the cube corners, a0 = 3.570A. Despite earlier reports to the contrary. the compound apparently remains ordered at least to 1000oC.5 It exists over a narrow range of compositions (-3 pct wide) whose limits and temperature dependencies thereof have been reviewed recently by Taylor and Floyd.&apos; Most previous workers have found Ni3Al to be both hard and brittle7,8 although Maxwell and Grala8 obtained 2.5 to 6 pct tensile elongation at room temperature and Alexander and vaughan3 described the material as "hard and tough" on the basis of observations made while preparing metallographic specimens. In a recent study one of the present authors found that although considerable intergranu-lar brittleness was evident: individual grains of a coarse-grained inert atmosphere arc-melted specimen were readily deformable.l0 The same study also showed Ni3Al to have an anomalous temperature dependence of hardness in that such a curve exhibits a number of peaks between room temperature and 800°C which could not be associated with any microscopically observable precipitation reactions or polymorphic transformations. It was, therefore, proposed in the present study to elucidate the anomalous high-temperature strength behavior and to examine the solid-solution alloying of this compound and the effects of such alloying on hardness. Sample Preparation and Experimental Procedure-Samples, both of pure Ni,Al and of various alloyed solid solutions based on Ni3Al, were prepared by arc melting 100-g buttons in an inert atmosphere arc furnace. All buttons were heat treated for 4 hr at 1275oC and examined metallographically for homogeneity. Microstructures were typically homogeneous and single phase with ASTM grain size about 1 and unusually clean grain boundaries. Chemical analyses showed good correspondence (within a few tenths of a percent) with intended compositions. Microindentation hardness data were obtained from room temperature to 800°C in vacuo using a balanced-beam type loading system in an instrument to be described in detail elsewhere." In general a 100-g load, a loading rate of 1 mm per min and an indentation time of 15 sec were used. Various indenter materials were used according to the materials bein tested because of specimen-indenter interaction. Fl Measurements of the Vickers-type indentations were made at room temperature. Hardness-temperature curves were run at least twice on each sample to ensure that surface work-hardening effects had been eliminated." Precision lattice-parameter measurements were made on powder samples using a back-reflection

Jan 1, 1960

By J. A. Rowland, C. W. Funk

THIS investigation is a part of the United States Bureau of Mines work in conserving the Nation&apos;s resources. The isothermal sections presented were developed as a guide to a comprehensive investigation of the properties and fabricating characteristics of copper-base alloys containing manganese and tin. These alloys are being investigated as possible replacements for the commercial bronzes containing substantial quantities of tin. Development of the isothermal sections has, therefore, been limited to the area between 0 to 20 pct Mn and 0 to 25 pct Sn. Since Heusler1 reported the ferromagnetic properties of the Cu-Mn-Sn alloys in 1903, the system has been the subject of several investigations. These studies were largely concerned with correlation of magnetic properties and crystal structure.2-6 Vero,&apos; however, established the solidus surface of the system, between 20 pct Mn and 40 pct Sn, by thermal and microscopic methods. The a, ß, and y solid solutions were also observed by Vero and reported to extend into the interior of the ternary system. For the present study, the annotated diagram of the Cu-Sn system prepared by Raynor&apos; and the Cu-Mn diagram by Dean and coworkers9 were used as the binary borders of the ternary system. Procedure The metals used in this investigation were electrolytic manganese, washed cathode copper, and three-star tin. The manganese, produced at the Boulder City pilot plant of the Bureau of Mines, had a purity of 99.9 pct; the copper contained less than 0.004 pct Bi and 0.001 pct S; the tin contained 0.070 pct Pb, 0.070 pct Sb, 0.050 pct Fe, and 0.065 pct Si. Heats of 1 lb were melted in alundum crucibles with a 3-kw induction furnace. The heats were chill-cast in iron or copper molds to form 6-in. ingots 3/4 in. in diameter. Molds were washed with graphite or zirconia, depending on the manganese content, and preheated to 150°C. The maximum impurity content of any alloy was 0.03 pct Fe, 0.02 pct Si, and 0.02 pct Al; in most cases, they contained less than half this quantity of any of these impurities. Composition of the alloys and their treatment prior to homogenizing are shown in Fig. 1. Designations adjacent to each composition refer to the heat numbers. With a few exceptions, ingots containing 20 pct Sn or less were hot-swaged. These ingots were swaged, after a 24-hr preheat at 650°C, to obtain a total reduction of 70 pct in cross section. Intermittent reheating was necessary, and reductions were limited to either 0.025 or 0.050 in. in diameter per pass. Selection of a 200-hr homogenizing treatment was based on extensive experiments which indicated that virtual equilibrium was reached after 150 hr. Structures were retained by quenching in water. Grain sizes in the specimens homogenized at 350" and 450°C were generally small. These homogenizing treatments were preceded by a 50-hr anneal at 650°C to facilitate phase identification by increasing the grain size in these specimens. The specimens were furnace-cooled from this temperature to the homogenizing temperature, and the heat treatment was continued for the standard 200-hr period. The homogenized samples were bisected to provide an internal section for metallographic examination. Optimum definition of the microstructure was obtained by successive polishing and etching. Quarter-strength ASTM copper reagent No. 13 gave the most satisfactory results.10 Filings from interior sections of the samples were used for diffraction studies. These filings were annealed in evacuated glass tubes at homogenizing temperatures for periods exceeding 50 hr and quenched in water. During this treatment the manganese content was reduced by significant amounts, probably by vaporization. This change in composition was considered in applying the diffraction data, which were used only as a means of identifying phases where the data were consistent with the metallographic evidence. Phase Identification The a solid solution is readily distinguished by the copper-colored, twinned, polyhedral grains, as shown in Fig. 2, and a face-centered cubic X-ray pattern. Tukon indentations, using a 25-g load, indicate a Knoop hardness of 155 for a quenched from 650°C. Using white light developed by a Wratten 78 A filter, the ß grains in etched specimens have a dark brown tint in contrast to the lighter copper-colored grains of the a phase. Knoop hardness values of 284 were obtained for the ß structure in specimens quenched from 650°C. The ß structure was also distinguished from the a phase by its body-centered cubic lattice. The acicular structure appearing in Fig. 3 was evident in several alloys of higher tin content after quenching from either 750&apos; or 650°C. Since a trans-

Jan 1, 1954

By W. L. Phillips

Aluminum and copper single crystals were strained in shear, unloaded and rotated 60, 120, and 180 deg. The magnitude of the Bauschinger strain increased rapidly during the easy glide region and became constant as the easy glide limit was reached. An increment in stress was not required to continue slip in any close-packed plane or in the reverse direction during easy glide. A qualitative theory is proposed to explain the present results. FOLLOWING the initial work on tin by Bausch,&apos; there have been several investigatorsB7 who have employed the shear test to study the deformation of aluminum. In all cases the shear technique was employed in an attempt to obtain a more homogeneous shear than in the conventional tension test The direct shear technique has the obvious advantage, relative to the tension test, that a crystal may be stressed in any predetermined plane and direction, including the reversal of strain sense. In the present investigation the shear technique was employed with aluminum and copper single crystals and their behavior was studied with respect to deformation in all close-packed directions in the octahedral plane. In contrast to previous investigations, the shear technique was used in an attempt to obtain a stress-strain curve in shear which is identical with the corresponding tension curve and then to use the additional degree of freedom of the shear test to study the mechanical behavior of crystals in all close-packed directions in the octahedral plane. PREPARATION OF CRYSTALS Single crystals 1/2 in. in diam and 6 to 7 in. long, were prepared from aluminum containing 0.001 pct Si, 0.001 pct Fe, and 0.001 pct Zn, by spectro-graphic ana1ysis and from OFHC copper. Crystals were grown from the melt in graphite at a rate of 1 5 in. per hr. The copper crystals were grown under an argon atmosphere. No lineage structure was observed in Laue patterns, though small angle boundaries were present in the crystals. The aluminum and copper crystals were annealed at 640" and 900°C, respectively, for 24 hr and furnace-

Jan 1, 1962

By A. S. Keh, H. A. Wriedt

The Precipitation of nitrides in quenched Fe-N alloys, aged between 25° and 200°C, was studied by transmission electron microscopy. Different dislocation substructures were introduced into the materials by recrystallization, cold working, and by Phase transformation prior to the introduction of nitrogen. It was found that the precipitated nitrides, tentatively identified as Fe16N2, have (100) habit planes. Nucleation takes place Preferentially at dislocations and at subboundaries. It sometimes occurs in the matrix, probably on vacancy clusters, but rarely on high-angle boundaries. The process of nucleation on dislocations and subboundaries is very temperature dependent. At room temperature there is no particular orientation relationship between the dislocations involved and the (100) planes on which the nitride Particles formed. With increasing temperature, the nucleation sites tend to become restricted to those dislocation segments lying in (100) planes. The change of yield stress of the 0.022 and 0.043 pet N alloys during quench aging can be correlated semiquantitatively with the change in particle distribution. Strain aging in quenched alloys can be separated into two stages. The early stage is attributed to nitrogen segregation to dislocations and the later stage is associated with precipitation. When previously quench-aged specimens were strained and aged, no additional Precipitates could be seen, although a considerable amount of strain aging occurred. DURING the past two decades, the precipitation of nitrides from a iron has been extensively studied by various techniques. It has been established that two types of nitride exist. The stable nitride, Fe4N, forms at temperatures above 250oC and Fe16N2, the

Jan 1, 1962

By J. E. Hilliard, J. W. Cahn

calculation has been made of the standard deviations to be expected in the measurement of volume fractions by areal analysis, lineal analysis and four point-counting Procedures. The effect of experimetztal errors is not included. Our conclusion is that a point count using a two-dimensional grid will be the most efficient method of volume fraction analvsis providing the grid spacing IS coarse enough. The predicted standard deviation for such an analysis is shown to he in good agreement with that determined experimentally. A metallurgist wishing to estimate a structural property by means of quantitative metallography is often faced with a choice between several different procedures. In such a case, he will naturally wish to know: a) Which procedure is the most efficient in the sense of requiring the least effort for a given precision? b) For a given procedure, what are the conditions for maximum efficiency? c) Under these conditions, how many measurements are required to attain a given precision? It is our aim in this paper to provide at least partial answers to these queries as they relate to the estimation of volume fractions from measurements on a random two-dimensional section of an opaque specimen. The commonly used techniques1,2 for volume-fraction analyses are based on one or more of the following principles: i) For an areal or Delesse3 analysis: That the areal fraction of a three-dimensional feature intercepted by a random plane provides an unbiased estimate of the volume fraction of that feature. ii) For a lineal or Rosiwal4 analysis: That the fractional intercept on a line passing at random through a two-or three-dimensional feature provides an unbiased estimate of, respectively, the areal or volume fraction of that feature. iii) For a point-count analysis: That the fractional number of randomly or regularly dispersed points falling within the boundaries of a two-dimensional feature on a plane, or within the boundaries of a three-dimensional feature in a volume, provides an unbiased estimate of, respectively, the areal or volume fraction of that feature. The absence of bias referred to in these principles does not, of course, imply that the results of an analysis will be free of error, but only that the expected result is equal to the true one. Application of the foregoing principles provide the following six possible experimental procedures: a) An Areal Analysis—This involves the measurement with a planimeter (or other means) of the area of a constitutent intercepted by the plane of polish b) A Two-Dimensional Random Point Count—Apos-sible experimental procedure for this analysis is to superimpose a sheet of transparent graph paper on a micrograph, and then use a table of random numbers to select coordinates for the points. c) A Two-Dimensiotuzl Systematic Point Count-Similar to b) except that the points are distributed in a prescribed manner, usually at the corners of a lattice superimposed on a micrograph or the screen of a projection microscope. d)A Lineal Analysis—A measurement of the fractional line length intercepted. It is usually performed by traversing the specimen under the cross hairs of a microscope and recording the distance travelled in each constituent. e) 4 One-Dimensional Random-Point Count— This could be performed by traversing the specimen with stops at random intervals to identify the constituent then present under the cross hairs. f) A One-Dimensionul Systematic Point Count— Similar to e) except that the traverse is stopped at prescribed (usually equal) intervals. It will be noted that the point-counting procedures b) and c) can be regarded as methods of estimating the areal fraction. Similarly, e) and f) are indirect methods of making a lineal analysis. All of the six procedures described above involve at least one stage of sampling. Thus, quite apart from any experimental errors, there will always be a statistical uncertainty associated with the final results. It is with the calculation of this uncertainty that we will be concerned. To avoid unnecessary repetition, the statistical relationships that will be used in the calculations are collected together in the appendix. For a fuller discussion reference should be made to one of the many textbooks5,6 on statistics. For symbols relating to the specimen structure we will follow as far as possible the terminology used by Smith and Guttman.7

Jan 1, 1962

By R. E. Carter, F. D. Richardson

Self-diffusion coefficients have been measured for iron in wustite (700° to 1000°C) and for cobalt in cobaltous oxide (800&apos; to 1350°C) by means of radio-isotopes. Both sectioning and decrease of surface-activity methods have been used and a particular shortcoming of the latter method has been disclosed for wustite. The effects of oxygen pressure on the composition of cobaltous oxide and on the diffusion coefficient of cobalt in the oxide have been established. The effect of the preparation conditions of the cobalt-oxide specimens on the diffusion measurements has been determined. It has been concluded that diffusion of cobalt occurs via vacant cation sites and that a cobaltous-oxide disk reaches equilibrium with its environment in a time which is very short compared with that needed for diffusion of the radio-isotope. THE availability of radioactive isotopes in recent years has permitted the measurement of many self-diffusion coefficients which were hitherto impossible to determine. With the advent of radioactivity, the method of measuring diffusion coefficients has also changed. The earlier sectioning method by which the concentration profile of the diffusing species was determined in the diffusion specimen has often been rejected in favor of the decrease of surface-activity method first developed by Von Hevesy,&apos; and extended by Steigman, Shockley, and Nix,&apos; and Raynor, Thomassen, and Rouse." With the experimental arrangement where an infinitely thin deposit of tracer is diffused into a semi-infinite cylinder, the decrease in activity of the active face due to diffusion of the tracer into the specimen is measured by counting the active face before and after diffusion. Under these conditions, the appropriate solution of the diffusion equation is R = e*(l-<K/Z) [1] where R is the ratio of the counting rates after-to-before diffusion, $& is the Gaussian error integral of \/Z, and Z is defined as Z = p2Dt, p being the linear-absorption coefficient of the detected radiation in the diffusion specimen, D the self-diffusion coefficient, and t the diffusion time. In the older sectioning method the concentration distribution of the tracer is found after diffusion by analysing thin parallel sections cut from the active face of the diffusion specimen. As seen from the solution of Fick&apos;s second law for the distribution of an infinitely thin source after diffusion into a semi-infinite cylinder the log of the concentration plotted against x&apos; will be a straight line of slope log e/4 Dt. The diffusion coefficient thus can be calculated directly. The units of the tracer concentration are completely arbitrary, since any change in units does not change the slope of the log c vs x2 lot. Although both of the above methods have been widely used, a thorough experimental comparison does not appear to have been made. Birchenall and Mehl,&apos; during an investigation of the self-diffusion rate of a and r iron by the surface method using Fe" tracer, diffused and sectioned one welded couple. The diffusion coefficient from this run agreed with those by the surface method. Buffington, Bakalar, and Cohen," during an investigation on a and y iron

Jan 1, 1955

By P. G. Winchell, J. K. Jackson

The strtcture of body-centered Fe-Ni-C alloys (0 to 16.5 wt pct Ni) containing less than 0.015 wt pct C was investigated by measuring the carbon-diffusion peak at low frequencies with a torsion pendulurrr. By determining the Snoek damping in ferrite and in quenched alloys of identical composition, the effects of nickel content and heat treatment were separated, and the existence of a structure in which Snoek damping is greatly reduced was revealed in the quenched ternary alloys. The relative amount of this structure increases with increased nickel content and cooling rate. It is proposed that this structure may be highly imperfect or tetragonal. In Fe-Ni-C alloys containing only ferrite the addition of nickel (0 to 5.1 wt pet) decreases tke area of the carbon-diffusion peak and shifts its position to lower temperatures but broadens the peak only slightljl. The decrease in peak area with increasing nickel implies that short-range interactions with the nickel atoms prevent some of the carbon atoms from participating in the relaxation process. The shift in the peak position indicates a long-range effect of nickel on the movement of carbon atoms so that at 300°K the diffisivity of carbon in -ferrite is increased by about 35 pct by the addition of 5.1 tut pct Ni. The results are presented as distributions of relaxation strength 072 the activation energy under the assumption that the frequency factor is invariant. BODY-CENTERED iron-base solid solutions dissolve carbon interstitially in the octahedral sites. The occupation of an octahedral site produces a tetragonal distortion of the surrounding lattice. These sites fall into three sets according to the orientation of the tegragonal distortion associated with each and may be identified by their lattice coordinates as: type 1/2 0 0, type 0 1/2 0, type 0 0 1/2. At low carbon levels in cubic ferrite the carbon atoms are distributed at random1 among these sites. In high-carbon body-centered iron-base solid solutions, which are unstable with respect to several two-phase mixtures, a tetragonal structure occurs as the product of the diffusionless transformation of austenite. This transformation is commonly accompanied by appreciable microstrain and its product is called martensite. The tetragonality of martensite, a supersaturated solution of carbon in a iron, is maintained prior to decomposition even though local diffusion of carbon occurs as evidenced by carbide precipitation. The tetragonality is taken as evidence that one set of octahedral sites is preferentially occupied by carbon. (This preferential occupation is called ordering below.) Support for this view is presented by the absence of strain-induced carbon-atom diffusion in high-carbon solid solutions (martensite). Such diffusion occurs in ferrite and produces the 40°C internal-friction peak observed at 1 cps frequencies.&apos; Thus, there appears to be qualitative evidence favoring the view that at sufficiently high carbon levels tetragonality arises spontaneously as a result of the ordering of the carbon atoms; however, no experimental evidence exists concerning an equilibrium between cubic and tetragonal body-centered solid solutions. Moreover, attempts to estimate the energy of ordering7-&apos; predict a value too low to maintain the tetragonality observed at medium carbon levels, 0.3 to 0.6 wt pct C. In the present work an attempt using internal-friction measurements is made to establish the presence or absence of tetragonality in low-carbon body-centered solid solutions formed by quenching austenite. Fe-Ni alloys containing about 0.01 wt pct and 0 to 15 wt pct Ni were selected for the study. While the work was in progress Gilbert and Owen10 reported that the transformation which occurs in these alloys during quenching at rates up to 5000°C per sec is not martensitic but is more properly categorized as massive. Thus, the solid solution

Jan 1, 1964

By A. L. Titchener, M. B. Bever

The stored energy of cold work was investigated in drawn gold-silver wires by tin-solution calorimetry as a function of strain, strain rate, initial grain size, and temperature of defornation. The energy expended in the deformation process was measured. Hardness was also measured and was correlated with the stored energy as a function of strain and strain rate. Mechanisms of energy storage are discussed. THE variables which determine the amount of energy stored in a metal during cold work are of two kinds: those involving the metal and its condition and those involving the process of deformation. For example, the stored energy differs between one metal and another and between different degrees of purity of a single metal. It also depends on the nature of the deformation process and particularly the extent, rate, and temperature of deformation. Most investigators have reported the stored energy as a function of strain (or some similar geometric parameter) or of the energy expended during deformation. In general the stored energy increases with strain, rapidly at first, but more slowly at large strains until a saturation level is reached. Information about the effects of other variables, however, is limited. Some data are available on the dependence of the stored energy on the purity of a metal1 and on the composition of

Jan 1, 1960

By W. N. Carr, J. R. Biard, B. S. Reed

An analysis of the semiconductor injection laser is presented which is based on a phenomenological model using device and material parameters. The intent of the laser threshold analysis is not to predict from theory the actual threshold current density but rather to provide a logical means of interpreting experimental results. The device and material parameters have been selected such that they describe both the threshold conditions and quantum efficiency of the lasev. A technique is proposed whereby the current required to obtain a population inversion is predicted from device 1-V characteristics. Experimental data are presented which support the analysis. THIS investigation is concerned with the characteristics of GaAs lasers fabricated in this laboratory. However, the assumptions used in these calculations should not limit their application to GaAs alone. It is felt that the analysis is sufficiently general to be useful in describing the operation of other semiconductor-laser materials. ANALYSIS heavily doped semiconductors to be considered. The absorption and/or generation of optical radiation may be completely specified in terms of q(E) and the probability functions f,,(E) and f,(E) which define the occupancy of the valence-band and conduction-band levels, respectively. The symbols used in this analysis are listed at the end. Following the approach of all&apos; it can be shown that the effective gain coefficient, g, in the plane of a forward-biased P -N junction is g(E)=-ai@)[1-fv(E)-fc(E)] [1I In particular Eq. [I] holds for the photon energy at which lasing occurs, Ep. Throughout the following discussion Eq. [I] is considered only for the lasing transition E = Ep. For simplicity the functional energy dependence of the various factors is omitted in subsequent equations. Unless otherwise stated the symbols used represent the values at Ep. If the analysis is limited to the P-type side of the laser junction, the function q(l -f,,) is recognized as the band-to-band absorption in the P-type material at thermal equilibrium. Under forward-bias conditions the function f, is determined by the electron quasi Fermi level in P-type material. For sufficiently large forward bias the second term in Eq. [3] will become dominant and net gain will result; i.e., g will become positive. In order to simplify the subsequent analysis several assumptions are in order at this point. a) No conductivity modulation occurs. /„ is independent of f, and therefore go is a constant. b) The P-type material is heavily doped such that the degeneracy of the valence band permits popu-

Jan 1, 1964

By A. G. Guy

Thermodynamic activity rather than chemical composition is basic to the analysis of diffusion. This is the essential conclusion reached by Darken1-3 and by Birchenall and Mehl.4 If so, it is reasonable to expect that diffusion calculations should be carried out using activities. This paper will illustrate a method of employing activities in analyzing interstitial diffusion. For interstitial diffusion (of carbon in iron and probably generally) the diffusion of the solvent atoms, volume changes, and other such complexities not explicitly provided for in the usual diffusion equations cause a negligible error of the order of 1 pct. Therefore it is relatively easy to test the validity of an analysis based on activities in the instance of interstitial diffusion. The mathematical treatment of the more complex substitutional diffusion could be simplified once the adequacy of the activity method had been established for substitutional diffusion. Thus the present work has the purpose not only of providing a superior analysis of interstitial diffusion but also of indicating a possible approach to the more important question of an adequate treatment of substitutional diffusion. Development of the Diffusion Equation Although activities are convenient quantities to determine experimentally, they are derived from the more fundamental fugacities.9 Possible confusion in using activities can be avoided by basing equations involving their use on the relationship, dP = -D1d/ax dt [I] where, dP = the number of grams of solute crossing one cm2 in the time dt sec. D, = the diffusion constant for use with fugacities; it is assumed to be constant at a given temperature, and has units of secs. f = the fugacity of the solute in the solid solution; the units are those of pressure, dynes per cm2. x = the distance in cm. Eq 1 can be considered to be the basic form of the first Fick Law for one-dimensional diffusion. In order to convert Eq 1 into an equation involving activities, a choice of a standard or reference fugacity, f°, must be made: then. a = f [2] where a is the activity corresponding to the fugacity f. However, the value of a is also determined by the relation, a = a\c [3] where a is the activity coefficient and c is the concentration. For a given value of a (such as a = 1) it is evident that the units used for expressing concentration will affect the numerical value of a corresponding to a given amount of carbon dissolved in y-iron. Since weight per cent concentration is so widely used, this unit will be adopted as the standard in this paper. When the value off given by Eq 2 is substituted in Eq 1, the first Fick Law becomes, dP= -D,f°dl [4] Since D,f° is constant (at a given temperature), the second form of Fick&apos;s Law can be obtained in the usual manner, -dt D&apos;f° dx [5] where c&apos; is the concentration in g per cm3. In terms of weight per cent concentration, c, Eq 5 becomes, dc _ 100 d2a Tt = T /f w [6] where p is the density of the solid solution and will be assumed to be constant. In order to simplify Eq6, it is necessary to consider the standard state, f°. Standard states are chosen for convenience. There appears to be one especially convenient choice for the analysis of interstitial diffusion. If the activity coefficient, a, is set equal to unity at infinite dilution of the solute, 100 then — 100D1fl° is approximately equal P to the usual diffusion constant in dilute solutions. This is true since the activity, a, in Eq 6 can be replaced by the concentration, c, with little error in an infinitely dilute solution.* Here f1° is the standard fugacity necessary to achieve this standard state. It is proposed that a diffusion constant Da&apos; be defined, Da1=100/pD1f1° [7] where A is the metal whose diffusion is * The second derivative of a is equal to the second derivative of c in dilute solution only if -da/dr- is. zero. Although theory does not require this condition, in the systems C in Fe, and Zn in Cu, dc is in fact found to be essentially zero.

Jan 1, 1950