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By W. G. Pfann

The simultaneous segregation of two solutes during the directional solidification of an ingot is treated mathematically on the basis of simplifying assumptions. Expressions are derived for the difference in concentration of two solutes, and for the location and concentration gradient of a pn barrier formed in a semiconductor by the segregation of a donor and an acceptor. THE problem of normal segregation of a single solute during the freezing of an alloy has been treated mathematically by a number of investigators. Gulliver' showed that coring increases the quantity of eutectic above that to be expected at equilibrium, for a system having limited solid solubility and, on the basis of simplifying assumptions, calculated the fraction of eutectic to be expected. Scheuer2 expressed composition of solid solution in terms of a distribution coefficient and the fraction solidified and compared experiment with calculation for the systems A1-Cu, Al-Zn, Cu-Sn, Cu-Zn. Hayes and Chipman,3 in a detailed study of segregation in a low carbon, rimming steel ingot, calculated distribution coefficients for a number of solutes in iron, compared calculated and experimental segregation curves, and discussed the effects of process variables such as rate of solidification and stirring. In the present paper a mathematical analysis is made of the simultaneous segregation of two solutes during the orderly freezing of a solid solution system, with particular emphasis on the difference in solute concentrations. Although the analysis is quite general and can be applied to the segregation of minor elements in alloys, it is directed in particular at the solidification of a semiconductor containing a donor and an acceptor. Normal segregation has unique aspects in a semiconductor, because of the ways in which donors and acceptors affect the electrical properties. The difference between two solute concentrations becomes of importance, as does also the gradient of the difference where the difference goes through zero, this being the concentration gradient of excess carriers at a pn barrier. A primary object of this paper is to extend the mathematical treatment of segregation to include these newer aspects which are of particular significance for semiconductors. One property of interest is the electrical conduc- tivity, which arises from the presence of donors 'or acceptors in solid solution. The conductivity may be either n-type or p-type depending on whether donors or acceptors, respectively, are in atomic excess. For both germanium and silicon, elements of Group V of the Periodic System, such as P, As, and Sb, are donors and elements of Group 111, as B, Al, In, and Ga, are acceptors.4-6 If a donor or acceptor is present alone in solid solution, the conductivity is proportional to its concentration." If both a donor and an acceptor are present, then the conductivity is proportional to the difference in their atomic concentrations. If the donor and acceptor segregate at different rates then a pn barrier in certain circumstances may form at some point in the ingot. Accordingly, equations are derived and illustrated which express the effect of segregation on: 1—the concentration of a single solute with a discussion of the assumptions; 2—the difference between two solute concentrations and means for minimizing its variation in an ingot; and 3—the location and concentration gradient of a pn barrier. The analysis is applicable to processes in which the entire charge is melted and then progressively frozen from one end. The method in which an ingot is solidified in a crucible7 and that in which a solidifying rod is pulled from the melt8 both fall into this category. Segregation of One Solute If a cylinder of molten alloy is caused to freeze slowly from one end, a normal segregation of solutes will usually occur, producing a lengthwise concentration gradient in the ingot. Depending on whether a solute raises or lowers the melting point of the solvent, it will become concentrated in the first or last regions, respectively, to freeze. If it is assumed that freezing is such that there is no diffusion of solute in the solid, complete diffusion in the liquid, and that k, the distribution coefficient, defined as the ratio of solute concentration in the just-freezing solid to that in the liquid, is constant, then the

Jan 1, 1953

By W. Rostoker

Single isothermal sections were constructed for the titonium-rich corners of the systems Ti-Fe-O, Ti-Cr-O, and Ti-Ni-0 with a view to locating the shape and disposition of the ternary intermediate-phase fields for the compounds isomorphous with Fe3W3C. IN a previous note,' it was reported that a group of ternary intermediate phases existed which occurred in the general composition range Ti,X,O-Ti&O where X was one of the following elements: copper, nickel, cobalt, iron, or manganese. In two instances, binary intermediate phases occurred (Ti²Cu and Ti²Ni) which appeared to be isomorphous. The X-ray diffraction patterns of the phases could be indexed on a cubic lattice whose cell edge was of the order of 11 kX. Karlsson2 stated that the line extinctions and line intensities indicated that these phases were isomorphous with Fe³W³C and therefore belonged to the space group Oh. This author also noted that no unique phase of this type occurred at the composition Ti,Cr,O. This latter statement was accepted as true at the time the previous technical note was written.' However, work continued along these lines and that reported here has shown that a phase isomorphous with Fe³W³C does exist in the vicinity of the composition Ti³Cr³O but not at or near Ti4 Cr²O. The lattice parameter of this phase was found to be 13.01 kX. Metallographic examination of both as-cast and annealed specimens of these supposed ternary oxide phases disclosed structures which were far from single phase in many instances. It was considered a point of some interest to locate more positively the composition coordinates of the single-phase fields. To this end, single isothermal sections for the systems Ti-Fe-O (1000°C), Ti-Cr-0 (1000°C), and Ti-Ni-0 (900"C) were constructed. Phase relationships were studied by both X-ray diffraction and metallotrra~hic methods. Allovs were prepared as 10 g buttons by melting in a noncon-sumable electrode, water-cooled copper crucible arc-melting furnace using a helium atmosphere. Oxygen was introduced quantitatively by the use of weighed amounts of TiO² or TiO. Iodide titanium and the purest available alloying metals were used. All results are discussed in terms of the nominal composition of the melts. Previous experience with making oxygen-rich alloys had shown that oxygen losses were not significant if a homogeneous melt was produced. Weight-loss measurements were used to check loss in metallics during melting. Anneals were conducted in evacuated Vycor bulbs, and annealing times ranged from 24 hr at 1000 °C to 48 hr at 900°C. To achieve equilibrium in certain sluggish alloys, seven-day anneals were used. Debye-Scherrer powder patterns were taken in a 14 cm diam camera using filtered CuK radiation (K² = 1.541232 kX; K., = 1.53739 kX). Specimens were prepared by crushing and screening previously annealed buttons. In general, the X-ray diffraction results were used to identify the major phases, while the metallographic examinations were more useful in indicating the number of phases present. By the use of the combined results and general fundamental rules governing ternary equilibrium, it was possible to construct isothermal sections in good detail. In some instances where auxiliary lattice-parameter data were available, it was possible to lay down tie lines and to locate more positively corners of three-phase fields. Isothermal Section of the System Ti-Fe-O at 1000°C The binary systems Ti-O³ and Ti-Fe4 have been published. Binary phases, TiO, TiFe, and TiFe2, are likely to be encountered in the titanium-rich corner of the system. The structures are NaCl (Bl), CsCl (B2), and MgZn, (C14) types, respectively. Although preliminary X-ray diffraction studies indicated the presence of a ternary phase in the composition range Ti,Fe,O-TiJ?e,O, metallographic examination of two specimens at these compositions revealed the presence of other phases. Thirty-eight alloys were used to delineate the phase boundaries of the isothermal section illustrated in Fig. 1. The five almost single-phase ter-

Jan 1, 1956

By Ken-ichi Hirano, B. L. Averbach, Morris Cohen, N. Ujiiye

The influence of plastic deformation in compression on the self-diffisivity of a iron has been measured in the temperature range of 742º to 885°C. The diffusivity is enhanced in proportion to the strain rate, but this dependence decreases with increasing temperature. The magnitude of the strain is relatively unimportant in this connection. Strain rates from 0 to -2 x 10-3 sec-1 were investigated, with a corresponding increase in diffusivity up to 3 x 103 times. The results are analyzed in terms of vacancy diffusion and the excess vacancies introduced during dejornzntion. It is concluded that grain boundaries are the main vacancy sinks in polycrystalline iron and that the vacancy lifetime is therefore dependent on the grain size. A typical vacancy lifetime at 800°C with a nzean grain-boundary intercept of 0.3 cm is of the order of 1 sec. LATTICE imperfections are considered to be the dominant mechanism of diffusion in close-packed crystals. Vacant lattice sites are usually the most important defect, but the enhancement of diffusion along grain boundaries and other short-circuiting paths has been well documented. Inasmuch as plastic straining produces a variety of defects, it would be expected that the application of stresses sufficient to cause plastic flow during a diffusion run might result in an increase in the self-diffusivity. Enhanced self-diffusion in a iron under compressive creep has been observed by Buffington and Cohen' and subsequent coworkers.2 Lee and Maddin3,4 have also found an enhancement in the self-diffusivity of silver single crystals undergoing torsional deformation. On the other hand, Darby, Tomizuka, and Balluffi5 have not observed any significant diffusion increase in single crystals of silver deformed either in tension or in compression, although the results in this case may have been affected by the occurrence of recrystallization. With poly crystalline silver, however, Forestieri and Girifalco6 have found en- hanced diffusion during compressive creep. A small decrease in the self-diffusivity of zinc single crystals under compressive loading has been reported,7 but the applied stresses in these experiments were too low to cause appreciable plastic deformation. For the cases in which enhanced diffusion has been found, a number of common effects have been noted. Usually, the ratio of the diffusivity during straining, Ds, to the diffusivity in the unstrained condition, D,, varies linearly with the strain rate, and does not seem to be a function of the plastic strain, at least up to ? = 0.2. Values of Ds/Du of the order of 100 have been reported'-4 but the enhancement decreases with increasing diffusion temperature. The activation energy for diffusion decreases markedly with increasing strain rate and approaches a level of roughly one-half the unstrained value. Most of the current explanations of these observations assume that excess vacancies are generated by the deformation, but rather large concentrations of excess vacancies and/or long lifetimes are required by these theories.',',',' Since the original work on iron, there has been an important improvement in the compressive-loading technique.2 The early data were also analyzed without taking into account the motion of the coordinate system in the specimens undergoing plastic deformation. This problem has now been solved by a number of investigators (see Refs. 11, 12, and the appendix of the present paper). Consequently, the strain-diffusion runs on iron for the Ds measurements were repeated and extended over wider ranges of temperature and strain rate. In addition, the original diffusivity data9 on the unstrained iron have been corrected for the divergence of the beam in the counting system and new data points have been added, resulting in somewhat different values for the self-diffusi-vities in a iron.10 The new values for Du are used in this paper. A comparison of Ds and Du confirms the large enhancement in the self-diffusivity of iron during compressive creep, as previously reported.',2

Jan 1, 1963

By R. E. Grace, H. W. Schadler

DARKEN1 has established the theoretical relation between the self-diffusion coefficients and the Boltzmann-MatanO Or interdiffusion coefficient: D is the Boltzmann-Matano or interdiffusion coef-

Jan 1, 1960

By R. F. Mehl, C. E. Birchenall

SINCE Maxwell1 first considered the self-diffusion process in 1872 its importance in the kinetic theory of matter has been recognized. Until the discovery of isotopes in 1913, a direct measurement of this quantity seemed impossible. The only information that could be gathered indirectly was for gaseous systems for which the kinetic theory was well developed. When methods of direct observation be-came available, investigations on self-diffusion were carried out in condensed phases. In metals these data are presumably of potential importance in the study of recovery, recrystallization, creep, sintering, and related phenomena. Following the pioneer work of Von Hevesy2 and his collaborators, who determined the rate of self-diffusion in lead with the naturally occurring thorium B isotope, several metals have been investigated. Pb, Ag³, Au.6,5, and Cu6,7,8,9 are examples of isotropic crystals existing in only one phase modification. Anisotropy of diffusion has been demonstrated in Bil6 and Znll. This paper is a study of a single metal in two allotropic forms and also of self-diffusion in a body-centered cubic lattice. Despite the fact that the measurements in each of the papers cited on self-diffusion in copper seem to be internally consistent to about 10 to 15 pet, the curves reported by different authors differ by factors of 2 to 4 at the same temperatures. This uncertainty may account, in part, for the failure to establish a successful correlation between the self-diffusion rates and other physical characteristics of the metals. No satisfactory theory of metallic self- diffusion has as yet been proposed to account for all the existing data and to permit estimation in other metals. Experimental Part: Two units of radioactive iron were used in these experiments, each a mixture of Fe53 and Fe59 (12). Fe53 decays by K electron capture and the emission of an X ray with a half life of about 4 years. Fe59 emits two beta spectra, one with a maximum energy of 0.26 Mev, the other 0.46 Mev, and gamma rays of 1.1 and 1.3 Mev, with a half life of about 44 days. They were present in nearly equal concentration initially. The composite half life started at about 50 days and increased as the Fe59 decayed, leaving Fe55 relatively more abundant. After aging a year, the half life was too long to measure significant decay in a month. The absorption properties also changed with time. Because of the importance of the absorption coefficient in the diffusion calculations, it was necessary to determine this quantity frequently over the period during which these experiments were carried out. This correction was unsuspected at the time of publication of notesl3 on this work and accounts for the discrepancy in alpha iron and for part of the discrepancy in gamma iron.* * The data in the present paper supersede those given in the notes entirely. In one note the scale of log D was inadvertently reversed. In addition to the correction for the drift in absorption coefficient, the D values for gamma iron at low temperatures were high owing to insuficient correction for diffusion occurring during heating and cooling through the high temperature part of the alpha range at a rate much slower than that employed in the runs reported here. The high temperature alpha rates are much higher than the low temperature gamma rates and correspond to large equivalent times at these temperatures. These experiments were discarded and new measurements taken. Independent experiments with the same activity units indicated a radioactive contaminant in the iron, but exhaustive attempts to isolate and identify it chemically failed. These experiments? indicate † These experiments will be discussed in a later publication from this laboratory. that the contaminant represented only a trace of little importance in the diffusion results. The active iron was plated from an iron chloride solution. Enough of the solution was dropped on a 1% in. square of filter paper to wet it thoroughly.

Jan 1, 1951

By R. J. Borg, C. E. Birchenall

The self-diffusion coefficients for a iron have been deternzined between 980° and 1167° K using Fe55 as the tracer. With decreasing temperature the diffusivity was found to decrease more rapidly than predicted by the Arrhenius equation in the region of the Curie temperature. The diffusion coefficients obtained well above the temperature of the magnetic transformation are more precise ard differ substantially from all the previously reported values. In the temperature region beginning at least 50" above T,, the diffusion coefficient is given by D = 118 (±2) exp 1 HE self-diffusion coefficients for a iron have been reported previously as the result of four independent investigations The general lack of agreement and the poor precision of most of the previous work prompted the redetermination by the present authors. The possible effect of the onset of ferromagnetism upon the diffusivity5 provided an additional impetus for this investigation. The earlier measurements of Birchenall and Mehl1 gave low values for the diffusion coefficient at temperatures below the Curie temperature, Tc. However, the precision of these measurements is not sufficient to establish an indisputable correlation between the low values and the magnetic transformation. The results of this investigation along with all the previous ones are shown in Fig. 1. EXPERIMENTAL Small rectangular samples, approximately 1 by 0.5 by 0.3 cm, of high-purity iron (Vacuum Metals Corp., "Ferrovac") are first heated in hydrogen at 900°C for24hr in order to remove possible trace amounts of dissolved carbon and oxygen. The small

Jan 1, 1961

By P. G. Shewmon

Radioactive MgZA has been used to study the rate of self-diffusion in oriented single crystals of magnesium in the temperature range 468O to 635OC. The diffusion coefficients parallel and perpendicular to the c-axis are: Dl, = 1.0 exp (—32,200IRT) cm2 per sec and Dl = 1.5 exp (—32,500IR T) cm Qer sec. The ratio Dl/Dll was found to vary from 1.13 at 468 to 1.24 at 575 C. Assuming a vacancy mechanism, an explanation of this aniso-tropy of diffusion follows from a consideration of the difference in the saddle points for diffusion in and out of the basal plane. RECENT discovery of radioactive Mg2"'-' has made possible the study of self-diffusion in magnesium. The experimental procedure and the results of a study of diffusion in polycrystalline magnesium have been described in an earlier paper.V he present work is an application of the same techniques to the study of self-diffusion in oriented single crystals of magnesium. In the general diffusion problem the diffusion coefficient is a second order tensor relating the two vectors, the diffusion flux, and the concentration gradient. In a hexagonal lattice, such as magnesium, the complete determination of the diffusion coefficient D as a function of direction in the lattice requires a knowledge of D,, and D,, i.e., the diffusion coefficients parallel and perpendicular to the c-axis, respectively. It can be proven quite generally that in a hexagonal lattice D is independent of direction in the basal plane, and that out of the basal plane it varies with 0, the angle between the c-axis and the direction of diffusion, according to the equation D(B) = D,,cos2B + D,sin2B. [I] The proof of Eq. 1 depends only upon the symmetry elements of the hexagonal lattice and upon the transformation properties of a second order tensor." Therefore, Eq. 1 holds for any mechanism of diffusion and any c/a ratio. Experimental Procedure The experimental procedure used with polycrystalline specimens can be briefly outlined as follows. Radioactive Mg" was produced by bombarding a NaCl crystal with 350 mev protons and was chemically separated from the target as MgO. The Mg" was then vapor deposited on a specially cleaned magnesium specimen by heating the MgO on a tantalum ribbon in a vacuum. During the diffusion treatment, oxidation and vapor loss of the radioactive material were minimized by annealing the samples in pairs with the active faces in contact, each pair being inside a magnesium container, which was in turn surrounded by an argon atmosphere in a sealed Pyrex tube. The distance-activity profiles were obtained by measuring the activity of thin sections cut parallel to the original interface with a lathe. The only technique which was peculiar to this work was the preparation and orientation of the single crystal specimens. The single crystals used in this work were grown by E. C. Burke of the Dow Chemical Co., using a modified Bridgman method, in which the furnace and specimen were stationary while the temperature gradient moved.' In growing these crystals the starting material was distilled magnesium, the crucibles were machined from Acheson electrode graphite, and the furnace atmosphere was tank argon. The crystals were grown from sublimed magnesium with the following analysis: 0.0002 pct Al, 0.0017 pct Fe, 0.0009 pct Mn, 0.0001 pct Ni, 0.0006 pct Pb, and less than 0.01 pct Ca, 0.0001 pct Cu, 0.001 pct Si, 0.001 pct Sn, and 0.02 pct Zn. The two crystals used were roughly lh in. diam and eight in. long. The c-axis made angles of about 7" and 78", respectively, with the specimen axes. If the values of D obtained by the use of these two crystals are taken equal to D,, and D,, respectively, the error introduced by this assumption is less than 1 pct. This can be shown by combining Eq. 1, the identity sin% + cos'8 = 1, and the experimental fact that DJDn in magnesium was always less than 1.25.

Jan 1, 1957

By F. E. Jaumot, R. L. Smith

Self-diffusion in zinc at temperatures below 200°C has been studied using both single crystal and polycrystal samples. Anomalous results were obtained for single crystal samples, the data indicating that in some cases grain boundary diffusion predominated. These anomalous results are presumed to be due to low angle lineage boundaries in the single crystals. When volume diffusion occurred in either single or polycrystal samples, the values for the diffusion coefficient were as much as several orders of magnitude larger than would be expected from high temperature data. ONSIDERABLE work has been done on self-^-*diffusion at temperatures in the neighborhood of two thirds of the melting temperature (in OK) and higher, but very little has been done at lower temperatures. This is particularly true of self-diffusion in single crystals. This paper reports work on self-diffusion in zinc at temperatures below 200 °C where both polycrystalline and single crystal samples have been used. The measurements were made using both the sectioning and absorption techniques. The experiments on diffusion at low temperatures were initiated primarily for three reasons: to obtain data from single crystal diffusion covering a wide range of temperatures, to obtain values of the activation energy for grain boundary diffusion, and to obtain, if possible, a lower limit to the temperatures at which the absorption technique may be used. It is possible to determine whether volume or grain boundary diffusion predominates at low temperatures by analyzing the manner in which the concentration of the diffusing atoms varies with the depth of penetration. The analysis is possible only for the sectioning technique. For volume diffusion and for the boundary conditions of the present experiment, the curve of the logarithm of the concentration, C, vs the square of the penetration depth, x, is a straight line with the slope proportional to — 1/D. If grain boundary diffusion predominates, Fisher' has shown that a curve of In C vs the first power of the penetration depth should be a straight line. This linear dependence of concentration on depth has been observed experimentally by several investigators. (See, for example, refs. 3 and 4.) Following Fisher's analysis, the grain boundary diffusion coefficient can be written as where 8 is the grain boundary width, D, is the volume diffusion coefficient at the temperature involved, t is the time of diffusion, C is the concentration of diffusing material, and x is the depth of penetration. A more general analysis employing the same model as Fisher's but with fewer assumptions has been given by Whipple.' In contrast to Fisher's analysis the more general analysis of Whipple does not always yield a straight line relation between In C and x. Recently Turnbull and Hoffman have discussed both analyses and have applied the Fisher-Whipple model to a more refined dislocation model of the grain boundary. They have also discussed the possibility of a large effect of grain boundary direction on diffusion. In the case of the absorption technique, for which results are also given, one simply gets a value for the diffusion coefficient with no indication of the predominating mechanism. Experimental Techniques Single crystals of 99.99+ pct Zn were grown using a modified Bridgman method. Polycrystal samples were prepared in three grain sizes (small, about 4 mm2; medium, about 9 mm'; and large, about 25 mmz) by quenching, air cooling, and furnace cooling. The grains of the polycrystalline samples tended to be columnar, with some preferred texture such that the c-axis was inclined at about 20" to 25" from the normal to the diffusion face, particularly in the medium and large grain size samples. The samples were cut from the rod, polished, and heavily etched, and faces cut accurately parallel using a jeweler's lathe. They were again etched to remove the lathe smear and annealed for 3 hr at 250" to 300°C. After annealing, they were etched, examined, and the orientation determined by Laue back reflection photographs. They were then electroplated with Zn. The plated surfaces were examined for uniformity; nonuniform coatings were removed by etching and the samples replated without additional preparation. For the diffusion anneals, the samples were placed in pairs, with active faces together, in evacuated Pyrex tubes. Constant temperature baths were used, and the variation in temperature did not exceed ±l°C, Diffusion runs were made at l00°,

Jan 1, 1957

By G. C. Kuczynski

Two particles in mutual contact form a system which is not in thermo-dynamical equilibrium, because its total surface free energy is not a minimum. If such a system is left for a certain period of time, the bonding of the two particles will take place in order to decrease the total surface area, even though the temperature is lower than the melting point. This phenomenon of bonding of two or more particles with the application of heat only and at temperatures below melting point of any component of the system will be called sintering, although the powder metallurgists use this term in a broader sense, including the presence of molten phase and pressure. It is the objective of this paper to study this process and the mechanisms involved in it. This problem is of utmost importance to powder metallurgy and powder ceramics, and its technological aspects have been studied for a good many years. However, the powder metallurgical operations are too complex and include too many superimposing mechanisms, and too many variables for a direct study. It was therefore advisable for the purposes of this study to reduce the variables to a minimum. In this work the radius of the interface formed during bonding in a simple system composed of a spherical particle and a large block of the same metal was studied as a function of time and temperature. It is believed that the mechanism involved in this simple process is fundamental to any sintering operations. Previous Work J. Frenkell was the first to make a serious attempt to develop a theory of sintering. He assumed that the process consists of a slow deformation of crystalline particles under the influence of surface tension which reduces to a viscous flow where the coefficient of viscosity n is related to the self-diffusion coefficient D by the following equation 1 _ D D6kT [1] where 6 is interatomic distance, k the Boltzman constant, and T the absolute temperature. This type of viscous flow of a crystalline substance is essentially different from the ordinary plastic flow. The latter is a specific propcrty of crystals and cannot take place in amorphous bodies. According to Frenkel this viscous type of flow is due to the diffusion of the holes or vacancies arising in the lattice. He was able to derive an equation relating the growth of the interface between two spherical crystalline particles or between a particle and a semi-infinite crystal (Fig 1) to time t at constant temperature. This relationship can be written as follows: x²— 3/2 a/nt [2] where x is the radius of the interface assumed to be circular, a the original radius of the sphere and a surface tension of the material. The other assump- x tions are that x is less than 0.3 and that during the period 1 the original radius, a, of the metallic particle did not change appreciably. A. J. Shaler and J. Wulff2 followed closely the ideas of Frenkel in their theory of sintering of a mass of metallic powder. Neither Frenkel nor Shaler has validated his theoretical speculations with conclusive experimental data. Two measurements reported by

Jan 1, 1950

By W. C. Hagel

Previous inuestigators have repovted unusually low H* and Do values for self-dzf@szon in certazn bcc metals, e.g., chromium nnd y -uvanium. It has been postulated that this is nn experimental crl -tetion for the existpnce of a four -atom ring meclzanzsm, mid there could be. zmpovtant theoretrcal Implicatlons. By utilizing a precise sectioning technique with Cr51 as the tracer, self-dzffi-lsion measurements in large grains of 99.99-pct Cr were extended from 1200° fo 1600"C. A least- squares analysis of the data yields that D 0.280 exp (-73,20U/RT) cm2 sec-'. These AH* and Do unlues are not anomalous; there is little reason to believe that n vacancy mechanism is not opevatiue. Apparent differences at lower temperatures may result from preferential diffusion along short-circuititing paths and/or sample contamination. WITH continued technological effort, chromium and chromium-base alloys should ultimately become useful high-temperature structural materials. Accurate measurements of self-diffusion coefficients in the pure metal are necessary for a quantitative understanding of sintering, grain growth, creep, and other related rate processes. However, two conflicting determinations have been reported.* Using an Since careful diffusion studies normally yield Do values ranging from 0.1 to 10 cm2 sec-l, the Paxton-Gondolf results require independent experimental corroboration before an undue amount of speculation arises to explain the apparent anomaly. As discussed by Nowick6 and clearly shown by Hoffman and Turn-bull7 for self-diffusion in silver, a low AH* and Do can arise from preferential diffusion along grain boundaries or other short-circuiting paths. The most reliable data come from high-temperature measurements where volume effects predominate. Following a precise sectioning technique developed for determining silver diffusion in the intermetallic compound AgMg8 and cation diffusion in the corundum oxide Cr2O3,9 the purpose of this study was to measure true volume diffusion in high-purity, large-grain samples at higher temperatures than considered previously. Another point of interest is the frequent proposal10-12 that chromium undergoes an allotropic transformation between 1375" and 1800°C where the diffusion data might then show a discontinuity. I EXPERIMENTAL 1) Sample Preparation. The highest purity (99.997 pet) Cr currently available is produced by reduction of chromium tri-iodide. Three pounds of the resulting crystal bar were shipped in a sealed container with a complete chemical analysis from the vendor.* In conformity with the procedure used for

Jan 1, 1962

By R. E. Hoffman, R. A. Ward

The self-diffusion coefficient in high purity nickel has been measured over the temperature range 870' to 1248°C. The results are described by the relation D = 1.27 exp[—-66,800/RT 1cm2ec-1. The measured activation energy correlates satisfactorily with absolute melting point, heat of fusion, and heat of sublimation. KNOWING the rates of self-diffusion is important in the understanding of many metallurgical reactions. However, in spite of the rather widespread interest in nickel, both commercially and in fundamental investigations, no measurement of the self-diffusion in solid nickel has previously been re- ported." In connection with a program of measuring rates of diffusion in some nickel alloys, self-diffusion in pure nickel has been investigated over a rather wide temperature range by combining a sectioning technique and a surface counting technique.

Jan 1, 1957

By J. D. Meakin

The self-diffusion coefficients of ß tin have been deterttlltled using a plating and sectioning technique. The principal diffusivities pavallel and perpendicu1ar to the "c" axis are given by the Arrhenius relations Dc = 8.2 exp (-25,600/RT) sq cnl per sec and Da = 1.4 exp (-23,300/RT) sq cm) per sec. The results are discussed in terms of a vacancy mecharism of diffusion. In conjunction with an investigation of diffusion in dilute indium-tin alloys we have reexamined self-diffusion in tin single crystals. The published self-diffusion coefficients of tin, eensham') have always appeared anomalous in comparison to diffusion data for other metals and do not correlate with parameters derived from creep tests.' In addition to the necessity of confirming the self-diffusion values for use in analyzing the alloy work, self-diffusion in an anisotropic lattice is of considerable fundamental interest and confirmation of the published data seemed worthwhile. For a crystal with an axis of symmetry three-fold or higher the diffusivity at an angle 6 to the axis is given by D? = (Da - Do) sin2?+ Do [1] where Da and Do are the principal diffusivities perpendicular and parallel to the axis. For a particular diffusion mechanism, again in a crystal of axial symmetry, these principal diffusivities are [2] Do - ? ?i2 COS?i In these expressions ?i is the displacement undergone by the ith atom and 8, is the angle this displacement makes with the symmetry axis. In this, as

Jan 1, 1961

By H. W. Mead, C. E. Birchenall

SELF-DIFFUSION of iron in austenite is a process which may play a significant role in some of the practically important reactions which occur in solid irons and steels. It also provides a system in which the interaction of vacancies on substitutional sites and atoms in interstitial positions may be investigated. For both purposes, accurate values of the diffusivities are required. Two sets of previous measurements exist, each containing features which require checking. In the course of an investigation to determine the rate of diffusion of iron in certain steels, Garwick and Rosenqvist' measured the self-diffusion coefficient of iron in a 0.10 pct* C steel and obtained a dicated that while at lower temperatures D increased with increasing carbon content, at the higher temperatures it first increased and then decreased. At the same time the activation energy decreased linearly from a value of 69,000 cal per g-atom for pure iron to 33,000 cal per g-atom at 1.06 pct C. This rapid decrease in activation energy implies that iron at higher temperatures within the austenite range would diffuse more slowly in high carbon steels than in steels of lower carbon content. Experimental Procedure The radioactive iron used in this work was obtained from Oak Ridge National Laboratory. The unit consisted of about 1 microcurie each of Fe7 and Fe It was allowed to decay for two years. At the end of this time most of the Fe with a half-life of 46.3 days, had decayed, while most of the original Fe(half-life 2.91 years) still remained. The iron as chloride was extracted into an ether layer, using the method of Ashley and M~rray. After evaporation of the ether the iron chloride was dissolved in 250 ml of aqueous solution containing 2.4 gpl Fe. The compositions of the steels used are listed in Table I. Disks of ¾ in. diam and about ¼ in. thick were machined from these steels for use in diffusion couples. The disks were surface ground on both sides to ensure parallel opposite faces. One face of each disk was polished through 000 emery paper. Half of the prepared disks were plated with radioactive iron. For this, 1 ml of the radioactive iron

Jan 1, 1957

By L. Himmel, R. F. Mehl, C. E. Birchenall

The rates of self-diffusion of iron in artifically prepared wustites of various compositions have been determined using the decrease in surface activity technique. Similar measurements are reported for artificial magnetites of nearly stoichiometric composition and for natural hematite single crystals. These data, together with appropriate thermodynamic data derived from the present study or from the literature, are used to calculate the rates of oxidation of iron and its oxides, taking as a basis the theoretical rate equations developed by Wagner. The calculations are compared with experimental rate constant data for these same reactions and are shown to provide essential confirmation for the Wagner theory. For the most part, the results are also consistent with the previously proposed transport mechanisms in the oxides of iron. WHEN a metal is exposed to an oxidizing atmosphere at elevated temperatures, a superficial oxide layer is formed which then thickens according to some characteristic rate law. Provided that the oxide layer is dense and continuous and does not present any paths for gaseous diffusion, continued growth must inevitably involve the migration of the reactants—either the metal or oxygen or both—through the lattice of the solid oxide. This is true whether the reaction rate is controlled by the diffusion process itself or by a slow step in a reaction at one of the interfaces. If the overall rate is limited by diffusion, and if the concentrations or thermodynamic activities of the reactants at the boundaries of the oxide layer are independent of time, the familiar parabolic rate law is observed. Under these conditions, Wagner1 has shown that the rate constant may be expressed in terms of the mobilities of the reacting components and the appropriate concentration or activity gradients which are established across the growing oxide layer. In presenting the phenomenological basis for the theory, Wagner has emphasized the role played by lattice defects in diffusion processes in oxides. The essential validity of this theory has been confirmed by experiment for a number of simple systems including the growth of Cu2O on copper.'-' One of the objectives of the present study has been to extend the Wagner approach to the more complex process of the oxidation of iron and its oxides, and particularly to wustite, which is normally the major constituent of the oxide scales formed on iron at temperatures above approximately 570°C. For this purpose the rates of self-diffusion of iron in w?stite, magnetite, and hematite have been measured by tracer techniques and the necessary thermodynamic data have been obtained from equilibrium studies involving the oxides and their atmospheres. These data have subsequently been used to test the underlying assumptions of the Wagner theory as well as the detailed mechanism of the scaling of iron previously proposed by Davies, Simnad, and Birchenall.5 Lengthy treatments of diffusion and conduction phenomena in ionic crystals have been published." * The latest, by Jost,8 also summarizes much of the existing experimental data. In general, however, the oxides of iron have not been considered specifically and hence a brief review of the available

Jan 1, 1954

By H. I. Aaronson, H. A. Domian

The self-diffusivity of Ag10 has been measured as a function of temperature and composition in AgMg. a CsCl-type intermetallic compound with a substitutional defect structure on both sides of the stoichiometric (50 at. pct Mg) composition. These measurements show that In D increases linearly, and both In Do and Q decrease linearly with deviation from stoichiometry. The slopes of these lines are significantly steeper in magnesium-rich alloys. Equations for the variation of Q with composition have been derived by application of the theory of random walk processes on the assumption that diffusion in compounds of this type takes place by the six-jump cycle mechanism of Huntington, Elcock, and McCombie. The favorable agreement obtained with experiment furnishes fresh support for the correctness of this assumption. A principal result of this treatment is that. although the concentration of vacancies is considerably higher in the silver sublattice than in the magnesium sublattice, the acfivation energies for movement of the magnesium sublattice vacancies are so much lower that these vacancies are responsible for effectively all diffusion of silver in this compound. A tendency for vacancies of both types to be kinetically bound to wrong atoms is also noted. THIS study was undertaken to examine the influence of substitutional defects upon self-diffusion in an intermetallic compound, AgMg, with a CsCl-type crystal structure. The CsCl structure is exception- ally simple. At the equiatomic or stoichiometric composition in a binary alloy, it consists of two interpenetrating simple cubic sublattices, each of which is exclusively populated by the atoms of one species. When atomic size and other factors are favorable, deviations from stoichiometry can be entirely accommodated by the substitution of atoms of the species present in excess of 50 at. pct on the sublattice of the other species on a one-for-one basis. Under this condition, the only vacancies present are those formed by thermal activation. The excess atoms thus incorporated into the sublattice of the minority species may be considered as substitutional defects in this sublattice. Several X-ray studies have shown that the defect structure of AgMg is entirely of the substitutional type on both sides of the stoichiometric composition.1"3 The thermodynamic investigation of Kachi indicates that, aside from a low level of thermal disorder and these constitutional defects, order is maintained to the solidus temperature. This compound is therefore a particularly useful vehicle for a study of the influence of substitutional defects upon diffusion kinetics. Other advantages of AgMg for this purpose include: the broad composition region, as much as approximately 10 at. pct on either side of the stoichiometric (50 at. pct Mg), in which the compound is stable; the comparatively low temperature range of the solidus (820°C and below),5 making diffusion anneals near this range experimentally convenient; the reasonably good machinability of the compound, permitting utilization of the standard sectioning method of radiotracer analysis; and the availability of a good tracer, A for one of the constituent species. While this investigation was in progress, Hagel and westbrooks reported data on the self-diffusion

Jan 1, 1964

By Appendix by A. S. Goldoni, R. E. Tate, E. M. Cramer

The diffision coefficient for self-diffision of plutonium in the temperature range 350" to 440°C has been measured by using puZ3 as the tracer isotope. Autoradiopaphic techniques were used to inzlestigate the possibility of grain boundary diffusion, hut only evidence for volume diffusion was found. The least-squares fit of the data gives the .following equation for the diffision coefficient: The computer least-squares technique for fitting the nonlinear equations is outlined. DETERMINATION of the self-diffusion coefficient of the fcc 6 phase of plutonium is in principle a straightforward experimental task. However, the chemical reactivity, the intense @ activity, and the toxicity of plutonium put limitations on the experimental techniques. The technique selected included roll bonding for preparation of the diffusion couples and pulse-height analysis of the @-particle activity to determine the distribution of the PU tracer in the diffusion couples. EXPERIMENTAL PROCEDURE Couple Preparation. Cylinders about 0.5 in. in diam were cast from two special stocks of plutonium, one of which had been enriched in puZ3', as shown in the isotopic analyses listed in Table I. For each roll-bonded composite sheet, a cylinder 0.437 in. in diam and 0.190 in. thick was turned from each kind of plutonium on a lathe in a 98 pct He atmosphere. The two freshly machined cylinders were positioned face to face in a tube of commercially pure aluminum which had been sealed at one end by welding and the assembly was evacuated on a vacuum manifold overnight. The elapsed time between machining the faces of the plutonium cylinders and evacuating the loaded tube was about 15 min. After overnight evacuation of the assembly the indicated vacuum was 1 X 10"5 torr or better. The aluminum tube was then warmed and pinched off with a cold-welding tool. The pinched-off weld was also fusion-welded as an additional precaution against leakage of air into the evacuated assembly. The assembly was immediately heated for 30 min in a 250°C furnace and reduced in thickness by being passed through a rolling mill with rolls heated to 200°C. The rolling schedule (four passes of 100 mils each with a 10-min reheat after two passes) reduced the thickness of the assembly to 100 mils. The rolled assembly was allowed to cool normally in air. The rolled assembly was sheared at the edges and the aluminum peeled from the composite plutonium sheet. The elliptical sheet was about 0.060 in. thick and usually three 0.383-in.-diam disks could be punched from its central portion. The sheet was heated on a hot plate to the ductile low 0 range (140°C as measured by temperature-indicating pellets) before each disk was quickly punched. The quality of the bond in the disks was evaluated by metallographic examination of the scrap sheet adjoining the hole left by the punch. Only well-bonded specimens without oxide in the interface (as indicated by metallography) were considered satisfactory for further use. More than half of the specimens so examined contained sufficient oxide in the interface to be rejected. Diffusion Anneal. Each disk to be diffusion-annealed was wrapped in 1-mil-thick tantalum foil and sealed within a Pyrex capsule evacuated to 1 x 10~5 torr or better. This capsule was then sealed within another Pyrex capsule at a similar pressure. The diffusion anneals were carried out at temperatures between 350" and 440°C in Marshall furnaces adjusted to have a temperature gradient of not more than *1/2"C over a 5-in. length. This gradient was then further smoothed by using a nickel tube as a liner in the furnace. The liner was divided into three longitudinal cavities by a septum of nickel sheet to which two calibrated Chromel-Alumel thermocouples were attached. One thermocouple was used for controlling the furnace temperature by means of a Brown Pyrovane controller equipped with a Capaciline anticipation circuit; the second thermocouple was monitored twice daily with a Leeds and Northrup K-2 precision potentiometer.

Jan 1, 1964

By D. L. Feucht, R. L. Longini

The semiconductor heterojunction is considered in terms of simple models which may lead to an understanding of move complex heterojunctions. Metallurgical and electrical properties of hetero-junctions aye discussed including the interface structure, energy -band diagram, and carrier transbovt across the interface. It is found that in a heterojunction all mechanisms such as injection, tunneling, and junction recombination found in simple junctions play modified voles. INTERFACES between materials (grain boundaries, the electrical junction between two differently doped materials in a single crystal, the oxide-metal interface, or metal-metal junctions) are of considerable importance in many situations. These various interfaces all have one very fundamental thing in common. Quantum mechanically speaking, the wave functions of the electrons in one material may penetrate the other material but, in general, only to the extent of angstroms. From an electrical point of view the conduction mechanism changes as a current passes through such junctions. In some cases the change is tremendous, in others almost negligible. The interface, then, is the locus of a change of conduction mechanisms. Some of these, particularly in semiconductors, are well-understood. The ordinary p-n junction in a single crystal can be the locus of an injection mechanism or a tunneling process, depending on conditions. The mechanisms are probably best understood in semiconductors because of the possible simplified view of particlelike conduction. The bands are either nearly filled or nearly empty and band overlap is seldom involved. The same fundamentals are probably important in other situations too but they are very difficult to look at naively. Although the simple look at the semiconductor case only gives us a relatively rough picture which must then be refined, the other systems, which involve a more complex situation, immediately are in many ways too difficult. There are too many initial choices of complex systems and therefore it is not possible to be even reasonably certain of any one model. Because of the relative simplicity of semiconductors, their good and controllable structure, and because of the ability to make many measurements on them not normally available to either metals or insulators! they are probably the best understood materials. It is therefore desirable to use them as a tool to further the understanding of interfaces in general. Semiconductor-heterojunction concepts were first proposed by kroemer1 in 1957. This was followed several years later by reports on the fabrication and experimental characteristics of heterojunction structures by Anderson2 and Diedrich and jotten.3 I) THE HETEROJUNCTION STRUCTURE To get down to hardware, when we refer to a semiconductor heterojunction we imply that there exists an intimate contact between different semiconductor materials. We could put two pieces of material together, complete with oxide layers, we could remove the oxides, or we could even melt the interface and hopefully get wetting and a good "bond" on solidifying. In fact we could by some means grow a crystal of one material using the other as a seed. Essentially we are interested only in the last two because they are the simplest to look at analytically. The degree of perfection of fit varies greatly and is reflected somewhat in the arc welder's joint strength. The lattice match of the two materials, their orientation, and so forth. is obviously necessary for a good bond but so is the continuity of any polar bonds which are involved such as in the III-V semiconductors. The mechanical misfit between two similar lattices can be described in terms of edge dislocations. The edge-type dislocations must be very close together for the usual misfit and there must be dislocations for each of several different Burger's vectors in order to produce a lattice match. The .'dangling bonds'' resulting will be involved in producing interface charge. Order of magnitude estimates of the charge density extrapolated from low densities of dislocations in homogeneous materials give 5 x 1013 cm-2 Ge-Si and 1 X 1012 cm-2 Ge-GaAs electronic charges. Edge dislocations also act as very active recombination centers between holes and electrons. One lattice "matching" difficulty usually exists even if two structures have essentially the same lattice constants as they will have different coefficients of therma1 expansion. Thus, on cooling from the usually high temperature of fabrication to room temperature, dislocations are produced, a good fit not existing at both temperatures. In brittle materials this shrinkage may even result in cracking. For the Ge-Si interface the mismatch is about 2 x 10 -6 per degree whereas it is less than 10"7 per degree between germanium and GaAs. The exact effect of the misfit is dependent on the thickness of the materials involved. For a very

Jan 1, 1965

By R. Maddin, S. K. Tung

SUCCESS of the dislocation theory in formulating the transitional lattice theory proposed by Har-greaves and Hill in 1929' is well established for low angle grain boundaries. The theoretical work of Burgersa and Read and Shockley,3 together with the experimental observations of Aust and Chalmers4 and Parker and colleagues,6 leaves little doubt as to the structure of grain boundaries with differences of orientations up to between 15" and 20". Beyond these angles, the structure of the boundary is not so clear, and is still a matter for conjecture. There have been notable experiments devised from which the data could be used to calculate energetics for possible arrangements of atoms in forming a large angle boundary.6 Most of these experiments deal with diffusion through and along the grain boundary. The recent work of Rhines and Cochart7 and Rhines, Bond, and Kissel18 was concerned with the relative movement of grains in aluminum bicrystals under constant tensile loads with their boundaries aligned at 45 " to the tension axis. The gliding along the grain boundary was cyclic and began after an incubation period. The velocity of gliding increased with the sum of the angular difference, 8, between the operative slip direction in the conjugate crystals and the angular difference between the active slip planes as measured between their traces upon the grain boundary. The initial gliding rate varied with temperature, and the logarithm of the average gliding taken over a considerable period was a linear function of the reciprocal of the absolute temperature: the slope gave an activation energy of 11,000 ±1,000 cal per mole. Since the rate of grain boundary shear is sensitive to the misorientation of the grains themselves, as shown by several workers,7,8 it is suggested that further work of this type might be fruitful in adding information regarding the behavior of boundaries under stress. The techniques devised by Chalmers" added impetus to the facility for observing grain boundary behavior, in that they provided a simple means for producing bicrystals of predetermined orientation so that the angle of the boundary could be controlled,

Jan 1, 1958

By Robin O. Williams

The textures which are produced by simple shear in poly crystalline samples of copper, brass, aluminum, iron, and zirconium have been determined. For the fcc materials, there are two major textures, both of which are oriented for slip on the usual slip systems. It is shown that each grain will, in general, give rise to both orientations through the production of deformation bands. This is considered a clear demonstration that such bands aye produced by the heterogeneous deformation which takes place instead of the five-slip-system model considered by Taylor. The results on iron and zirconium are more limited but would appear to exhibit certain additional complexities. It is suspected, however, that the general consideration of deformation bands may apply here also. One finds some correlation between the textures produced by this single shear and the conventional textures although it would appear- that additional data and analysis are requived to produce a strong correlation. It is somewhat remarkable that of all the texture investigations only a very few have been concerned with the action of simple shear on polycrystalline aggregates. Of these limited investigations, all used torsional strains and only in Backofen's' and Backofen and Hundy's "ork was sampling carried out so as to derive a texture in terms of the shear direction. Some rather remarkable conclusions re- sulted from Backofens'1,2work: 1) the rate of texture formation in copper is high, being almost complete at a shear strain of unity; 2) the texture does not change if the sample is sheared back, although the surface grains return to their original shape; and 3) the texture is complex in that it has four components, three of which have [110] as a shear direction. These conclusions were based on both qualitative and quantitative data. Shear deformation is intermediate in complexity between simple shear of single crystals and the usual deformation of polycrystals; hence, one might expect to learn considerable from the extension of this work. CONVENTION Deformation in simple shear has the lowest possible symmetr3, less than the other simple modes of deformation (rolling, drawing, and compression); hence, one cannot necessarily expect the textures to have the same degree of symmetry. The geometry of the process is represented by Fig. 1 along with its sltereographic projection. One has the shear plane, the shear direction, the mirror plane normal to the shear plane and containing the shear direction, and a neutral plane normal to both the shear plane and mirror plane. These planes divide the projection into four quadrants. These quadrants are not, however, identical, since all directions within the upper left and lower right are being shortened by the shear process, and the other directions are being lengthened. The first kind are called compression quadrants and the direction of greatest contraction rate, at 45 deg, is labeled with a minus sign; the corresponding direction of greatest extension rate

Jan 1, 1962

By George R. Cowan

A number of studies hare been reported of the effects produced in metals subjected to deformation by shock waves with maximum pressures ranging from tens to hundreds of kilobars. On the basis of the equations for the flow of mass, momentum, and energy through a stationary shock front, the macroscopic stress-strain curve for the resulting shock deformation can be calculated within narrow limits from the experimentally determined Hugoniol curve. In relatively weak shocks which are preceded by an elastic wave, the stress rises above the clastic limit only as plastic deformation proceeds cold thus the shock has a long toe. In strong shocks that override the elastic wave a high stress is applied without prior plastic deformation. A more important effect of increasing the shock pressure is the generation of shear stresses, called supercrilical shear stresses, that exceed the strength of the perfect lattice. A change in the mechanism of deformation is expected to result from the onset of supercritical shear. The shock disordering of ordered Cu3Au in strong shocks appears to be an example of such a change. It is suggested that the formation of fine twins in copper and nickel and the formation of structures which enable visible twins to be formed in the rarefaction ware, observed in copper and presumably in disordered Cu3 Au, are related to the occurrence of supercritical shear in shock dcformation. In recent years several studies1,2 have been made of the changes in structural and mechanical properties of metals produced by the passage through the metals of strong shock-compression waves ranging from about 50 to 800 kbar pressure. Recent work involving dynamic measurements of the shock compression "Hugoniot" curves 3-8 of many metals has developed techniques and provided data required to obtain the shock pressure and the (transient! plastic deformation produced in the shock-conlpression experirnents.9 Shock deformation has been found to be much more effective than slow deformation in changing the mechanical properties of metals, when the two are compared on the basis of equal plasti strain, Holtzman and Cowan9 made quantitative estimates of the shearing stress occurring in a shock front in a metal by assuming that the shearing stress is similar to that occurring in a shock front in a viscous, heat-conducting fluid, with the addition of a yield stress. Taylor's solution9 for a weak shock was used to estimate pairs of values of shearing stress and thickness of the shock front obtained by assumed choices of the ratio of effective kinetic viscosity to thermal diffusivity. It was noted from these values that. unless the shock front is extremely thin. heat conduction has slight effect, and the shearing stress is nearly independent of the mechanism of deformation. This mechanism does, however, determine the thickness of the shock front and the rate of strain. Furthermore, since the maximum possible shearing stress occurring in shocks of moderate strength does not greatly exceed the shear stress occurring in conventional slow deformation, the mechanism of deformation is not expected to be qualitatively different. The greater effectiveness of shock deformation in changing the mechanical properties of metals can be attributed partly to the fact that dislocations, when driven by near-conventional stresses, cannot keep up with the shock front, thus necessitating a higher dislocation density than required for an equivalent slow strain. The fast uni-axial strain occurring in the thin shock front would also be expected to cause a larger number of dislocation intersections to occur. In the upper range of shock pressures that have been studied the estimated values of the shearing stress exceeded the estimated shear strength of a perfect crystal. Under these circumstances it is reasonable to expect that the mechanism of deformation might be considerably different from that involved in slow deformation. Except for the observation by smith1 of twins in shocked copper, the effects of shock waves on metals did not show any obvious or large changes in properties that would indicate the onset of a change in the mechanism of deformation. The recent investigation of the effect of shock waves on ordered and disordered specimens of Cu3Au by Beardmore, Holtzman, and ever" showed a spectacular decrease in the amount of long-range order retained by initially ordered Cu3Au when the shock pressure was raised from 290 to 370 kbar. Since Dr. Holtzman and I suspected that this behavior probably was due to the onset of a shearing stress in the shock front in Cu3Au which exceeded the limiting shear strength of the perfect crystal. it was considered appropriate to examine directly the shock-front equations for a solid. and to obtain a sound estimate of the shearing stress occurring in the front using equation of state data obtained from shock studies. In this paper an estimate is made of the

Jan 1, 1965