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IN this survey of the progressive development of education for the mineral industries throughout the United States, the review of the history of each school has usually been completed wherever it is first mentioned, but the history of the schools in the cities of the eastern seaboard was not carried beyond 1870. It is necessary now to cover what happened during the period of approximately the last quarter of the nineteenth century in several of the schools in the east. PHILADELPHIA Of Philadelphia, it has been related in chapter 4 that the Polytechnic College of Pennsylvania began offering a curriculum leading to the B.S. in mining in 1853; that the school did not prosper, and finally closed about 1890. The establishment of a Department of Mining, Arts, and Manufactures at the University of Pennsylvania in 1855-56 was also mentioned, but its history was not carried beyond 1860. In 1859 J. Peter Lesley was appointed professor of mining in the University of Pennsylvania. The son of a cabinetmaker, born in Philadelphia Sept. 17, 1819, he had taken his A.B. degree at the University of Pennsylvania in 1838 and then secured a position on the Geological Survey of Pennsylvania under H. D. Rogers. When the Survey was discontinued, in 1841, he went to Princeton to study theology, was ordained a Presbyterian minister and followed that career until 1853, when the Pennsylvania Geological Survey was again started and he rejoined its staff. He published "A Manual of Coal and its Topography" in 1856, and studied at the École des Mines, Paris, in 1850-57. He published an 800-page "Iron Manufactures Guide" the year that he was appointed to the staff at the University of Pennsylvania. In 1863-64 a College of Agriculture, Mines, Arts and the Mechanic Arts was established, but it was not a separate school

Jan 1, 1941

United Engineering Trustees, Inc. TAMES L HEAD, '50 E C MEAOHER, '52 R M ROOSEVELT, '49 The Engineering Foundation F F COLCORD, '50 E A PRENTIS, JR , '48 R M ROOSEVELT, '49

Jan 1, 1952

By Pradeep K. Rohatgi, David N. French, Surendar M. Jain, Clyde M. Adams

The effect of solute concentration and diffussivity on dendritic solidification of dilute binary aqueous solutions has been investigated; chlorides of sodium, potassium, lithiurn, and hydrogen were used as solutes. Dendrite spacings were measured in spherical droplets and in unidirectionally solidified specimens. In the droplets, the freezing rate is uniform over the entire specimen and remains constant from the beginning to the end of solidification. In the unidirec-tionally frozen specimens the freezing rate varies within a given volume of liquid during the solidification cycle; in addition, it varies with location uithin the specimen. Mass transport analysis has been used to calculate the supersaturation, AC, supercooling, AT, and transcerse growth velocity, V , of ice dendrites during solidijication from the dendrite spacings measured in the droplels. The ice dendrite spacing in a given solute increases lineary with the solule concentralion at a fixed freezing rate. The ualues of AC, AT, and V increase with an increase in solule concentration. The value of AT in the aqueous solution is of the same order of magnitude as in Al-Cu alloys of the same concentration; the dendrite spacings in Al-Cu alloys also increase with an increase in solute concentralion. For the jamily of aqueozrs solulions incestigaled, the dendrite spacing increases linearly with solute diffusivity at fixed values of concertralion and freezing ,rate. The values of ?C and ?T decrease, whereas the value of V increases with an increase in solule diffusicity. The measured dendrite spacings , L, can be expressed by the relatiun: L = K;(dfa/db) 4 K2C +K3D - K4 where dfs/d? is-change in fraction solid with time (freezing rate), C is solute concentration, D is solute diffusivit K1 = (14 * 4)pmin1'2, K2 = (70 ± 10) pl (moles)- K3 = (24 r 3) . l05 psec (cm)-~. K. = -132 ± 10) 1. FREEZING of solutions is almost always an ehunmix-ing" process because the composition of crystallizing solid generally differs from that of the parent liquid. " Solute concentration and transport are therefore in]portant considerations in solidification of solutions. The influence of these factors on the dendritic solutions produced from metallic solutions is of practical significance since the properties of the solid aggre-gate are often strongly influenced by dendrite spacing.''' Alexander and Fthines3 found that dendrite spac- ings generally increase with solute concentration in several metallic systems. Michael and Bever1 have observed an increase in a aluminum dendrite spacings in dilute A1-Cu alloys. Rohatgi and AdamsJ have reported a linear increase in a aluminum dendrite spacings with increasing copper contents up to 28 pct in A1-Cu alloys. Bolling and Tiller"' have theoretically examined the effects of solute concentration and diffusivity on the growth of cells and dendrites. Their analysis of dendritic growth assumes a steady-state growth and the maximum possible velocity for dendrite propagation. In addition, the quantitative analysis is restricted to dendrites of rotational symmetry and is therefore not directly applicable to the plate-shaped dendrites of ice obtained in the present study. There are practically no experimental observations of the effect of solute diffusivity on dendrite spacings in metallic systems. The reason appears to be the lack of experimental data on solute diffusion coefficients in liquid metallic solutions. The present work is concerned with the effect of solute concentration and diffusivity on dendritic solidification of dilute aqueous solutions: the solutes examined are potassium chloride. sodium chloride, lithium chloride. and hydrogen chloride. There were several reasons for selecting these solutes. The variation of dendrite spacing with freezing rate in these solutiolls is similar to that observed in A1-Cu alloys.8 These aqueous solutions have almost identical phase diagrams at dilute concentrations. Fig. 1. and the diffusion coefficients show very little variation with solute Concentration.9,10 Dendrite spacings were measured as a function of freezing rate. solute concentration. and solute diffusivity. Dendrite spacings were measured in two different types of freezing systems. droplet and unidirectional, respectively. In the droplet freezing system

Jan 1, 1970

(Members are urged to send in for this column any notes of interest concerning themselves or their fellow-members.) Members and guests who called at Institute headquarters during the period Mar. 10, 1916 to Apr. 10, 1916: R. C. Blanchard, London, England. George H. Packard, Boston, Mass. F. Lynwood Garrison, Philadelphia, Pa. Harvey B. Small, Mangangue, Colombia, S. A. S. A. Horton, Altoona, Pa. H. De Witt Smith, Kennecott, Alaska. J. S. Lane, New York City. Richard Tarantous, Terlingua, Tex. Harry T. Mercer, Painesdale, Mich. S. A. Taylor, Pittsburgh, Pa. A. L. Blomfield, for many years in the employ of Bewick, Moreing & Co. as metallurgist, and for the last eight years mill superintendent of the Golden Cycle mill at Colorado Springs, has become associated with The Dorr. Company as consulting metallurgical engineer, with headquarters at Denver. He retains his connection with the Golden Cycle mill as general superintendent.

Jan 5, 1916

On. Tuesday, Apr. 8, the Rocky Mountain Club held its 12th annual dinner at the Waldorf-Astoria, New York. In reporting the dinner the Evening Telegram says: "General Coleman du Pont was sort of master of ceremonies, and he conducted the affair at the end of a Colt .45. It was the General's `forty-five' which filled the banquet hall of the Waldorf-Astoria with the brand of smoke which Uncle Sam showered so generously upon Germany's hordes. It was General du Pont's `forty-five' which bellowed and thundered and echoed through Fifth avenue, scaring horses and shell shocking pedestrians: But no one was hurt or badly shocked except those who could not discern the artistry of Miss Doraldino's dance. "There were Western men around the table who have just come hack from over there. For instance, there were Brigadier General G. H. McManus, General Shanks and a score of others. And there were Thomas H. Watkins, who is said to own half the State of Pennsylvania and is known as the `Coal Baron'-of Pennsylvania,, and Alba B. Johnson, president of the Baldwin Locomotive Works, each one a Westerner, with personality and type suggestive of some of the wonderful men Bret Harte wrote so much about. "It seemed as though General du Pont, who was chairman of the Reception Committee, tried to eclipse all former meetings, and he succeeded if congratulations are a criterion. " General du Pont startled the management of the Waldorf and surprised the diners by staging a three-round bout between Frankie Fay and Phil Wagner. The clash between these two featherweights made Waldorf history, because it was the first boxing contest ever held in the hotel. Tex Rickard, who is going to chaperone Jess Willard and Jack Dempsey in a fight in the near future,-was scheduled to referee the contest, but at the last moment, for some obvious reason, M. E. Schermerhorn was sent in as third man. The bout was a slashing one. Miss Amparito Farrar, lately returned from France, where-she sang for the 'dough-boys;' Eleanore Wetmore, the Rath Brothers and Phil Baker were among the others who entertained.

Jan 5, 1919

By R. P. Marshall

This note describes positive evidence that hydrogen in titanium alloys diffuses under the influence of a thermal gradient. The experiments confirmed the expected similarity of this system to the H-Zr system and produced qualitative information on the magnitude of the heat of transport for hydrogen in titanium alloys. This effect has practical importance in applications where titanium alloys operate in a thermal gradient under conditions such that hydrogen embrittlement may develop within the expected lifetime of the structure, for example in supersonic aircraft service. The theories of thermal-gradient diffusion may be found in recent reviews and experimentation in this field by Shewmon,1 Sawatzky,2 Markowitz,3 and Darken and Oriani.4 The basis for Soret thermal diffusion is the "heat of transport", denoted by Q*. This is an energy term for the amount of heat (energy) transferred by atomic diffusion, and can be either positive or negative signifying a driving force for atomic diffusion to the cooler or hotter region, respectively. Experimentally, Q* is determined by plotting the natural logarithm of the point-to-point concentrations after exposure to a thermal gradient vs the respective point-to-point temperatures in units of the inverse absolute temperature scale. The slope of the straight line is Q*/R, R being the gas constant. For the titanium experiments, specimens (3 by 1 by 1/16 in.) of a) commercially pure titanium, b) the 8 Al-1 Mo-1 V titanium alloy, and c) the 6 A1-4 V titanium alloy were exposed to thermal gradients of 6° to 115°C per in. for 18 days at 250° to 550°C. The temperature gradient was in the direction of the 3-in. dimension. The hydrogen concentrations existing after exposure were determined on sections of the specimens by vacuum extraction. Three sections of unexposed sheet of each alloy were analyzed as standards to allow a check for hydrogen

Jan 1, 1965

Canada's newest low-grade porphyry copper mine, Gibralter Mines Ltd. (N.P.L.), has reached production. The subsidiary of Placer Development Ltd., located at Williams Lake in the Caraboo District of central British Columbia, began open pit mining in mid-June on reserves estimated at 358 million tons assaying 0.373r: Cu and 0.016C; MoS,. Each of four separate zones will be mined in a predetermined sequence, in one or more stages, and with a throughput of more than 36,000 tpd. The sequence and stages of mining the zones have been determined by computer calculation and economic analysis and are designed to maximize recovery and minimize cost. At the very least, the mine is off to a promising start. Development costs leveled off at $67 million. 9'; below the anticipated budget.

Jan 7, 1972

By Jesus P. Manzano

The report of which this paper is a summary was made in 1890 for the Compania Investigadora Mexicana y Americana. Since the region examined covers 900 sq. kilom., chiefly of wooded arid rocky hills, almost without roads, and for the most part uninhabited, While the examination had to be made in two months, of which period more than 15 days were continuously rainy, I need not say that this account is not complete and thorough. I. Geography and Topography. The center of this zone, 19 kilom. SW. of San Luis Potosi, is situated in lat. 21° 24' N., and long. 2° 3' W. of Mexico City. The altitude ranges from 1866 m. (that of San Luis) to . 2816 m. (that of the Realito mountain), in the Sierra de Sail Luis. The topography is hilly—partly wooded, partly bare. The principal more level areas are the Bledos valley on the south, and portions of the haciendas of Tepetates, San Fran cisco and Santiago, on the west. The water-supply is scanty. The two streams that cross the Zone, the Santiago and Bledos rivers, carry water nearly the year through, though much reduced in the dry season. Other streams are torrents of less importance. All the waters of the valley and Sierra of Sail Luis arc absorbed or sink, not passing beyond Soledad de los Ranchos. The Continental di vide is within the Zone, some waters going west to the Pacific, others east to the Gulf of Mexico. II. Geology. Concerning the geology of this district I condense the fol lowing from the memoir, Las Minas de Guanajuato, prepared for the Minister of Fomento by Senor Pedro Lopez Monroy, mining engineer, with which I am entirely in accord. He says, ill substance:

Jan 1, 1902

Jan 1, 1967

By John Dilworth

THE word "development," as used in connection with .mining, is a rather general term and in most instances must be qualified or explained before the exact thought in the mind of the user is made clear. For instance, by the development of a mining district is often meant its entire facilities of mineral production, such as railroads, towns, mine plants, etc. In speaking of the development of a property its prospecting by test pits or diamond drill borings may be referred to; or it may be the mines, strip pits or quarries from which the coal, ore or rock is won. But, as a rule, when the development of an individual mine is spoken of the idea in mind is the system of underground workings by means of which the mineral is recovered and transported to the surface. It is this character f development that is considered in this paper. The discussion will be confined to the ordinary room-and-pillar, system of mining. practically horizontal coal seams that is used in most bituminous coal mines in this country, and to only two phases of this subject; viz., what is properly included under the head of "Development"' as it appears on the books of a coal company or the inventories of an examining engineer, and how its "value" should be determined -by the accountant or engineer. No absolute rule or definite agreement covers these points, yet these questions are far more than academic. Development is often an important part of a mining company's assets, affecting the amount of its invested capital (and hence its rate of taxation under present Federal income tax laws) as well as the value of its total assets and the amounts chargeable to cost of coal and to depreciation.

Jan 7, 1921

By L. F. Stutzman, George Thodos

A new graphical method, which is a modification of that proposed by Buckley and Leverett', is presented for the determination of the displacing fluid saturation at breakthrough for frontal drive mechanisms. This method is applicable to all types of fluids, which may be either soluble or insoluble in the oil at reservoir and injection conditions. This adaptation has unique advantages in handling problems in which either liquids or gases are the displacing fluids. INTRODUCTION In the displacement of oil from a reservoir with water or gas injected into the formation or by the expansion of a gas cap, the determination of breakthrough or flood advance is important to define the initial and subordinate phases of the operation. Thus, the determination of the displacing fluid saturation at breakthrough is fundamental to the establishment of over-all reservoir production. Methods for analyzing this saturation and for calculating production rates have been previously reported1,'*".". All of these methods utilize the rate equation for linear flow developed by Buckley and Leverett1. xs = QT - Qn/A? (?fu/?Su) .....(1) Sp where f, represents the volumetric fraction of displacing fluid in the total flowing fluid within the reservoir as is defined by: fo = 1/1+ ku/kd µn/µn .......(2) Since the time, ?, necessary for the flood front to travel distance ?X is unknown, Eq. 1 cannot be solved directly for the determination of the breakthrough point. Therefore, it becomes necessary to solve Eq. 1 simultaneously with an equation representing a fluid bal-ance on the reservoir behind the flood front. To develop such a fluid balance equation, it has been assumed that the fluid composition is uniform in the reservoir behind the flood front. Based upon this assumption, the follow-ing equation can be derived from a material balance on the displacing fluid: x = (?X) s Dx ? x=0 [ Snx - Sn1 ] dx = QT - Qn/A? It is readily observed that the displacing fluid satura-tion behind the flood front, Snx, may be determined through the simultaneous application of Eqs. 1 and 3. Although the approach may vary, this objective re-mains the same for all of the previously mentioned graphical or analytical methods. After Sox is determined, it is then possible to calculate the gas-oil and/or water-oil ratios and the rates during the initial and subordinate phases of production, and thereby define the production history of the reservoir. In this presentation only the de-termination of Snx is considered, and the reader is re-ferred to the several texts in reservoir engineering2,4,5 for a complete discussion on the methods for establish-ing the production history of a reservoir. PROPOSED METHOD FOR DETERMINING Snx Since at any time the quantity, QT - Qn/A? is a constant, Eq. 1 may be expressed in differential form as follows: dxsn = QT - Qn/A? d (? fn/? Sn) sn .....(4) Eq. 4 may be substituted directly into Eq. 3 to obtain: ? fn/? Sn ? [ Snz - Sn] QT - Qn/A? d ( ? fn/? Sn) = QT - Qn/A? ...............(5) Eq. 5 simplifies to the expression: ? fn/? Sn ? [ Snz - Sn] QT - Qn/A? d ( ? fn/? Sn) = 1 ....(6)

Jan 1, 1958

By B. D. Lichter

In a previous paper, Oelsen, Schuermann, and Hey-nertl pointed out the possibility of obtaining complete thermodynamic functions for alloy systems from calorimetric measurements alone. Specifically, if the heat content of a heterogeneous, multicomponent system is known as a function of temperature and composition, then it is possible to derive the heats, entropies, and free energies of formation of liquid phases and of solid intermediate phases of fixed composition. The authors present equations and graphical techniques applicable to eutectic systems and systems with intermediate phases without solid solubility at lower temperatures. eijering has considered possible extension of the treatment to systems with complete solid-state miscibility at lower temperatures; however, experimental difficulties practically restrict the method to systems with very limited miscibility in the solid state. The purpose of this note is to present an alternative derivation of the heats and entropies of formation of liquid phases and of solid intermediate phases from heat-content measurements alone. Fig. 1 shows three types of phase diagram to which the analysis may be applied. If the heat content is determined as a function of composition x, at temperature T, we have quite generally and the heat of mixing of the liquid is obtained directly from heat-content measurements. If the temperature T lies in a two-phase region (solid-liquid) then

Jan 1, 1967

Jan 1, 1970

Jan 1, 1967

By T. K. Smith

The comments contained in the discussion of the paper concerning the prediction of fracture gradients are of considerable interest, since the subject is of a very controversial and timely nature. It is regrettable and Fairhurst2 contradicts the author's statement in the third paragraph of Other Applications. These references create significant doubt that Eaton's Eq. 1 predicts fracture pressure during the injection of fluid into old wells. 3. No evidence is presented in this paper to support the author's Conclusions 2 and 3. While an empirical coefficient related to back-calculated field data is correlated with depth (Eaton's Fig. 4), there is no reason to suspect that this coefficient might be the Poisson's ratio of the various formations involved. 4. Until the field data supporting the empirical approach proposed by the author are presented and evaluated, the accuracy claimed in Conclusion 5 is without substantiation. References 1. Geertsma, J.: "Problems of Rock Mechanics in Petroleum Production Engineering", Proc., First Intl. Symposium on Rock Mechanics, Lisbon (1966) 585. 2. Haimson, B. and Fairhurst, C.: "Hydraulic Fracturing in Porous-Permeable Materials", I. Pet Tech. (July, 1969) 811-817. that the comments are those of a critic and do not offer an alternative and better method for the prediction of fracture gradients. Nevertheless, a reply is in order.

Jan 1, 1970

University of New Mexico, Albuquerque, N M Requests for publications should be addressed to the Librarian of the University Papers of the Geological Series of the University are bound in volumes: Vol 1, vol 2, pt 1, and vol 2, pt. 2, and vol. 3 are out of print, but separates of the following papers may be had at a uniform cost of 25 cents each. Vol. 1? No. 1, Geology of the environs of Albuquerque (1898); 2, Occurrence of copper and lead in the San Andreas and Caballo Mountains (1898), 3, Geological reconnaissance in western Socorro and Valencia counties (1898); Geology of the San Pedro and Albuquerque districts (1898). , Vol 2: No. 1, Geology of the Albuquerque Sheet (1900); 2, Miscellaneous economic papers by Herrick (1900); 3, Geological reconnaissance in western Socorro and Valencia counties (1900); 4, Geology of the White Sands of New Mexico (1900); 5, Identifi¬cation of an Ohio coal measure's horizon in New Mexico (1900); 6, Cyanide process in New Mexico (1900). Vol. 3. No 3, The oil situation in New Mexico (1920), 4, Geology of Sandia Mountains (1922); 5, Oil and gas m New Mexico in 1923.

Jan 1, 1933

By H. K. VAN POOLLEEN, W. Hurst, H. C. Bixel

In an earlier publication* I showed the development of the instantaneous point source solution for a well producing at a constant rate at the center of a system of two radial, adjoining sands of different permeabilities. I also showed how a wellbore radius could be incorporated if this were the development required. For the purpose of that paper and the point I wished to make, this solution served. It yielded a simplifying solution of sands in series identified by Eqs. VI-26 and VI-27. (These equations are reproduced in part by Bixel and van Poollen as their Eq. 3, Page 302.) I am well aware that the simplified solutions are approximations. The facts that these Ei-functions satisfy the diffusivity equation and meet the boundary conditions both at the well and at the interface between sands, and that the earliest time considerations yield zero pressure drop, are evidence that the solutions are good approximations. This was the purpose for these formulas, which I interpret that where the permeabilities are of finite order of magnitudes in the respective media, the relations give satisfactory answers. However, from the authors' point of view there are reservations, which leads me to consider their entire generalization as illustrated in their Figs. 1 through 14. What I understand from their storage capacity, relationship Fs, which is the ratio of the product of porosity and compressibility of Sand II to Sand I, that is employed here by the authors as a correlative means of identifying one figure from another, there is nothing in the exact mathematics that supports such a term as a means of correlation. This is taken out of context, unsustained that this applies. Here the mathematics are known, where the true correlation is the diffusivity term of one sand to another. This entails permeability that refers back again to the term M in their paper for the transmis-sibility of fluid from one sand to the other that should have also been a part of this storage capacity. In Figs. 8 through 14, which purported to show the simplified equation following this correlation, there are apparent discontinuities for M equal to or less than 0.1000 that differ from the trends for greater values of M. In this connection there is no reference radius given for this cumulative pressure drop in the paper that would permit me to make these calculations. However, what is clear for M equal to 0.1000 or less we are treated to almost a closed barrier at the interface; a condition entered by the writers without qualification. This across-the-board treatment of these simplified formulas, whatever the motive of the authors, enters into a range of permeabilities that even I as the contributor would have reservations that such could apply. Under comparable conditions I would be looking at a closed reservoir or some facsimile for such. The extent to which the approximation or the exact equations pertain is not evident in the generality offered by Bixel and van Poollen in their paper. Only when they or other writers show the comparison of the pressure distributions within the formations can one be able to judge the rational limits to subscribe to the simplified formulas.

Jan 1, 1969

By Cleland N. Conwell

Coals are found in all parts of Alaska, ranging in geologic age from Carboniferous to Tertiary, and grading from lignite through anthracite. Only the Matanuska and Nenana coal fields have been extensively developed. Nenana Field Today roughly 94% of the coal mined in Alaska comes from the Nenana field near Healy. The coal-bearing horizons in the Nenana field are found in a sedimentary sequence of Tertiary age approximately 2800 ft in thickness. The sediments rest directly on schists of possible Precambrian age. Individual coal beds in the field have been designated by both letters and numbers, starting with the alphabetical designations from A to G, and the numbers 1 to 6. Of these coals, the principal beds that are mined are F, 1, 2, and 3. These vary greatly in thickness, e.g., bed 1 ranges to over 50 ft, and beds 2 and 3 to 30-40 ft.

Jan 10, 1972

By Sherwin F. Kelly

Twenty five years of exploration has shown that the "geophysical dollar" is worth $1780 in the Orange Free State, $1488 on the Witwatersrand, $85 in Sweden, $28,000 in the Transvaal and $3500 or more in the U.S.

Jan 6, 1951

By M. Bevis, A. F. Acton, P. C. Rowlands

Defor~nation twinning and transformation twinning modes most likely to be operative in Fe-Ni and Fe-Ni-C martensites have been determined using a new theory of the crystallography of deformation t~inning.~ This analysis shows that potentially important conventional and nonconventional twinning modes1 have been omitted in previous analyses. Discussion is given on the relevance of the predicted twinning modes to the lattice invariant shear associated with the martensite transformation in steels and to anomalous deformation twinning in Fe-Ni-C martensites. THE two most important criteria which appear to govern operative twinning modes in metallic structures1 are that the magnitude of the twinning shear should be small and that the twinning shear should restore the lattice or a multiple lattice in a twin orientation. The latter criterion ensures that the shuffle mechanism required to restore the structure in a twin orientation is simple. These criteria have been adhered to in the prediction of twinning modes2"6 in bcc and bct single-lattice structures with axial ratios in the range y = 1 to 1.09 as, for example, encountered in martensite occurring in steels. Refs. 2 and 3 in particular consider the martensite transformation in steels and the twinning modes in these cases relate to transformation twinning, and hence the lattice invariant shear associated with the martensite transformation. The list of twinning modes which can be compiled from these sources is incomplete and the ranges of magnitude of shear considered could be unrealistically small, particularly in the case of deformation twinning. The latter consideration is supported by the fact that twinning modes with magnitudes of shear large compared with the smallest shear consistent with a simple shuffle mechanism have been established in, for example, the single-lattice structure mercury7 and the multiple-lattice structure zirconium.&apos; In addition the anomalous deformation twins reported by Ftichrnan4 to occur in a range of Fe-Ni-C martensites still remain unexplained. It is clear that a comprehensive analysis of twinning modes likely to be operative in martensite In steels is required. The results of the application of a new theory of the crystallography of deformation twinningg to these structures are presented in this paper. The theory has been used to determine all shears which restore the lattice or a multiple lattice in a new orientation with magnitude of shear up to a required maximum. The orientation relationships between parent and twinned lattices are not restricted to the classical orientation relationships of reflection in the twin plane or a rotation of 180 deg about the shear direction. PREDICTED TWINNING MODES Twinning modes which restore all or one half of lattice points to their correct twin positions will be referred to as m = 1 and m = 2 modes, respectively. These modes are the most likely to describe operative modes in single lattice structures. The bcc m = 1 and m = 2 modes which have magnitudes of shear s in the range s < 2 and s < 1, respectively, have been given10 and are reproduced here in Tables I and 11. Detailed discussion of the crystallography of these modes and cubic modes in general will be discussed elsewhere (~evis and rocker, to be published). The four twinning elements Kl, &,ql,7)2 as well as the magnitude of shear s are given for each twinning mode, and the twinning modes are given in order of increasing shear. Two twinning modes are given in each row of the tables, the twinning mode Kl, Kz, ql, q2 and the reciprocal twinning mode with elements Kl = K,, Ki = Kl, q: = q2, and 17; = ql. The m = 1 and m = 2 twinning modes which describe twinning shears with small magnitudes of shear and simple shuffle mechanisms in bct crystals with -y = 1 to 1.09 are given in Tables I11 and IV, respectively. On increasing the symmetry of the tetragonal lattice to cubic, that is making y = 1, all modes listed in Tables 111 and IV must reduce to crystallographically equivalent variants of the modes given in Tables I and 11, respectively, or become twinning modes with both shear planes as symmetry planes in the cubic lattice and hence not considered in Tables I and 11. With the exception of this last type of mode only those tetragonal twinning modes which reduce to modes 1.1, 1.2, 2.1, and 2.2 of Tables I and I1 are considered in Tables 111 and IV. For values of y in the range -y = 1 to 1.09 the tetragonal modes and the corresponding cubic twinning modes have approximately the same magnitude of shear. The twinning modes listed in Tables 111 and IV are therefore by the criteria given above the most

Jan 1, 1969