Shock Waves in Condensed Matter - 1983

Shock Waves in Condensed Matter – 1983 covers the proceedings of the American Physical Society Topical Conference, held in Santa Fe, New Mexico on July 18-21, 1983. The book focuses on the response of matter to dynamic high pressure and temperature. The selection first elaborates on the review of theoretical calculations of phase transitions and comparisons with experimental results; theoretical and experimental studies of shock-compressed benzene and polybutene; and theory of the iron equation of state and melting curve to very high pressures. The text then ponders on nonhydrostatic effects in stress-wave induced phase transformation of calcite; Bauschinger effect model suitable for use in large computer codes; and strain rate sensitivity prediction for porous bed compaction. The manuscript takes a look at flaw nucleation and energetics of dynamic fragmentation, shock loading behavior of fused quartz, and aluminum damage simulation in high-velocity impact. Shock wave diagnostics by time-resolved infrared radiometry and non-linear Raman spectroscopy; Raman scattering temperature measurement behind a shock wave; and experiments and simulation on laser-driven shock wave evolution in aluminum targets are also discussed. The selection is a dependable reference for scientists and readers interested in the response of matter when exposed to dynamic high pressure and temperature.

• Language: English
• Publisher: Elsevier Science
• Release date: Dec 2, 2012
• ISBN: 9780444600172

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CHAPTER I

KEYNOTE

Outline

Chapter I : 1: THUNDER IN THE MOUNTAINS

Chapter I : 1

THUNDER IN THE MOUNTAINS

John W. Taylor,     University of California, Los Alamos National Laboratory, P. O. Box 1663, Los Alamos, NM 87545

This paper summarizes the pioneering work leading to the development of scientific studies of the physics of shock-compressed matter at Los Alamos and culminating in the publication of the article Compression of Solids by Strong Shock Waves, by M. H. Rice, R. G. McQueen, and J. M. Walsh, Solid State Physics, Vol. VI, 1958. The work had its beginning during World War II when it became clear that development of a plutonium weapon would probably require the use of high explosives. It was immediately obvious to the staff that an entirely new level of sophistication in explosives technology would be required and that the equations of state of metals must be thoroughly understood. Following a suggestion by C. Critchfield, R. W. Goranson started a program to obtain equation-of-state data from shock-wave experiments. This program, begun late in 1944 and made possible only by the current and subsequent developments in explosive fabrication technology and electronic instrumentation, was continued after the war by Goranson and his successors. Nevertheless, the program remained relatively fallow until about 1950 when an influx of enthusiastic young staff were able to take advantage of a maturity of technical facilities. Centrally important to the new thrust were, (1) an optical diagnostics group which had developed a thorough familiarity with the technique in studies of shocked gases, (2) a charter to work in various aspects of hydrodynamics in unusual areas, and (3) the overall supervision which, in consonance with general Laboratory policy, looked favorably on research without restriction.In the summer of 1952, J. M. Walsh with M. H. Rice and C. M. Fowler, developed the flash-gap technique that made possible rapid and ultimately essential mass production of shot assemblies. Their recognition of the potential of impedance-matching techniques enabled them to begin a highly efficient major experimental program. The investigations of R. G. Shreffler and W. E. Deal on explosively driven plates prompted the group, strengthened by R. G. McQueen and S. P. Marsh, to expand the work into the megabar region over the next few years. Meanwhile, a discrepancy between flash-gap data and older pin data prompted D. Bancroft, E. Peterson, and F. S. Minshall to investigate iron in detail and discover the 130-kbar phase transition. Also at that time, relaxed security restrictions made publication possible. Prominent in the early work was the involvement of D. S. Hughes from the University of Texas and the reinvention of the dc capacitor by M. H. Rice. This capacitor permitted improved velocity resolution over that of pins and required less assembly time.

1 INTRODUCTION

On September 18, 1957, at approximately 10:00 p.m., I arrived at Los Alamos driving a rusty 1953 Rambler station wagon with all my worldly goods packed in the back end. This was the culmination of a three-day marathon drive from Ithaca, New York, which began the day after my doctoral orals and a very intense farewell party. I had accepted an offer to join Group GMX-6, located in Ancho Canyon in the explosive firing area about 15 miles from the center of town and two miles from Bandelier National Monument. As I drove north from Cline’s Corners toward Santa Fe, I began to see lightning bolts outlining the distant mountains on a scale I had never deemed possible, and by the time I entered Santa Fe the peals of distant thunder had merged into a nearly continuous roar. I remember thinking then that nature seemed to be in harmony with the sort of activity I had decided to try as a career. At the time I arrived, the active research group in GMX-6 consisted of John M. (Mac) Walsh, the Group Leader; Bill Deal, his alternate; Bob McQueen; Stan Marsh; Jerry Wackerle (who had arrived in the spring from Kansas); Max Fowler (also having arrived as permanent staff in the spring); and Wray Garn, a Los Alamos veteran of the war years. Mel Rice was back at Iowa State finishing his PhD program. There was also a rather remarkable, pugnacious, but nonetheless lovable consultant whose name was Darryl Hughes, but whom everyone called Doc with varying degrees of affection, fear, hatred, and respect. Up on the mesa at R-Site were Eric Peterson, Stan Minshall, Elizabeth Gittings (Marshall), and Stan Landeen. Russell Duff had by then become interested in chemical reactions in gas shocks and had his own small group at TD-Site on the edge of the mesa. In addition, of course, there was Wayne Campbell’s excellent group of detonation researchers at Kappa Site and Bill Woods’ group of theoretical physical chemists in the Ad Building in town, right next to the GMX Division Office, where reigned Duncan P. MacDougall, lest anyone ever forget it.

There was also a mildly mysterious place called S-site, which seemed to have hundreds of acres of land, populated by hundreds of people in widely scattered buildings, surrounded by rings of concentric fences, and from which trucks laden with explosives issued frequently unto all the firing sites. Also, in addition to a very efficient and cooperative small machine shop captive in Ancho Canyon, there was a not very mysterious but enormous shop facility in the main technical area.

Explosives, magnificently quality controlled, very accurately machined, and in all interesting shapes, sizes, and compositions, came from S-Site merely because one wanted them. Prototype metal components were custom fabricated in the local shop on the basis of quick sketches and hasty verbal requests, and the main shops rained mass-produced assemblies with a turnaround time that was then frustrating, but would now be regarded virtually everywhere as miraculous.

For acquisition of some of the more exotic materials, there was available the cheerful cooperation of the excellent metal fabrication group in CMB Division and, failing success there, the efficient Los Alamos purchasing department.

Why and how did all these elements come together in such a perfect combination to give birth to a new field of scientific inquiry?

The present history is a synthesis based on several types of sources. Perhaps the most valuable of all have been freely granted and enthusiastically delivered interviews with numerous persons who were present during the years in question. Next most valuable were original LA (Los Alamos) and LAMS (Los Alamos Manuscript) reports, and Los Alamos Division and Group progress reports from the period 1943–1956, in particular. The open literature is, of course, available to all and was consulted, but was in fact of little value to this history because of wartime and immediate postwar secrecy requirements.

Perhaps this is the appropriate point to discuss the major impacts of secrecy on this subject. Shock-wave physics as we know it today owes its genesis and its entire formative period to a minor industrial revolution in the technologies of precision casting, pressing, and machining of explosives. This revolution and in addition the original motivation and long-term support for equation-of-state research were entirely the result of the World War II crash program to develop a nuclear weapon. From the earliest days and until the Rosenberg-Greenglass-Gold-Sobel espionage case publicly began in early 1950, it was forbidden for Los Alamos employees even to admit the general nature of their interest in explosives. As a consequence, everything written was classified Secret because of fact of interest. Bill Mautz noted to the author that when he was interviewed in June 1949, Wayne Campbell said, I am permitted to say that we are interested in explosives, including high explosives, but no more.

Equally suppressed were all references to interest in the high-pressure equation of state of solids, whether experimental or theoretical methods were employed. Under such restraints, recruiting of staff at the Laboratory was, during the years 1946–1950, a difficult task at best. Consequently, the available staff were constantly overworked simply to solve day-to-day problems of weapons development. Further, the virtually zero prospect of publication reduced incentive to do systematic general interest research.

Fortunately for the purposes of history, many of the formal and informal reports pertaining to this subject have subsequently been declassified. The reader is assured that from the point of view of scientific history, the present report is virtually completely documented, although many of the cited references are not generally available. They are, however, unclassified. The reader is further assured that the real historical value of those classified documents that have been consulted is to provide an important thread of continuity. This thread has been used without citing reference throughout the body of text.

2 EXPLOSIVES TECHNOLOGY

During the hectic, almost frantic war days at Los Alamos it was clear that only a very limited quantity of ²³⁵U would be available for a weapon and that, if possible, plutonium must be used. It was equally apparent that the plutonium weapon would have to be assembled by high explosives.

The history of shock-wave equation-of-state work in Los Alamos is intimately connected with and dependent on the history of development of explosives fabrication, quality control, and inspection technology. The initial experiments by Seth Neddermeyer and co-workers in the spring and summer of 1943 (1) had revealed, among other things, that precision in these matters on a scale hitherto not contemplated would be absolutely essential. When George Kistiakowsky arrived in early 1943, he rapidly concurred and recommended in the autumn of 1943 that an explosives casting and assembly facility be constructed at Sawmill Site, later to be called simply S-Site. This facility was operated by Group X-3, under the direction of Major John Ackerman, US Army. It became operational in May 1944. The explosive of choice was Composition B (approximately 60% RDX/40% TNT) because it was both more energetic and safer than the more familiar pentolite. Meanwhile, explosive lenses for wave shaping were being developed at Bruceton, Pennsylvania, in E. H. Eyster’s group under Duncan MacDougall. For this purpose, the low-energy explosive baratol (Ba2NO3 in TNT) was first developed.

To the considerable frustration of everyone concerned, although all the principles appeared sound, neither X-3 at S-Site nor the group at Bruceton were able to achieve quality control sufficient to the purpose at hand. In the autumn of 1944, there arrived a contingent from the metallurgical laboratory at Chicago. Four members of this delegation, D. Gurinsky, M. L. Brooks, L. M. Foster, and B. Turvolin, concluded that of the outstanding problems of the day the explosive fabrication problem was most critical. They joined Group X-2, the lab at S-Site, and proceeded to approach the explosives problem as metallurgists. There were in fact chemical engineering issues such as proper wetting agents, etc. The crucial breakthrough was, however, due to the introduction of metallurgical casting procedures and standard machining techniques. Product quality control was greatly enhanced when G. H. Tenney introduced X-radiography techniques.

It should be noted that meanwhile new precision-timed detonators were being developed by K. Greisen and L. B. Seeley and coworkers.

This crash program left Los Alamos, at the end of World War II, uniquely in possession of the most advanced explosive fabrication technology on earth and with a mission to improve on it further. This development has never stopped, but the important point for this story is that, at an accelerated pace through the period 1945–1950, the Laboratory developed an in-house capability to fabricate high-quality, precision-finished blocks and other shapes of virtually every practical explosive. Further, because of the ongoing interest in the response of inert materials to shock loading, several sizes of plane-wave explosive lenses based on Baratol and Composition B were standardized and produced as needed.

3 EQUATION OF STATE CONSIDERATIONS

Neddermeyer and Bradner’s demonstration in early 1943 that an explosive-driven implosion was possible led the Theoretical Division under H. A. Bethe to launch a two-pronged program. It is interesting to observe in Figures 1 and 2 the evidence upon which this program was based. As a major effort, they taught themselves and each other hydrodynamics, with due regard for its applicability to materials exhibiting finite strength.

Figure 1 Reproduction of Neddermeyer’s sketch of his experimental arrangement for studying cylindrical implosions.

Figure 2 Reproduction of Neddermeyer’s photographs of his first and seventh steel implosion experiments. His caption for the series of photographs reads, "The detailed data are given in Table 3. Experiment numbers are marked on the samples in the photographs. In each photograph the original size of the sample is shown by a ring or a complete cylinder. All cylinders are seamless 30-40 carbon steel tubing with a static yield point of about 55000 lb./in². Four-point detonation at center unless otherwise stated. Loading density for TNT, 0.87; for Comp. C. about 1.5."

Simultaneously, they began intensive theoretical work on equation of state, using the published theoretical and experimental background as a guide. In addition, they cast about for practical ways to obtain more experimental data. The same group devoted considerable attention to the theory of detonation and the properties of explosives, including modifications to idealized conditions as indicated by experimental data as it came in from the firing sites.

The period from midsummer 1943 to midsummer 1944 was characterized by a large amount of work and the development of ideas. Results were shared informally, but there was no time for even informal report writing. This situation was largely corrected in the late summer of 1944 and the winter of 1944–1945.

In his first report, Neddermeyer had written: The pressure existing in a detonated high explosive is of the order of 10⁶lb/in², which is much higher than the yielding strengths of any ordinary metals. In treating the motion of a collapsing shell, we, therefore, assume that there is plastic flow against a constant yielding stress throughout the shell. For the simple preliminary treatment, we ignore volume changes and, therefore, any effects of elastic waves.

Nothing was taken for granted, however. Every aspect of the broad subject of detonations, the interaction of explosives with inert materials, the possible effects of materials strength, and supersonic hydrodynamics was examined extensively. There were individual efforts resulting in short reports on limited subjects, but these were ultimately consolidated via a seminar series into an overall position and understanding. While reading the reports, one is caught up in the feeling of urgency and excitement that existed.

Perhaps this can be conveyed to the reader by noting some of the report titles. Felix Bloch wrote on Momentum Transfer by High-Explosives Layers. (2) Sir W. G. Penney discussed, The Energy Dissipated in Plastic Deformation. (3) As late as May, 1945, R. E. Peierls and P. R. Stein wrote on The Effect of Rigidity and Shock Waves in Solids. (4) Figure 3 is a reproduction of Figure 1 from their report, illustrating their explanation of how the Hugoniot curve of a material with rigidity would compare with a hydrostatic curve.

Figure 3 Peierls’ and Stein’s graphical representation of the Hugoniot curve of a material with rigidity. Point T represents the Hugoniot elastic limit. Curve TSQPC is the locus of states that will be achieved in actual experiments, with P representing the lowest pressure that will propagate as a single shock front.

H. A. Bethe, J. Von Neumann, K. Fuchs, W. G. Penney, and others, systematically studied nonlinear hydrodynamics, including consideration of spalling phenomena. The shock hydrodynamics lectures were documented by J. O. Hirschfelder (5)

Theoretical equation-of-state work was conducted by a stellar team. A major initial effort by R. F. Christy (6) enabled the theorists to begin design calculations. N. Metropolis, reporting work by R. R. Davis, G. Marvin, and N. Metropolis, wrote on Equations of State and Associated Shock-Wave Variables for Certain Metals. (7) The work was refined by Ashkin, Keller, Bethe, Davis, Feynman, Keller, Metropolis, Peierls, and Teller (8) and again by Ashkin, Daves, Keller and Peierls. (9)

The overall plan had been to use available data (pressures to 0.04 Mbar) as reported in Birch’s physical tables, investigate and generalize the Thomas Fermi Dirac model presumed to be valid for pressures above 50-100 Mbar, and interpolate in between by some appropriate procedures. It was noted that because of goephysical measurements on the core of the earth, there was some data for iron in the range of a few Mbar, so this element became the standard for interpolation techniques.

Meanwhile, splinter groups advanced ideas for alternative methods and cross checks. In September, 1944, E. Teller, E. Fermi, and S. Ulam published An Equation of State in the Condensed Phase for Arbitrary Pressures and Moderate Temperatures. (10) This research note is a derivation of an equation of state from assumptions that are commonly known as Mie-Gruneisen theory.

Meanwhile, Group G-8 (Gadget Division) had been organized in May 1944 under D. K. Froman to explore electronic techniques for monitoring high-velocity metal motion. This work, especially the pin technique, was developing nicely by the late summer when C. L. Critchfield (11) of the Theoretical Division on August 26 published a document entitled, Determination of the Equation of State of a Metal from Shock Wave Measurements. The abstract reads:

It is proposed to investigate the equation of state of a metal by observing the impact of a projectile and target both made of the metal. Shock velocity and pressure of iron and of uranium are evaluated from experimental results through relationships developed for homogeneous impact and the hypothetical equations of state involving only pressure and density; heterogeneous impact is also considered. The effect of the reflected rarefaction wave and the effect of yaw are treated, and conclusions involving experimental procedure are drawn. The relationship, involving the effect of entropy change, of the static adiabat to the Hugoniot equation is discussed.

Froman’s original group, consisting of himself, A. C. Graves, and J. J. Wechsler, had by now grown to include, among others, Dennison Bancroft. Bancroft was familiar with the work of Roy Goranson, who had in 1942 been principal investigator in a program by NRC, sponsored by the Bureau of Ships, to devise scientifically based methods for estimating the damage to ships’ hulls caused by underwater explosions (12) He and co-workers had used tourmaline crystals for water shock-pressure measurements and electrical contact pins for metal-plate motion studies. Goranson had been a student of P. W. Bridgeman, and was, therefore, well-grounded in much of the lore of experimental and theoretical high pressure physics of the day. His explosion-damage research revealed for the first time that metal plate motion could result in cavitation in the water. It was a considerable contribution to the general subject.

By December 1944, Froman et al. were able to publish an RSI type report on the existence of acceptable pin technology (13) and Goranson had joined the group. Figure 4 is a reproduction of one of Froman’s first oscilloscope records and has his original figure caption. Figure 5 shows one of his first metal plate velocity records. While the main group pursued further pin technique development and their principal mission, Goranson set out to make shock-wave equation-of-state measurements on iron, aluminum, and tuballoy (depleted uranium obtained from Tubes, Ltd, of England). Iron was of interest not only for reasons stated previously, but also because R. Peierls and P. Stein (4) had noted that at the shock pressures of interest it was the only one of the three metals expected to exhibit an elastic precursor wave.

Figure 4 Froman’s first typical record. His caption reads "Typical oscillogram with the contact made there are six flat positive and five negative steps representing contacts to eleven pins placed under a flat steel plate. In this case the charge was a conical lens and the object of the shot was to determine the flatness of the detonation wave. The blanking spaces are at 0.5-μs intervals.

Figure 5 One of Froman’s first metal plate velocity records obtained by the pin technique.

In September 1945, Goranson and Blechar published a report on progress to August 15, 1945, confirming Peierls’ and Stein’s hypothesis about elastic waves and demonstrating to the Theoretical Division that their interpolation methods were fairly accurate in the range of 300–400 Kbar for aluminum and uranium (14). Figure 6 shows Goranson’s first reported pin data and Figure 7 his first reported Hugoniot data. Goranson and collaborators then directed their attention toward using their techniques to investigate the structure of detonation waves and the pressure gradients in explosives. The method was to make measurements on metal plates in contact with the explosive as a function of metal to explosive thickness ratios, just as is still common. The abstract of this report states: A method, which is here described, has been devised and utilized for the study of detonation waves in high explosives. Those investigated are pentolite, Composition B, baratol, and granular TNT. Pressure-volume data have been obtained for the detonation head and the Chapman-Jouget point. The peak pressure of the detonation head is found to be about 2.5 times that at the Chapman-Jouget point. Furthermore, assuming the adiabatic expansion of the burnt gases can be represented by pvγ = constant, it was found that the mean square deviation of any of the explosives from the value γ = 3 was about 4%, which, therefore, makes the formula p = (ρOD²/γ −1) a useful first approximation in determining the Chapman-Jouget point. If this picture of the detonation head is correct, then abnormal spall effects, i.e., not present in thicker specimens, should occur in thin plates. These have been observed. Estimates have also been made of the width and duration of the reaction zone. They are found to lie in the intervals 0.8 ± 0.2 mm and 0.2 ± 0.1 us, respectively. It is a bit curious, considering Goranson’s interest in the elastic wave in iron, that he did not take advantage of a concurrent parallel instrumentation development. On April 11, 1945, D. Marshall et al. in the same group published a condenser microphone method of studying motion of conducting surfaces. The abstract states (15): The following is a discussion of a method of measuring the velocity of a metal surface accelerated by high explosive. Another metal surface is placed so that the two surfaces form the plates of a condenser. The variation of the capacity of the condenser is used to study the motion of the moving surface. Displacement-time relationships have been obtained for spherical moving surfaces with an accuracy of about 5%. Preliminary measurements recording directly the velocity as a function of time are described.

Figure 6 Goranson’s first reported pin data.

Figure 7 Goranson’s first formally reported Hugoniot data.

There follows a complete discussion of the theory, design, and use of a dc capacitor gauge for both space and velocity measurements, with reproductions of oscillograph records. It was noted that considerable improvements in accuracy would be required before precise measurements could be made. Perhaps it was this problem and the time urgency of the measurements that caused this technique to lie fallow until reinvestigated by Rice in the late 1950’s. Figure 8 shows Marshall’s first reported High Resistance Method Capacitor Data.

Figure 8 One of Marshall’s first oscilloscope traces from the high resistance capacitor method.

In 1946, the Laboratory was reorganized. Kistiakowsky’s X Division under Max Roy and M Division under Darryl Froman absorbed the functions of the old G (Gadget) Division. Group G-8 became M-4 under Alvin C. Graves. Graves was succeeded after his near fatal exposure to neutrons (in the accident that killed Louis Slotin) by J. M. Burris. The firing site was moved from Beta Site on the mesa above Pajarito Canyon to the new R-Site, close to S-Site. Much effort was devoted to upgrading, standardizing, and improving the efficiency of the electronics. In the M-4 progress report for December 1946, Burris wrote, It is now possible, probably for the first time, to make accurate measurements of equation of state parameters for metals in the range of a few hundred kilobars. In January 1947, he wrote, "Because of the large number of possible problems upon which work was being done, the work in M-4 has been reorganized this month. Priorities have been set upon the various problems, and it was decided to press the work most vigorously in two general phases: (1) equations of state of metals under shock conditions and (2) censored. Equation-of-state investigations cover the range from about 0.01 to 0.4 Mbars and include development work on methods of measuring the temperature in shocked metals.

Technique development was pursued by R. Schwanery, E. Houston, and T. Blechar. S. Minshall devoted his efforts to attempts to measure temperature in shocked metals with thermocouples. This program was continued over the next two years with tantalizing, but ultimately entirely frustrating, results. Goranson had left to return to university life in the summer of 1946, but Bancroft remained until 1948, when he too rejoined academia. The group’s efforts in the equation-of-state area were largely directed toward taking a few data points on each of a variety of materials of interest to the theoretical division for potential weapons applications.

During the war years, W. S. Koski had formed a small group, X-6, whose mission was to employ high-speed framing and sweep cameras in analysis of the gadget. It will be recalled that Koski was later the US Government’s principal technical witness at the Rosenberg Trials (16). The cameras were developed by B. Brixner, whose expertise was optics, and B. Webster and J. Serduke, who designed the air-driven turbines.

When Kistiakowski left, X-Division continued under M. F. Roy, and Koski remained as group leader of X-6. In addition to its principal mission, the group devoted considerable effort to the study of high-speed jets and began to think about the equation of state of gases. Shortage of personnel and the pressures of the moment prevented extensive work, however.

4 THE MODERN EQUATION OF STATE PROGRAM

In the summer of 1948 Bradbury reorganized the Laboratory. The reogranization amounted to sorting out the functions of the old G, M, and X divisions and reforming as GMX and W divisions. D. P. MacDougall, a physical chemist who had been at Bruceton during the war and had also been a consultant to Kistiakowski, was appointed division leader of GMX. M. F. Roy became W-Division leader.

GMX Division contained all explosive research, fabrication, nondestructive and destructive testing, and application facilities, equipment and expertise in the Laboratory. In addition, Brixner’s high-speed camera development and maintenance group was attached to the Division.

MacDougall’s charter was to build up the Division’s facilities and its technical strength, with the objective of intensifying its research capabilities as a solid basis for its mainline weapons responsibilities. One of his first actions was to convince Roy Goranson to return to Los Alamos as a member of his staff. Koski had left and Frank J. Willig replaced him as group leader of GMX-6.

Bancroft was engaged as a consultant. MacDougall introduced a Division seminar series with the object of educating the staff on the general subject of nonlinear hydrodynamics and establishing technical communication between groups. A set of lectures by Goranson in 1949, entitled GMX Seminar Notes, attests to the depth of some of these seminars. In 1949, R. G. Schreffler joined GMX-6 and began to interest himself in the equation of state of gases. W. E. Deal and J. M. Walsh arrived in October and November of 1950, respectively. In the summer of 1952, C. M. Fowler came as a consultant and M. H. Rice as a summer graduate student. It was during that summer that the basic flash-gap technique was developed. Possibly the most important point about the flash-gap technique was that it enabled relatively inexpensive mass production of experimental assemblies. Anyone who has ever set pins or seen it done will appreciate the difference. As noted previously, another significant event that summer was the declassification of the existence of the implosion concept, which rapidly led to relaxation of the security restrictions on revealing the nature and extent of interest in high explosives at Los Alamos. Thus, it transpired that by the time the group had accumulated and analyzed sufficient experimental data to warrant publication in the open literature there were no substantial classification barriers.

The pin technique seemed to work well with some materials but not so well with others. However, by late 1949 E. F. Gittings (Now E. F. Marshall) was exploring the possibility of obtaining higher pressure data using explosively accelerated brass plates as strikers. Figure 9 is her sketch of a pin-shot assembly for use in free-run shots. Houston was developing and using tourmaline crystal detectors, and a number of materials had been explored in a survey fashion. A very long period of technique development followed. Minshall managed to get data sufficient to resolve the elastic plastic transition in iron and tungsten, but high-pressure work awaited a technique for avoiding ionizationinduced pretriggering of pins. This sort of effort went on into 1952. The only significant data produced was a fairly complete description of the 24ST duralumin Hugoniot for the pressure range achievable with contact explosives.

Figure 9 Gittings sketch (1949) of a pin shot assembly for use in free run measurements.

The first mention of the optical technique appears in the GMX-6 progress report for March 1952. It is noted that A program to investigate the feasibility of using optical methods for equation-of-state studies has been undertaken. Experiments thus far have been concerned with an investigation of the possibilities of making precise shock velocity measurements in solids. In May it was reported that shock and free surface velocity measurements on 24ST duralumin with 1% precision appeared possible and that the results would be compared with the GMX-4 pin data. This work was done by J. M. Walsh and R. H. Christian. In August the use of lucite flash cutoffs was reported and the first streak camera traces were published. Figure 10 is a reproduction of their typical record. In November, Walsh and Christian reported a summary of their data on 24ST duralumin and the GMX-4 group summarized theirs. Figure 11 is Walsh and Christian’s first summary data on zinc and copper, and Figure 12 is the 24ST duralumin summary.

Figure 10 One of the first typical records obtained by Walsh, Christian, and Rice as the flash-gap technique was developed.

Figure 11 Walsh and Christian’s first summary data on zinc and copper.

Figure 12 The 24ST duralumin data as summarized in November, 1952.

At this point the direct interests of the two groups began to diverge. The pin group, noting that the optical technique was better for mass producing data and cheaper to implement, looked for other areas of interest.

Minshall had for several years been studying the elastic wave in iron and realized that his methods were better suited than flash gaps to resolving two-wave structure. In addition, the pin group had ideas about how to get multimegabar data by a technique that remains classified.

During 1953, Walsh and Christian expanded their interests in several directions. They explored various aluminum alloys, began work with uranium, zinc, and copper, and worried about the precision of the free-surface approximation. In March they performed experiments with sintered aluminum and after a short period of surprise, realized that porous materials could be used to explore entirely new regimes of entropy.

By July, they had delivered a paper on aluminum, copper, and zinc at the APS meeting at State College, Pennsylvania, including a detailed theoretical consideration of the free surface approximation.

Max Fowler, who had been in graduate school with Shreffler, spent the summer with the group and interested himself in heat-treatment effects in steel in the low-pressure regime. This interest was concurrent with Walsh and Christian’s decision to get serious about Armco iron. The first experiments were performed in late November, and Walsh wrote in the December progress report that the material was unbelievably compressible and that their data were totally at variance with theoretical estimates, Bridgeman’s data, and all older shock-wave data. This immediately interested Minshall, who assembled a duplicate experimental geometry for a pin shot. Quite fortuitously, in the interest of maximum precision, he included enough free-surface velocity pins to notice one wave with an amplitude of approximately 0.6 mm/μs and a second slower wave with an amplitude of 0.9 mm/μs. This was reported in February 1954 and the data were correctly interpreted in March by Walsh.

What seems to have happened over the years was that until Walsh and Christian became interested in iron, the pin experiments at higher pressures had detected the final free-surface velocities only, and these had been associated with the plastic I shock velocities. When data scatter was factored in, no one realized that an anamoly existed. By June, 1954, Marshall, B. L. Burton, and S. A. Landeen reported eleven experiments in which the two-wave structure in iron was thoroughly investigated by the pin technique. This effort focused Minshall’s interest on iron and steels, and over the next two years he investigated 1020, 1040, and 1055 steels as a function of heat treatment, paying attention to both the elastic wave and the plastic I (Transition wave). At the suggestion of Russel Duff, he verified in January 1956, that 348 stainless steel has no high-pressure phase transition and began to attempt to measure the temperature dependence of the transition pressure in Armco iron. It is clear from the reports of the era that Minshall coined the terms Hugoniot Elastic Wave and Hugoniot Elastic Limit sometime in late 1956.

To illustrate the magnitude of the effort involved in sorting out the iron transition problem, some of the data are shown in Figures 13, 14, and 15. Figure 13 shows some of the oscilloscope records from tourmaline crystals that showed qualitative features well but were very poor in precision. Figure 14 shows one typical plot of pin data taken for precise free surface velocity measurements. The painstaking care required in setting pins is evident. Figure 15 shows a final summary of the iron Hugoniot curve in the transition region.

Figure 13 Oscilloscope records from tourmaline crystals in the GMX-4 study of the phase transition in iron. The various wave fronts are indicated by letters.

Figure 14 A (truly) typical plot of pin data from the shot series in which the iron phase transition was studied in detail.

Figure 15 The summary data in the iron transition from June 1954.

Also, in late 1956, Minshall began to study the effect of nickel content on the phase transition in steels. This work was later expanded to a multicomponent alloy survey by C. M. Fowler and E. G. Zukas with guidance from Cyril Stanly Smith.

In the spring of 1954, Russell Christian left Los Alamos and went to Livermore. Fred L. Yarger joined the group in April. In May Willig left and R. G. Shreffler became group leader. The data production line had become quite efficient, and a survey of all readily available elements, with three data points for each, was well under way. In July data were reported for beryllium, cadmium, carbon, nickel, silver, tin, protactinium, and Dow metal, all impedance-matched to 24ST duralu min. In August tungsten, sulfur, molybdenum, bismuth, and cobalt were added. In October gold and brass were added, with more data on molybdenum, cobalt, tungsten, and bismuth.

In the summer of 1952, Mel Rice had joined the group as a summer graduate student. He had the honor of assembling the first successful equation-of-state shot. He returned in 1953 and 1954 and with Walsh began to investigate the Hugonoit of water, a program that was completed in the summer of 1955. He also took an interest in the theory of Equation of State and in 1956 rediscovered the Gruneisen formulation, assertedly without knowledge of Teller and Fermi’s work twelve years previously. It is perhaps worth noting that the need to know aspect of security made general access to the classified library very difficult to obtain. Under such conditions it is frequently preferable to reinvent. In fact the only unclassified text on shock waves available at the time was Courant and Fredricks. (18) In February 1955, R. G. McQueen joined GMX-6 after several years as a weapon designer. Shortly thereafter, S. P. Marsh transferred from the same environment. During the three previous years Shreffler and Deal’s interest in the equation of state of gases and explosives had automatically resulted in a large body of observations of metal plates accelerated by explosives. McQueen and Walsh realized that although there was much left to be learned about the elements at pressures available from in contact systems, there was a whole new frontier in higher-pressure work. One concern was the possibility that errors would arise due to driver-plate spalling, but Max Fowler’s theoretical work of the previous summer convinced them that this was a subject to be studied, not to be worried about. By April, 1955 the freerun program was well under way, and in June the first data on uranium were reported. By July they had uranium data to 2.26 Mbar.

Meanwhile, several refinements of the subject were of interest to the group. They procured several single crystals of zinc to see if any anisotropy effects would be manifested. They initiated a series of experiments using impedance matching techniques to map the iron isentrope down from where it crosses the Hugoniot at 365 kbar. They procured a 6.5-in.-diam. smooth bore gun to do better controlled low-pressure and spall studies. The program for release wave pressures between 365 and 80 kbars was completed in one month. In May 1955 a summary report on data from 23 solids was published.

In the summer of 1955, Professor Darrell S. (Doc) Hughes from the University of Texas at Austin came as a consultant. Hughes had worked for Shell Oil interpreting seismic exploration data in the 1930s, had been a consultant to Los Alamos since the war years, and had taught both Walsh and Deal when they were graduate students. He was interested in geophysics, guns, and everything about physics. Noting that Landeen and Houston were having only limited success with pins in obtaining rock Hugoniot data, he persuaded McQueen (easily) to obtain some data on dunnite, San Marcos Gobbro, and Braunite Gabbro. He also took a direct and forceful interest in the 6.5-in. gun and supervised having it rebored and the end faced to high precision at Watervliet arsenal. Further, he knew propellant experts, developed loading procedures, and designed the first control room facility for the gun. This occurred over a period of approximately two years. The group was so busy with other things that the gun kept falling to second priority when Doc wasn’t around.

The year 1955 was somewhat complicated by the fact that as the new techniques had developed it had been deemed imperative to apply them to plutonium. Plutonium work had to be done at the weapons test site at Nevada, and the group personnel were somewhat oversubscribed between conducting experiments at home and on a commuting basis. I have not found any participants except Bill Deal who claim to have enjoyed the Nevada exercise.

In January 1956, two tons of J-1 brass arrived in Ancho Canyon. This material had been selected as the high-pressure standard primarily because the machinists asserted that it was the best material from the cost and precision aspect. It took nearly a year to realize that the very source of machinability was also the source of local density fluctuations. For several years thereafter, J-1 brass plates were used as winter ballast in the group’s pickup trucks. This experience taught the group to select materials on the basis of material properties other than machinability. The new high-pressure standard was high-purity copper. The machinists were less happy, but McQueen was pleased.

During 1956 the major thrusts of the group were rare metals, alkali metals (Rice); liquified gases (argon, oxygen, and nitrogen) (Marsh) liquid TNT (Walsh and W. B. Garn), and high pressures and dynamic tension by free run (McQueen). Other programs were of course carried along. For example, data on sintered copper had been accumulating but the group was concerned about whether thermal equilibrium was sufficiently complete to warrant theoretical interpretation. Meanwhile, by October sufficient high-pressure data had been gathered that McQueen decided it was time to look at graphs of primary data rather than derived data. It was this exercise, with a large body of high-quality data, that revealed that most Hugoniots are straight lines in the shock velocity-particle velocity plane.

In autumn 1956, Rice returned to graduate school and did not return to Los Alamos until the spring of 1958. In September 1956, Shreffler left to become a member of MacDougall’s personal staff and Walsh became group leader. In October, Jerry Wackerle arrived and interested himself in ferroelectric materials.

The last half of 1956 was primarily devoted to writing up data for publication [principally the Solid State Physics article (19)], and to accumulating data in various programs. The group included an excellent staff of technicians, who had been trained on the job and were eager to do their very best for the staff. Frank Jackson, who had participated in the earliest experiments, was familiar with every aspect of the work from shot assembly through firing point and darkroom work. Felix dePaula presided over several assembly technicians. Austin Bonner, Delbert Meadows, and Ruben Ortega knew their firing site capabilities thoroughly. Johnny Chavez and Dorothy Stamm were outstanding at precision film reading with optical comparators. As long as nothing significantly new in concept was needed, this team simply mass produced data.

Early in 1957, McQueen decided that his experimental program to study spall was sufficiently productive that more elaborate calculations were necessary. He and Marsh, both of whom had previous experience with computers, began to develop a plane-wave hydrodynamics code. Walsh continued to oversee the liquid TNT program and was otherwise burdened with administrative duties. Fowler joined the group as a permanent staff member and devoted his principal efforts to the longstanding program (originally suggested in 1950 by Teller) to use explosive-driven assemblies for the production of large magnetic fields. Deal continued his interest in the equation of state of detonated explosives.

The group was increasingly involved in weapons development diagnostic work and in supplying equation-of-state data on various compounds, alloys, and composite materials, all this requested by the Theoretical Design Division.

When I arrived in September 1957, it was very difficult to decide which of the many urgent and exciting programs to learn about first. One very interesting and important question was the temperature achieved in shocked materials and another was the extent of the influence of elastic strength on all shock-wave data. There was also the extremely important contribution the group was making to the forthcoming weapon test series. Hughes was overseeing preparations for installation and use of the 6.5-in. gun that was due to arrive in the early summer of 1958 and without which very little could be done in the low-pressure regime.

I tackled the temperature-measurement problem, plunged into the weapon diagnostics development work, and learned as much as I could absorb from the old hands in the group. Hughes and Wackerle, after a learning curve of nearly a year, turned the gun facility into a working tool except that no truly satisfactory diagnostic technique was found. Early in 1959, Rice rediscovered the dc capacitor technique and devised an ingenious method of calibration. When the first data on aluminum and then Armco iron showed that the whole concept of sharp elastic and plastic waves had to be discarded, we realized that we were launched into another whole program. I remember that as we looked at them around a conference table, Walsh and I agreed that We’re in the metallurgical mud.

By this time the 1958 review paper had made its impact on the technical community, and experimental equation of state by shock wave techniques was an established branch of science.

CONCLUSION

This history begins with a few hastily constructed demonstration experiments and ends with a new field of scientific inquiry. It is significant to make that at about the same time there began to appear numerous papers on the subject in Soviet journals. Undoubtedly there is a similar story to be told from their viewpoint. An important point to note is that by the end of 1944, when the first experimental work was beginning, the entire field of nonlinear hydrodynamics theory had been thoroughly documented. Furthermore, theoretical equations of state that proved sufficient for the purpose at hand existed. The great contributions of the experimental program were to provide precision to a degree otherwise unachievable, to discover the numerous phase transitions in materials and to document the effect of rigidity. The moral seems to be that every time one looks carefully at nature, something unexpected and valuable is learned.

ACKNOWLEDGEMENTS

For background on the war years and the late 1940s, conversations with Norris Bradbury, J. J. Wechsler, Melvin L. Brooks, and John H. Russel have been most helpful. John M. (Mac) Walsh, Melvin H. Rice, William E. Deal, and Duncan P. MacDougall provided very useful recollections of the early 1950s. Many members of Los Alamos service facilities have been most helpful, but special mention goes to Judith Young of the Laboratory Report Library.

REFERENCES

1. Bradner, H., Neddermeyer, S., and Streiker, J. F., The Collapse of Hollow Steel Cylinders by High Explosives, LA-18, August, 1943.

2. Bloch, Felix, Momentum Transfer by High Explosive Layers, LA-36, November 9, 1943.

3. Penney, Sir W. G., The Energy Dissipated in Plastic Deformation, LA-155, October 7, 1944.

4. Peierls, R. E. and Stein, P., Effect of Rigidity on Shock Waves in Solids, LA-271, May 4, 1945.

5. Hirshfelder, J. O., Shock Hydrodynamics, LA-165, October 28, 1944.

6. Christy, R. F., Preliminary Equations of State for Aluminum, Cadmium, Iron, Uranium. LAMS-164, November 24, 1944.

7. Metropolis, N., Equations of State and Associated Variables for Certain Metals, LA-208, January 24, 1945.

8. Keller, J. M. and Metropolis, N., Equations of State of Metals, LA-385, September 12, 1945.

9. Ashkin, J. Equation of State of a Compressed Metal According to the Thomas Fermi Dirac Models LA-393, September 1945.

10. Teller, E. and Ulam, S. M., An Equation of State in the Condensed Phase for Arbitrary Pressures and Moderate Temperatures, LA-142, September 20, 1944.

11. Critchfield, C. L., Determination of the Equation of State of a Metal from Shock Wave Measurements, LA-131, August 26, 1944.

12. Goranson, R. W., Garten, W., and Crocker, J. A., The Measurement of Large Transient Stresses, NDRC report No. A-45, April 9, 1942.

13. Froman, Darol, Contact Electrical Method of Studying Implosions, LA-182, December 7, 1944.

14. Goranson, R. W., Pressure, Compression, Shock, and Particle-Velocity Measurements in the Neighborhood of One-Third Mega-bar, LA-384, August 15, 1945.

15. Marshall, D. G. Condenser Microphone Method of Studying Motion of Conducting Surfaces, LA-259, April 11, 1945.

16. Nizer, LouisThe Implosion Conspiracy,. New York: Double Jay and Company, 1973.

17. Goranson, R. W., GMX Seminar Notes, LAMD 153, May 13, 1949.

18. Courant, R., Fredricks, K. O.Supersonic Flow and Shock Waves,. New York: Interscience publishers, Inc., 1948.

19. Rice, M. A., McQueen, R. G., Walsh, J. M., Solid State Physics. Compression of Solids by Strong Shock Waves,; Vol. VI. Academic Press Inc., N.Y., 1958:1–63.

CHAPTER II

EQUATION OF STATE

Outline

Chapter II : 1: A REVIEW OF SOME RECENT THEORETICAL CALCULATIONS OF PHASE TRANSITIONS AND COMPARISONS WITH EXPERIMENTAL RESULTS

Chapter II : 2: DENSE MOLECULAR THERMODYNAMICS

Chapter II : 3: SHOCKED FLUIDS AT HIGH DENSITIES AND TEMPERATURES

Chapter II : 4: THEORETICAL AND EXPERIMENTAL STUDIES OF SHOCK-COMPRESSED BENZENE AND POLYBUTENE

Chapter II : 5: PREDICTION OF PRESSURE INDUCED STRUCTURAL PHASE TRANSITIONS AND INTERNAL MODE FREQUENCY CHANGES IN SOLID N2

Chapter II : 6: MEASURED HUGONIOT STATES OF A TWO-ELEMENT FLUID, O2 + N2, NEAR 2 Mg/m3

Chapter II : 7: EQUATIONS OF STATE FOR THE ALKALI METALS, WITH GENERALIZATIONS TO NaCl

Chapter II : 8: STATIC P-T-V MEASUREMENTS ON MgO: COMPARISON WITH SHOCK WAVE DATA

Chapter II : 9: A THEORETICAL EQUATION OF STATE FOR URANIUM

Chapter II : 10: THEORY OF THE IRON EQUATION OF STATE AND MELTING CURVE TO VERY HIGH PRESSURES

Chapter II : 11: COMPRESSIVE WAVES AND SHOCK WAVES IN ANHARMONIC SOLIDS

Chapter II : 12: REFLECTED SHOCKS IN SIO2

Chapter II : 13: PRECISE ULTRAHIGH-PRESSURE EXPERIMENTS

Chapter II : 14: SHOCK IMPEDANCE MATCH EXPERIMENTS IN ALUMINUM AND MOLYBDENUM BETWEEN 0.1-2.5 TPa (1-25 Mbar)

Chapter II : 15: SHOCK HUGONIOT MEASUREMENTS ON Ta TO 0.78 TPa

Chapter II : 16: HUGONIOT MEASUREMENTS IN VANADIUM USING THE LLNL TWO-STAGE LIGHT-GAS GUN

Chapter II : 17: RAREFACTION VELOCITIES IN SHOCKED TANTALUM AND THE HIGH PRESSURE MELTING POINT

Chapter II : 18: THE VELOCITY OF SOUND BEHIND STRONG SHOCK WAVES IN 2024 Al

Chapter II : 19: HUGONIOT MEASUREMENTS ON UNSINTERED METAL POWDERS

Chapter II : 1

A REVIEW OF SOME RECENT THEORETICAL CALCULATIONS OF PHASE TRANSITIONS AND COMPARISONS WITH EXPERIMENTAL RESULTS*

Marvin Ross,     University of California, Lawrence Livermore National Laboratory, Livermore, California 94550, U.S.A.

We review some recent studies on structural phase transitions, metallization and melting. Theoretical methods have been increasingly successful in predicting these properties. A major difficulty remains in the determination of effective metal pair-potentials for finite temperature calculations in non-simple metals.

1 INTRODUCTION

Theoretical predictions of phase transformations are difficult because of the need for a very high accuracy in determining the free energy of the competing phases. Recent advances in theoretical and computational methods have brought with them an improved predictive capability and a deeper understanding of why many transformations take place. These advances have been stimulated by the appearance of the diamond anvil cell (DAC) technique which has made possible accurate measurements of pressure, density an optical properties. In this paper we emphasize the results of calculations and the comparisons with experiments.

2 ELECTRONIC TRANSITIONS AND PHASE STABILITY IN METALS

The study of s-d electronic transitions, f electron delocalization and structural phase stability has advanced our understanding of the systematic trends in metals both as a function of atomic number and of pressure. By changing the lattice parameter, electrons may be transferred between states of different s, d and f electron character and, in effect, transform elements into approximations of their neighbors. This feature, dubbed pressure tuning, has proved extremely useful for better understanding the materials at ambient conditions. It has become increasingly apparent that many phase transformations are closely linked to these changes in electronic structure.

The classic example of an s-d transition is the behavior of Cs in the pressure range 0-50 kbar. Bridgman,¹ investigating the compressibility, found a large volume uncertainty (9%) without a change in crystal structure at 42 kbar. In comparison to the other alkalis, the compressibility below this transition is surprisingly high and above the transition unexpectedly low.

The interpretation of this transition was first presented by Sternheimer on the basis of Wigner-Seitz calculations.² He showed that at normal density the highest occupied band was 6s-like and the lowest unoccupied levels were 5d-like.

Under compression the empty d-like states decreased in energy relative to the s-like, crossing the Fermi surface near the density in the observed transition. This suggested that the high compressibility and the isomorphic transitions resulted from the promotion of s-like electrons into the empty d-like levels. Since then numerous theoretical investigations have determined that these features are characteristics of most metals and are illustrated in Fig. 1. A similar description is expected to be valid for the Rb and K rows. For the transition elements the p band is contracted and lies entirely below the d. In the rare earth’s many of the conduction levels are degenerate with f states.

Figure 1 Schematic energy level diagram dT and dB are respectively the top and bottom of the d band.

The feature to note is that with compression the bottom of the d level (dB) moves below the bottom of the 6s band while the 5p band is broadened. This leads to a pressure induced increase in the d electron of metals. At very high compression the 5p to 5d gap closes and some 5d and 5p states become degenerate. For Xenon with a filled 5p band this leads to metallization. The negative volume derivatives of these electron energy levels may be roughly viewed as partial pressures. Thus, the 5d electrons have initially lower partial pressures than do either the 5p or 6s electrons. As a result, an electron transfer into a 5d state, either by thermal excitation or compression, leads to a decrease in that electron’s contribution to the total pressure, and will be observed as a softening of the pressure-volume curve. At sufficiently high density, the 6s band is emptied, and the isotherms are dominated by the repulsive pressure of the 5p and 5d electrons.

One feature of the continuous s-d transition is that different metals may be characterized as being at different stages along the transition. As a result the crystal structure sequence as a function of pressure and atomic number is related to the d band occupancy. Pettifor,³ Macintosh and Anderson⁴ have explained the structural trend observed across the transition metal series by calculating the difference in structural energy (ΔE) as the difference between the sums of the one-electron conduction band energies (εj).

(1)

These calculations were made by fast band methods, primarily the linear-muffin-tin-orbital (LMTO).⁴ It is about 100 times faster than the older d-plane-wave (APW) method and as a result is becoming very widely used. The APW is in principle more exact but the speed of LMTO compensates by permitting calculations for several orders of magnitude more points in the Brillouin zone. The higher precision is critical for calculating phase stability. The results shown in Fig. 2, demonstrate the systematic variation of crystal structure with calculated d-band occupancy (nd). Some difficulty is observed near nd ˜ 9,10 but in general the overall trend is apparent. Skriver, also using LMTO, calculated the structural energy differences for the alkaline metals and the divalent rare earths, Eu and Yb.⁵ Except for Yb, he correctly predicts the low pressure structure. For Yb the fcc structure was found to be marginally more stable than the observed hcp. Figure 3 shows the alkaline metal sequence as a function of nd. The results from Fig. 2 suggest that nd ˜ 1 should lead to the hcp structure indicated in the upper right corner (Fig. 3). The pressure induced increase in nd should lead to this sequence of structures. Several of these have been observed by McWhan and Jayaraman⁶ and Holzapfel and coworkers.⁷ These are: Ca(fcc → bcc) at 19.5 GPa (21); Sr (fcc → bcc) at 3.5 GPa (4) and Ba(bcc → hcp) at 5.5 GPa (10). The values in parenthesis are the pressures predicted by Skriver.

Figure 2 Systematic variation of observed and calculated crystal structures of the transition metals. Calculations are referenced to Efcc = 0. Lowest curves are the most stable. Mn, Fe and Co are magnetic and not considered.

Figure 3 Calculated d band occupancy (nd) for the alkaline earth metals. Data from Ref. [5].

McMahan and Moriarty⁸ predict phase transitions for: Mg(hcp → bcc) at ˜50-57 GPa, and Si (hcp → fcc) at ˜76-80 GPa (following lower pressure transitions to hcp) and Al (fcc → hcp) in the range 120-360 GPa.

It is now established that the four f electrons in the rare earths and late actinides (beginning with Am) are well localized and play no role in bonding and may be considered as part of the ion core. Except for Eu and Yb all the rare earths are trivalent and have an (sd)³ conduction band. In these elements the f⁷ configuration is favored by Hund’s Rule and as a consequence they all have properties similar to Ba (sd)². It follows that the elements with the (sd)³ configuration have similar properties. A feature of the trivalent rare earths are the sequence of pressure-induced phase transitions hcp → Sm type → dhcp → fcc which is also observed for decreasing atomic number from right to left across the series. Duthie and Pettifor⁹ have calculated the difference in total energies and shown that it is the degree of d band occupancy or the s-d transition that drives the lattice through the observed crystal structure sequence without any significant participation of f electrons. This result suggests that the high pressure sequence of structures in 4d transition metal yttrium and 3d scandium both (sd)³ might also follow this pattern. Yttrium was studied up to 34 GPa¹⁰ and the sequence hcp → Sm type → dhcp → fcc was observed with increasing pressure in total agreement with Duthie and Pettifor. However, an hcp to tetragonal transition was observed in SC around 20 GPa.¹¹ This structure is not found in any of the rare earths and suggests a limitation to the parallel possibly because the 3d band electron is more localized than the 4d. The 4d must be orthogonal to the 3d which acts to repel 4d and keeps it relatively further outside. Duthie and Pettifor did not make calculations for compressed Sc and these would be useful.

The high pressure phases of C, Si and Ge at T = 0 K have been studied by Cohen and coworkers.¹² They use a self-consistent pseudopotential method within the local density formalism. The pseudopotential represents the electron-core interaction and is chosen to reproduce the corresponding all electron excitation energies and eigenvalue and wavefunctions outside the core region. By not explicitly including core states the calculated total energies are usually two orders of magnitude smaller than those associated with all electron calculations. Hence the computational precision permits an accurate resolution of total energy differences between structures. Calculations for Si and Ge predict the cubic diamond structure will transform to a metallic β-tin phase at 99 kbar and 96 kbar respectively.¹²a The corresponding experimental values are 125 kbar and 100 kbar. The results for carbon are particularly interesting because from among the fcc, bcc, hcp, sc and β-tin structures the diamond phase is predicted to be stable until 23 Mbar when it will transform to the simple cubic structure.¹²b Yin and Cohen believe that if diamond does transform to some other metallic phase it will be at a pressure of at least 10 Mbar. These figures may be regarded as upper bounds for pressures achievable by diamond anvil cells. In practice the limitations result from stress fracture below 1 Mbar.

These predictions for diamond are consistent with Shockwave studies. The formation of dense modifications under shockloading has been observed in Si (112 kbar) and Ge (143 kbar)