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Submitted to the ANS/NT (2021). LA-UR-21-20443
Equation of State: Manhattan Project Developments and Beyond
Scott Crockett, Franz J. Freibert
Los Alamos National Laboratory
Los Alamos, NM 87545
Abstract: The hydrodynamic response of materials under extreme conditions of pressure, temperature and
strain is dependent on the equation of state of the matter in all its states of existence. The Trinity plutonium
implosion device development required the Los Alamos research community to advance the understanding
of equations of state further and faster than ever before. The unpredicted high yield from the Trinity fission
device explosion and the push to design the “Super” thermonuclear device initiated 75 years of unprecedented
research and technological progress in equation of state development. This article describes the progress
made on equation of state development during and since the Manhattan Project at Los Alamos.
Keywords: hydrodynamics, thermodynamics, shock, allotrope, multiphase
INTRODUCTION TO EQUATION OF STATE AND which we will examine. When we explore condensed
ITS PURPOSE matter behavior in extreme environments, the EOS
The Equation of State (EOS) efforts at Los Alamos have defines our understanding in a material’s physical
long been responsible for the development and delivery response to high-strain rate, high temperature, and high
of EOS utilizing the best available experimental and pressures conditions. The condensed matter EOS is
theoretical data. This fact applies broadly to the elements, defined in terms of thermodynamic and physical
other metals, alloys and materials in general and actinides properties1, i.e., specific volume, thermal expansion,
in particular due to the Los Alamos National Laboratory specific heat, and elastic moduli and sound speeds. Solid
mission. As discussed here, the EOS serves as an integral state EOS formulations1, such as those of Grüneisen and
part to solving the hydrodynamic equations used in Birch-Murnaghan, contain key connections between
simulations of high strain rate phenomena, but have fundamental thermo-physical properties and response to
significance well beyond that application. A traditional environmental extremes, whether those extremes are
EOS represents the thermodynamic states (pressure P, intentionally or accidentally driven.
internal energy E, and entropy S) of matter at a given Developing the fundamental theory behind an EOS
density (the reciprocal of specific volume V) and requires the interdisciplinary engagement of experts in
temperature T. These concepts can be generalized to more key topics such as Density Functional Theory (DFT),
complex response of states of matter (i.e., non-isotropic Quantum Molecular Dynamics, Thermodynamics, and
plastic deformation in response to an anisotropic stress Statistical Mechanics. The development of accurate and
tensor, irreversible atomic diffusion, magnetic hysteresis, predictive EOS theories and advanced modeling leads to
etc.); however, that requires a deeper technical discussion reduced uncertainties, improved predictive capabilities
than will be introduced here. Ensuring a high-quality, first from integrated simulations, and accurate physical
principles informed EOS is essential to predictive representations of matter in nature.2 These theoretical
hydrodynamic simulations and extreme environment efforts must be closely coupled with experimental efforts
modeling. Small variations in thermodynamic quantities, in Shock Physics, High Pressure Physics and High Energy
such as the initial position in (, T)-space and latent heat Density Physics facilities providing the necessary
or change in enthalpy H of phase transitions, can have a laboratory data to improve, verify and calibrate theoretic
large propagated effect on simulated trajectories and state models. A close collaboration between the experimental
endpoints. As will be discussed, the phase and and theoretical EOS research community has existed
thermophysical properties of matter whether solid, liquid, since the early days of the Manhattan Project and
gas, or plasma and transitions within and interactions continues today at Los Alamos National Laboratory and
amongst these phases, define the EOS for matter and its other laboratories around the world.
behavior. BRIEF HISTORY OF EOS AND DEVELOPMENTS
There are numerous technical fields requiring EOS LEADING TO THE TRINITY TEST
knowledge, in particular, geophysics, planetary sciences, Prior to World War II, key scientists in the Manhattan
and astrophysics; chemical, aeronautical, and aerospace Project established many of the fundamental ideas
engineering; and that of nuclear physics, metallurgy, required in understanding EOS theory beyond ideal gases.
materials engineering, and national security applications Many of the models used then and now were established
1Submitted to the ANS/NT (2021). LA-UR-21-20443
in the late 1800’s and early 1900’s by Rankine3, Van der thermal expansion of each phase, and the density shift
Waals4, Hugoniot5,6, Einstein7, Lindemann8, Debye9, between the phases along the one atmosphere isobaric
Grüneisen10, Thomas11 and Fermi12. As we examine the (constant pressure) curve. Also, observable in Fig.1 is
history of the EOS effort it will be pointed out how the thermal expansion hysteresis larger than T=150K in the
methods of the past established present modeling phase transformation, which represents the excess
philosophies. Early EOS models started from ideal gas strain energy associated with contracting the lattice and
approximations which were then extrapolated to include increasing the density proceeding from -phase
internal degrees of freedom and interactions of the gas (15.92g/cc) to -phase (17.14g/cc). Many refinements of
molecules4 to match available experimental data of the these and other thermophysical properties measurements
time. Limited in temperature, the models would be further and theory for plutonium have been implemented since
extended to include high-temperature thermal electronic then.18
responses. These efforts represent foundational principles
in thermodynamics, statistical mechanics and quantum
mechanics.
Prior to the establishment of Los Alamos, in 1940 Bethe
and Teller collaborated on the paper “Deviations from
Thermal Equilibrium in Shock Waves”13 examining
oxygen, nitrogen and shock waves in air. Bethe and Teller
studied how phenomena such as excitation of molecular
vibrations and dissociation connect with statistical
equilibrium and how it deviated from known
experiments.13 Continuing in this work, Bethe in 1942
outlined hydrodynamic theory of one-dimensional shock
waves in terms of the laws of conservation of mass,
momentum and energy.14 The study was focused on air
and water EOS. Bethe elegantly rederived the Rankine-
Hugoniot jump conditions.3,5,6 The thermodynamics of
phase transformations and the resultant impact on shock
behavior were described by Bethe in detail14, and this
understanding plays an important role for our most
modern EOS nearly eight decades later.
When characterizing any new material, certain
thermodynamic and physical quantities must be known to
Figure 1 First dilatometry trace of Martin and Selmanoff17 for
properly determine an EOS. In general, one must at a unalloyed plutonium showing five allotropes. Later work on
minimum have the following information: a reference higher purity plutonium revealed the sixth allotrope known as
density, an atomic weight and number, phase ’. Reproduced with permission from Los Alamos National
composition, an understanding of the compression Laboratory.
response, and phase boundaries (i.e., melt line, solid-solid
transformation boundary, etc.). In early 1944, the initial The need for EOS research and hydrodynamic theory and
density measurements of gram quantities of the newly modeling became essential to the Los Alamos mission
produced plutonium wildly varied from 16 g/cc to 20 with the transition from a gun assembled device to a high
g/cc.15 The University of Chicago “Met Lab” and Los explosively imploded plutonium device.19,20 Although the
Alamos Laboratory metallurgists recognized the fact that idea of implosion originated in 1942 by Tolman21, its
multiple allotropes of elemental plutonium existed via necessity became evident in early 1944, as it was realized
density measurements and volumetric dilatometry.15 The that the hydrodynamics of implosion was necessary to
lower density allotrope or -phase was malleable, but speed the assembly of available plutonium due to its
stabilized at higher temperatures; whereas the higher isotopically and elementally impure state and high
density allotrope or -phase was stable at ambient neutron background to achieve fission yield and avoid the
temperatures, but brittle.16 The pioneering plutonium fizzle of predetonation.22 No one had ever employed high
dilatometry work by Martin, Selmanoff, et al. showed five explosives (HE) to assemble a collective of matter before
different allotropes and -phase of plutonium surprisingly and to do so with precise timing and geometrical
exhibited a negative thermal expansion17 (Fig.1). High uniformity would demand perfect coupling of
temperature dilatometry was essential to understanding experimental and theoretical EOS research and
the temperature at which phase transitions occur, the development.
2Submitted to the ANS/NT (2021). LA-UR-21-20443
The early Manhattan Project implosion plutonium bomb cold curve enables an approximate lower bound of the
design was based on a hollow plutonium shell for the compression energy for a given state of matter density.
device core. However, implosion experimentation of 2D A memo dated April 16, 1945 detailed the first of
hollow objects, such as terminal observation from the HE Bridgman’s compression experiments on plutonium and
flash photography of imploding hollow cylinders, was followed up with additional measurements reported
indicated an asymmetry of collapse.23 These asymmetries April 25, 1945 of different samples up to ~0.1Mbar.29,30
upon implosion are resultant from hydrodynamic related Other materials data were reported soon afterward, e.g.,
phenomena such as jetting, spalling, and Rayleigh-Tayler Peierls discussed shock wave experiments on Al, Cd, Cu,
instabilities and led to inconsistent assembly of mass. The Fe, U in May 1945.31 The original work was performed
Christy Gadget, a tampered solid core device was chosen using a smooth-bore gun, due to geometric constraints
and shockwave interactions exhibited by the HE lens
moving forward based on its design and functional
systems19 under development by Goranson, et al.32 By the
simplicity.20 As such, concern over accuracies in EOS and
July 1945 Trinity Test and subsequent Nagasaki blast, T-
hydrodynamic modeling were lessened, but not
Division devoted significant time to the hydrodynamics
alleviated. interpretation of the blast measurements and to the
Early hydrodynamic methods applied to the Christy radiation hydrodynamics of the implosion fission bomb.33
Gadget development were based on a realistic multiphase During the same period, shock and material velocities had
EOS model that assumed imploding material changes been measured in six metals and six non-metals,34 and
discontinuously; however, complications arose when a with this work, Ancho Canyon in Los Alamos became the
multiphase EOS was employed. The numerical solution birth place of modern shock physics. Shock dynamics and
of the partial differential equation describing the the effects on materials were studied from the beginning
hydrodynamics of implosion were too difficult for Los of the Manhattan Project as discussed by Marshak35 and
Alamos computing staff to hand calculate. Considerable summarized by Taylor36. An extensive discussion on the
amount of effort was put into the multiphase model, but history of the experimental shock program at Los Alamos
the results proved very difficult to interpret. In the spring was written by Taylor.36 According to Taylor, gas-guns
of 1944, newly purchased IBM machines were configured used for measurements of Hugoniots only came into use
to solve the implosion problem, a theory effort led by by early 1955 and later in 1958. Hints of plutonium
Teller in Theory (T-1) Group.23 Teller and collaborators experiments (restricted to the Nevada Test Site) occurred
derived approximations to result in a simplified EOS used around 1955.
for IBM calculation and first results of the implosion In September 1945, Keller, et al.37 made refinements to
simulations were extremely satisfactory. Implementation the EOS from the work of Metropolis27 which aimed to
of multiphase EOS models under extreme conditions24 better incorporate temperature effects. These temperature
was delayed until modern computational capabilities and effects leveraged the cold curve response from the
experimental validation and verification could support Thomas-Fermi-Dirac (TFD)38 and Bethe-Marshak
approach for finite temperature to the electron density.
such developments.
The ion model follows the previous discussed work of
Teller and Ulam submitted “An Equation of State in the
Teller and Ulam.25 Special care was made in interpolating
Condensed Phase for Arbitrary Pressures and Moderate
between the Bridgman data which goes to 0.1 Mbar and
Temperatures” in September 1944.25 In this paper, they
the TFD calculation which started at ~75Mbar. They used
approximated the EOS by splitting internal energy into
a simple analytic form where the results compared
two parts: a low temperature thermal ion (vibrational) part
favorably with the initial work of Christy.26 The first
which is temperature dependent and a compressional
comparisons were made of uranium shock data32 to 1/3
energy part which is volume dependent via Birch’s
Mbar showing only a 7% deviation from the experiment.
parameterization. The basic approach allowed them to
Present day work differs mainly in that the experimental
approximate the low-pressure shock response of the
data and theoretical calculations overlap more closely,
material and had assumptions similar to the Mie-
thereby reducing or eliminating intermediate regions of
Gruneisen theory. By November 1944, preliminary
interpolation. It is now standard practice for condensed
interpolated curves representing EOS for Al, Cd, Fe, and
matter EOS determination to initiate a three-term
U were produced26 and by January 1945 Metropolis
decomposition of Helmholtz free energy F into
generated zero temperature compression “cold” curves
for a number of metals leveraging equations of Fermi and F=ES+FV+Ee- (1)
Thomas.27 This Fermi and Thomas model, now
considered the simplest example of a Density Functional for which ES is the T=0K static lattice cold curve typically
Theory, was published shortly after Schrodinger’s paper derived from DFT, FV is the vibrational energy of the ions
introducing his wave equation.28 Having the effective from thermal excitation as elucidated by the Debye
3Submitted to the ANS/NT (2021). LA-UR-21-20443
model, and Ee- is the energy of free electron density from microstructures and morphologies, which was a means to
thermal excitation as described in TFD theory. 1,2 study porous materials. This and related work has great
value in geophysics1 and seismic based nuclear non-
POST-TRINITY EOS RESEARCH proliferation monitoring and treaty verification. Further
Soon after the Trinity Test, Los Alamos was reorganized utilization of EOS during this time included the Bethe and
to address the next greatest driver for the laboratory which Tait 1956 fast reactor safety analysis which considered
was the development of the “Super”, a thermonuclear the internal energy of the system determined by the
fusion device. The July 16, 1945 Trinity Test led to the fission rate in the molten liquid fuel and the EOS of the
interest concerning radiation hydrodynamics and the fuel material, specifying the relationship between the
implosion fission bomb.33 The higher than anticipated nuclear aspects of excursion and the dynamic response of
yield of the Trinity explosion33,39 resulted in a the core.45
revitalization of earlier speculations regarding the The first publicly available post-war review of shock
simplifying assumptions and disregard for radiation built physics and EOS was published in 1958 by Rice, et al.
into the original implosion efficiency calculations. entitled “Compression of Solids by Strong Shock
Researchers began to take notice of these Waves”.46 This work provided a review of research and
“simplifications” as interesting areas of research. For technological advancement that had occurred to date
example, it has long been recognized that the Rankine‐ within the shock physics community. In particular, Rice,
Hugoniot relations for shock waves and the empirical et al. discussed the evolution of the experimental methods
relation between the shock‐wave and particle velocities used to measure and produce strong shock waves and
define an incomplete thermodynamic description of the provided the Hugoniot curves for twenty-seven materials.
states along the Hugoniot curve.40 This thermodynamic The experiments utilized a flash gap approach with a
limitation drove shock experiments diagnostic photographic method to measure the shock fronts. The
development to measure temperature under shock authors also compare the shock data with the static work
conditions in metals which led to such challenging of Bridgman. Rice, et al. summarize the capabilities that
experiments as the use of thermocouples for rapid existed between the experimental and theory efforts.
temperature determination.36 On Aug 26, 1947, Mayer Post-war shock and EOS research represented the most-
suggested the use of a short duration burst of x-rays to rapid expansion of knowledge in this field. That
observe the shock and detonation front in materials41 from precipitous growth and accumulation of data brought
which to estimate temperature excursion. The application recognition of the need to consolidate and document those
of flash x-ray radiography techniques has been utilized to data sets for use in computing. In the 1960s, Cowan
observe shockwave phenomena for six decades; however, established Maple, the first tabulated EOS database; and
the ability to make accurate estimate of temperature using in early 1970s, Barnes was instrumental in creating the
the flash x-ray technique are only possible in recent years SESAME I database. The SESAME database was intended
at facilities such as the Los Alamos National Laboratory to be a standardization of equation of state, opacity, and
DARHT facility. Theory developments led to the 1956 conductivity information. Barnes and others leveraged
Cowan extension of temperature range in Thomas-Fermi- the now ample amount of shock data and recalibrated the
Dirac models42 which furthered this model to allow for Maple tables. Barnes et al. subtracted the zero
calculations of temperature into the keV range. The temperature cold curves and replaced them with
methods which Cowan used were simplified by Liberman calibrated cold curves that matched experiment. The
around the late 1960s and this adaptation of the TFD thermal components (ion and electron) remained the
model is what we currently use for free electron density same. By this time, many of the capabilities to generate
contributions to EOS.43 the thermal components of an EOS had been lost, at which
Other EOS developments post-Trinity have included point the EOS project was rebuilt by Barnes, with Kerley
focus on metal, nuclear materials and minerals. By 1953, as the project lead. For much of the 70’s, 80’s, and 90’s,
Cowan reanalyzed previous work37 and made Abdallah, Albers, Bennett, Boettger, Dowell, Holian,
improvement to the aluminum and uranium EOS by Johnson, Liberman, Lyons, Rood, Straub, Barnes and
accounting for error in the 24ST aluminum used in the Kerley contributed to the creation of the SESAME II
initial shock experiments. As the understanding of the database.47,48 SESAME II was made available to the world
EOS of standard materials improves, the validation of in 1979. The researchers listed here utilized the models
new experimental platforms improves and uncertainty in developed by our project founders and solidified the
measurements is reduced. Continued refinement and methodology we employ today.49
improved fidelity in our simulation capabilities requires
higher precision EOS and hence improved experimental Modern EOS Methods
resolution. Fickett and Cowan also explored the idea of A modern EOS model is built by leveraging data collected
analyzing shock compression of sintered versus cast over experimental conditions ranging from ambient,
samples44 to better understand the impact of varied
4Submitted to the ANS/NT (2021). LA-UR-21-20443
static compression, and shock regimes, and by developing Lawrence Livermore National Laboratory
an integrated theoretical approach to ensure dataset PURGATORIO, a novel implementation of the
inclusion. These modern models cover compressions of 0 INFERNO equation of state physical model.50
to 106 volumetric strains and temperatures from 0 to 109 Examining the compression of materials at zero
Kelvins. Once optimized across multiple data sets, the temperature, DFT calculations are leveraged with
model forms naturally extend to known thermodynamic Diamond Anvil Cell (DAC) measurements and then
limits. We start, however, with ambient data along the extended to high compression by matching TFD
one-atmosphere isobar. The EOS relevant thermophysical calculations. DAC is a measurement of static
properties data includes information for the reference compression typically measured at room temperature. By
density, thermal expansion, specific heat, and bulk heating the DAC, it can be leveraged to measure phase
moduli. X-ray diffraction measures the initial crystal boundaries to moderate temperature. Quasi-isentropic
structure and density. Dilatometry measures the thermal compression experiments, pioneered by Sandia National
expansion. Resonant Ultrasound Spectroscopy is used to Laboratories on the Z-Machine in the early 2000s,
measure the adiabatic bulk modulus. Calorimetry is a provides a revolutionary means by which to measure
measurement of the enthalpy and specific heat. compression of high Z (atomic number) materials
Experimental methods also provide the temperatures of including actinides to extreme pressures. Isentropic
phase transitions (solid-solid, solid-liquid, liquid-gas, compression, which leaves the entropy of the system
solid-gas). These isobaric data provide constraints to the constant, achieves higher compression and lower
thermal components of an EOS model. Then theoretical temperature states than single shocks. These experiments
calculations are used for constraining our models in can be combined with shock (Hugoniot) experiments
regions where data is often absent. For that we rely on allowing access to unique regions of thermodynamic
DFT, Quantum Molecular Dynamics, and Quantum space as shown in Fig.2.
Monte Carlo (QMC) calculations. These methods are Shock compression experiments test the EOS of a
computationally intensive, but better match experimental material through the Rankine-Hugoniot relations, based
results for most materials describable by a multiphase on conservation of mass, momentum, and energy. These
EOS.24 Such a modern multiphase EOS generated at Los relations are temperature independent, and as a result
Alamos National Laboratory for aluminium is shown in even a poor-quality EOS can produce the correct
Fig.2. Other modern EOS models include that of the Hugoniot as this is not a unique relationship. The
difference in quality of EOS lies within how the EOS
predicts the derivative quantities off the Hugoniots such
as the sound velocity and release response, hence the
importance of combining shock data with other sources in
order to calibrate an EOS. Shocks can be obtained by
multiple platforms. Gas-guns, laser, high-explosives,
magnetic, and nuclear are all means of delivering energy
into a sample to produce high strain rate compressive
states.
EOS modelers must be careful when interpreting
experimental data of solids. So-called EOS measurements
in a solid, whether by dynamic or static means, often
contain deviatoric effects such as strength, kinetics, or
other non-hydrostatic mechanical or non-equilibrium
responses that are accounted for not in the EOS, but by
other physics models. Therefore, comparisons with
experiment must leave room for other models to come in
and provide a complete picture.
CONCLUSIONS
Figure 2 The phase diagram for aluminum overlaid with key
The Manhattan Project efforts conducted at Los Alamos
thermal dynamics curves. The isobar (constant pressure),
isotherm (constant temperature), adiabat or isentrope represent the most prolific era in the history of equation
(constant entropy), and the Hugoniot are standard thermal of state development. The shift from a gun assembled
dynamics paths that can be measured via experiments. The plutonium device to an implosion assembled plutonium
labels fcc, hcp and bcc represent regions of face-centered device demanded a rapid advancement in equation of
cubic, hexagonal close-packed and body-centered cubic phase state experiment and theory to support the hydrodynamic
stability, respectively. simulations necessary for device development. However,
5Submitted to the ANS/NT (2021). LA-UR-21-20443
over the intervening 75 years, many of the original EOS 11. Thomas, L. H., "The calculation of atomic fields",
simplifications employed to yield a tractable Trinity Mathematical Proceedings of the Cambridge
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14. Bethe, H. A., "On the Theory of Shock Waves for an
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Alamos Laboratory, LA-00318, (1945). (1998). The SESAME EOS database in its current
36. Taylor, J. W., Thunder In The Mountains, '83 APS, format has been utilized by academia, industry, and
Shock Waves In Condensed Matter Los Alamos other national laboratories around the world since
National Laboratory, LA-UR-83-1971, (1983). 1972.
37. Keller, J. M., Metropolis, N. C., Equations Of State 50. Wilson, B., Sonnad, V., Sterne, P., and Isaacs, W.,
Of Metals, Los Alamos Laboratory, LA-00385, PURGATORIO — a New Implementation of the
(1945). INFERNO Algorithm. Journal of Quantitative
38. Feynman, R. P.; Metropolis, N.; Teller, E., Spectroscopy and Radiative Transfer, 99(1), 658
"Equations of State of Elements Based on the (2006)
Generalized Fermi-Thomas Theory", Physical
Review 75(10), 1561, (1949).
39. Selby, H.; et al., "Trinity yield from radiochemical
analyses", Nucl. Technol. (2021). The contemporary
assessed yield of the Trinity Device was 21 kT;
however, this recent Los Alamos Trinity Device
assesment is 24.8+/-2 kT that is based on analyses
that take advantage of advances in high precision
mass spectrometry to study the debris.
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