A method of structural expansions for use in determining the equation of state of metallic hydrogen (and indeed other metals) up to the 4th order in the perturbation theory was developed. The electrical and thermal transport properties of the planetary interior of Jupiter were calculated. The nature of the interaction between molecules at short range and the importance of multicenter terms in arriving at an adequate description of the thermodynamic functions of condensed molecular hydrogen were also investigated

Ashcroft, N. W.

English

NASA Technical Reports Server

Astrophysical Materials Science:
Theory
NASA-AMES Agreement
#NAG 2-159
Final Technical Report
(NASA-CR-177021) ASTEOPHISICftI H&TEBI&LS
SCIENCE: THEOBY. . final Technical Report
(Cornell Univ.), 13 p CSCL 03B
N86-3Q6510
Onclas
G3/90 - 43230 --•
N. W. Ashcroft
Professor of Physics
Cornell University
Ithaca, NY 14853
https://ntrs.nasa.gov/search.jsp?R=19860021138 2020-03-20T13:26:00+00:00Z
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Astrophysical Materials Science:
Theory
Final Technical Report
I. Introduction
Since the initial award of Grant NGR-33-010-188 in summer of 1972, the
aim of the project "Astrophysical Materials Science: Theory" has been to
develop analytic methods to better our understanding of common astrophysical
materials particularly those subjected to extreme physical conditions. The
program was administered in the past by the staff of the Lewis Research
Center, National Aeronautics and Space Administration, Cleveland, Ohio. In
October, 1978, the project was administered by N.A.S.A. Washington,
reappearing under the same title as NSG-7487. Beginning DecemberSl, 1981,
the project was transferred for administrative purposes to NASA, Ames
Laboratory, Moffett Field. The contract ended on December 1, 1983.
This document briefly summarizes the research discoveries and work
carried out on these various contracts. Hydrogen and helium constitute by
far the most abundant of the elements, and it is no accident that the
research continues to be focused heavily on these elements in their
condensed forms, both as pure substances and in mixtures. It will be seen
below that the research has combined the fundamental with the pragmatic.
r- . •
The proper'and complete understanding of materials of astrophysical interest
requires a deep appreciation of their physical properties, especially when
taken into the unusual ranges of extreme conditions. Fundamental
theoretical condensed matter physics has played a very important part in the
research to date, and will continue to be a dominant element in any research
carried out in the future. The collaboration with the experimentalists at
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Cornell (Professor Ruoff and his group) have also been exceedingly
beneficial.
The research will now.be summarized.
II. Research Synopsis
Attached to this document are three Appendices listing the cumulative
publications of work supported by NASA under Grants NGR-33-010-188,
NSG-7487, and NAG 2-159. Papers #1-13 give the background necessary to
place the research supported by Grants NSG-7487 and NAG-2-159 in proper
context. The themes developed in this work can be described briefly as
follows.
Paper #1, on the ground state energies of simple metals, developed the
method of structural expansions for use in determining the equation of state
of metallic hydrogen (and indeed other metals) up to 4th order in
perturbation theory. Previously, work in the Soviet Union and elsewhere had
made predictions on the nature of the structure of metallic hydrogen based
on lower perturbation theory. Paper #1 called this into question, at least
for static lattices.
Paper #2 concerned itself with the nature of the deep interior of
Jupiter, particularly with respect to the transport properties. We were
able to calculate both the electrical and thermal transport properties of -
{.
the planetary interior and hence comment on the origin of the Jovian
magnetic field.
Paper #3 is devoted to a problem in molecular hydrogen, specifically
the nature of the interaction between molecules at short range and the
importance of multi-center terms in arriving at an adequate description of
the thermodynamic functions of condensed molecular hydrogen.
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Paper #4 returned to the subject of Paper #1 and took up the question
of proton dynamics, again arriving at a method applicable to many metals.
In accounting for the structural energies in a dynamic lattice, we also
obtained a method for determining x-ray structure factors (particularly
diffuse thermal scattering) which has been very useful.
Paper #5 addresses a problem raised in Paper #2, namely, are metallic
hydrogen and metallic helium mutually soluble under the conditions
prevailing in the deep interior of Jupiter? The results of the calculations
presented in Paper #5 show fairly convincingly that almost complete phase
separation is to be expected, and this has interesting consequences in the
transport properties as a function of depth into the planet.
Paper #6 tackles a question emerging from Paper #4, namely, can the
proton and electron degrees of freedom really be separated when dealing with
the thermodynamic functions of hydrogen, or should they be treated as
coupled systems? The latter is found to be the case, and the structural
consequences are really quite important. Simple structures are favored by
this approach, rather than the grossly anisotropic structures proposed by
the Soviet groups.
Paper #7 continues the work of Paper #5, but continued into the domain
of liquid rather than solid solutions of hydrogen and helium. The
miscibility gap in the solid is found to persist in the liquid alloys unless
the temperature gets exceedingly high. This has application in some stellar
exteriors.
Paper #8 begins a study of molecular hydrogen and its band-structure
and continues the work begun in Paper #3. The ultimate intent is the
determination of the thermodynamic functions of the molecular phase, and
then the estimation of the metallization pressure. The results of the
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calculation introduce the notion that metallization by isostructural band-
overlap may be a possibility.
Paper #9 deals with the quantum aspects of ground state defects in
hydrogen and asks whether "quantum-defectons" can be present in metallic
hydrogen crystals, and if so, whether they can coagulate into macroscopic
voids whose surfaces may then be unstable to molecule formation. This
prospect is ruled out by calculation: again a general method for dealing
with systems other than hydrogen is introduced.
Paper #10 introduces a new idea: that the ground state of metallic
hydrogen might be a quantum liquid. To obtain the ground state energy of
such a system, it is then necessary to extend the theory of liquids somewhat
and the paper deals with a method for obtaining the necessary distribution
functions.
Paper #11 then takes up the idea of Paper #10 to calculate the ground
state energy of a proposed liquid phase of metallic hydrogen and indeed
finds that up to third order at least (in the electron-proton interaction)
such a state is a very strong possibility. It also examines the likelihood
of partially ordered magnetic phases, and notes that some of the ordering
energies are quite characteristic of superconducting ordering energies.
Paper #12 extends the notions discussed in Paper #2 and discusses both
the metallic and insulating form of hydrogen and helium in the context of
models of the interior of Jupiter and Saturn.
Paper #13 is also concerned with Jupiter and Saturn, but from the
standpoint of dynamic aspects, specifically convection and the influence on
it of composition gradients in the mixture of hydrogen and helium.
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III. Research 1978-81
Section II gives the background to the following more recent
developments:
Paper #14 introduces a new way of describing dense molecular and dense
metallic hydrogen from a common density functional framework. Since the
same theoretical picture is used in both phases, systematic errors are
eliminated. Metallization pressures in the neighborhood of 1.8 Mbar are
predicted.
Paper #15 treats a theory of the liquid state applicable to insulating
systems and metallic systems (for instance, liquid metallic hydrogen in the
classical regime). It introduces a powerful reinterpretation of what is
known as the reference hypernetted chain model (RHNC). In particular, it
shows that all classical simple liquids are, to a high degree, universal
when viewed in terms of the behavior of their bridge functions.
Paper #16 extends some of the ideas of universality (Paper #15) and
gives an iterative argument establishing the insensitivity of the bridge-
function to the choice of pair potential.
Paper #17 extends the ideas of Papers #16 and #17 even further to the
interesting class of dense fluids characterized by very soft pair
potentials. The high density classical plasma (e.g., hydrogen) is a case in
point to which this theory can be applied.
r.
Paper #18 deals with a dense one-component plasma but in a new context:
the plasma confined to a two-dimensional surface. It is a simulation study
in which the aim is to examine the thermodynamic functions, correlation
functions, and ordering.
Paper #19 examines the role of the pair potential in a liquid metal in
determining the details of the static structure. Normally the pair
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potentlal is adequately represented in pseudopotential theory as a screened
Coulomb interaction between ions. For certain systems, 'however, the
fluctuating dipole contributions (dynamically screened by the electron gas)
can lead to important corrections having noticeable structural consequences.
The effect is present, in principle, in the light elements at high density.
Paper #20 extends the Monte-Carlo simulation methods developed in Paper
#18 to the three-dimensional context. Specifically it treats the case of
high density metallic hydrogen and comes to the interesting conclusion that
over a certain density range the system may prefer to be a liquid metal at
T = 0. This would be a new state of matter.
Paper #21 takes up this idea (a zero temperature liquid metal) and
examines the physical properties of such a novel quantum-liquid. The paper
generalizes two-component Fermi-liquid theory to the case of two components
(electrons and protons).
Paper #22 begins to tackle the problem of the melting of hydrogen (in
the density range where it is a solid at T = 0) by developing a hard-sphere
crystal theory which largely parallels in intent the hard sphere liquid
theory so successfully used in the treatment of classical liquids. The idea
is to use a hard-sphere solid as a basis for perturbation treatments of real
dynamic crystals.
Paper #23 is a discussion of possible ordered phases of liquid states
of metallic hydrogen and deuterium near T = 0. It includes
superconductivity and superfluidity. Again, a superconducting liquid metal
would be a new state of matter.
Paper #24 extends the method introduced in Paper #14 to include a
discussion of proton dynamics in the destruction of the paired state as
pressure increases. It has direct relevance to two measurements of Raman
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scattering from ultra-pressure hydrogen, one of which it supports and the
other it does not.
Paper #25 applies some of the ideas of Papers #15-17 to the case of
inhomogeneous fluids, for example, the surface of a classical fluid, or the
boundary layer between a classical fluid and a container.
Paper #26 further extends the discussion presented in Paper #21 on the
Fermi liquid properties of two-component Fermion systems. In particular, it
presents stability criteria for these systems which must be obeyed if the
fluid is to remain 'normal' (i.e., no pairing)..
Paper #27 presents the necessary theory for a liquid superconducting
state. It begins with the Eliashberg equations and because of the
translational and rotational symmetry of a liquid (not present, of course,
in a crystalline solid) can reduce them to manageable forms. The predicted
transition temperatures for superconductivity are still very high.
IV. Research 1981-83
For the background on this phase of our research, the reader is
referred to sections II and III, above. The papers now to be described were
supported by the NASA-Ames Agreement (NAG 2-159). Appendix III lists the
appropriate citations.
Paper #28 reviews the progress in elucidating the theoretical issues
/.
raised by the prospects of two new quantum liquids namely liquid metallic
deuterium and liquid metallic hydrogen. Both would be novel states of
matter.
Paper #29 gives a rather complete theory of the transport properties of
a two component Fermi liquid starting with the generalized Landau-Silin-
Boltzmann equations. It's shown that the transport properties of liquid
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metallic hydrogen are dramatically different from those of its crystalline
counterpart.
Paper #30 addresses a variational approach to the groundstate energy of
dense molecular and metallic hydrogen within the spin-restricted Hartree-
Fock approximation. It is explicitly demonstrated that bond-separations
within molecules change rapidly at the onset of the metal-insulator
transition.
Paper #31 proves an interesting cusp theorem that relates the zero-
separation value and slope of two particle positional correlation functions
in quantum many-body systems with Coulombic interactions. The theorem was
applied to atomic-orbital based models of both molecular hydrogen and
metallic hydrogen.
Paper #32 examined the behavior of the critical fields of liquid
superconducting metallic hydrogen. The system is precicted at certain
densities to pass from dirty to clean and from type-II to type-I behavior as
temperature is lowered. The change in type is associated with a strong
temperature dependence of the basic electron transport time.
Paper #33 addresses two possible types of pairing involving the protons
in a low-temperature liquid phase of metallic hydrogen. One is the many-
body generalization of the basic electron-proton pairing familiar in the
atomic context. The other is a proton-proton pairing analogous to that
occurring in the ordered phases of liquid He3.
Paper #34 further examines the energetics of some liquid-like states of
dense hydrogen, and specifically addresses the difficulties of obtaining
accurate estimates when using many-body perturbation theory.
Paper #35 comments on a derivation of acoustic plasma modes in two
component Fermi liquids which appeared in the literature but which was
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easily shown to be incorrect. Metallic hydrogen is an example of such a
system and which has but a single longitudinal acoustic mode (in contrast to
three such modes, as was recently claimed).
V. Conclusions
Metallic hydrogen, metallic deuterium, and metallic helium have
considerable importance in the content of planetary interiors and stellar
interiors. In such surroundings the systems are undoubtedly in the
classical regime, and in this range of densities and temperatures their
properties are extremely interesting. Even more fascinating, however, are
their properties in the quantum regime. It is not yet established firmly
that metallic hydrogen and deuterium can exist as liquids at low
temperatures and at certain densities. However, the prospect looks very
probable and, if true, the range of predicted properties for metallic
hydrogen will greatly exceed the initial expectations. Thus, the highly
quantum aspects of high density hydrogen and helium will remain a source of
interesting physics and astrophysics for some time to come.
N. W. Ashcroft
Ithaca, NY
April, 1984
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Appendix I
CUMULATIVE PUBLICATIONS: NGR-33-010-188
(Supported by NASA)
1. "Ground State Energies of Simple Metals," J. Hammerberg and N. W.
Ashcroft, Phys. Rev. B9, 409 (1974).
2. "Conduction in Fully Ionized Liquid Metals," D. J. Stevenson and N. W.
Ashcroft, Phys. Rev. A£, 782 (1974).
3a. "Aluminum Under High Pressure: I. Equation of State," C. Friedli and
N. W. Ashcroft, Phys. Rev. B12, 5522 (1975).
4. "Thermal Diffuse X-ray Scattering in Simple Metals," D. M. Straus and
N. W. Ashcroft, Phys. Rev. B_14, 448 (1976).
5. "Phase Separation of Metallic Hydrogen-Helium Alloys," D. M. Straus and
N. W. Ashcroft, Phys. Rev. B15_, 1914 (1977).
6. "Self-Consistent Structure of Metallic Hydrogen," D. M. Straus and N.
W. Ashcroft, Phys. Rev. Lett. J38, 415 (1977).
7. "Thermodynamics of Thomas-Fermi Screened Coulomb Systems," B. Firey and
N. W. Ashcroft, Phys. Rev. A15, 2072 (1977).
8. "Combined Representation Method for Use in Band-Structure Calculations:
Applications to Highly Compressed Hydrogen," C. Friedli and N. W.
Ashcroft, Phys. Rev. BJJ5, 662 (1977).
9. "Einstein-Kanzaki Model of Static and Dynamic Lattice Relaxation:
Application to Vacancies in Metallic Hydrogen," J. F. Dobson and N. W.
Ashcroft, Phys. Rev. BJL6, 5326 (1977).
10. "Analytical Treatment of.Hypernetted Chain and Percus-Yevick Equations
for Bosons," S. Chakravarty and N. W. Ashcroft, Phys. Rev. B18, 2978
(1978). ~~
11. "Ground State of Metallic Hydrogen," S. Chakravarty and N. W. Ashcroft,
Phys. Rev. BJL8, 4588 (1978).
12. "The Dynamics and Helium Distribution in Hydrogen-Helium Fluid
Planets," D. J>. Stevenson and E. E. Salpeter, Astrophys. J. Suppl. 35,
239 (1977).
13. "The Phase Diagram and Transport Properties for Hydrogen-Helium Fluid
Planets," D. J. Stevenson and E. E. Salpeter, Astrophys. J. Suppl. 35,
221 (1977).
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Appendix II
CUMULATIVE PUBLICATIONS: NSG-7487
(Supported by NASA)
14. "Molecular and Metallic Hydrogen. A Unified Density Functional
Approach," in High Pressure Science and Technology, Vol. 2, eds. B.
Vodar and Ph. Marteau (Pergamon Press, 1980) p. 591.
15. "Theory of Simple Classical Fluids: Universality in the Short-Range
.Structure," Y. Rosenfeld and N. W. Ashcroft, Phys. Rev. A20, 1208
(1979). ~~
16. "A New Theory of Simple Classical Fluids," Y. Rosenfeld and N. W.
Ashcroft, Phys. Lett. _73A , 31 (1979).
17. "Mean Spherical Model for Soft Potentials: The Hard Core Revealed as a
Perturbation," Y. Rosenfeld and N. W. Ashcroft, Phys. Rev. A20, 2162
(1979). ~
18. "Monte Carlo Simulation of the Classical Two-Dimensional One-Component
Plasma," R. C. Gann, S. Chakravarty, and G. V. Chester, Phys. Rev. B20,
326 (1979).
19. "Core Polarization and the Structure of Simple Metals," N. W. Ashcroft,
G. V. Chester, and K. K. Mon, Phys. Rev. B_19, 5103 (1979).
20. "Simulation Studies of a Model of High-Density Metallic Hydrogen," K.
K. Mon, G. V. Chester, and N. W. Ashcroft, Phys. Rev. B21., 2641 (1980).
21. "Low Temperature Properties of a Possible Liquid Metallic Phase of
Hydrogen," J. Oliva and N. W. Ashcroft, Bull. Am. Phys. Soc. 25, 436
(1980). ~~
22. "Cluster Variation Theory of the Hard Sphere Crystal," B. Firey and N.
W. Ashcroft, Bull. Am. Phys. Soc. ^5_, 350 (1980).
23. "Dense Phases of Hydrogen," in Modern Trends in the Theory of Condensed
Matter (eds. A. Pekalski and J. Przystawa), Springer-Verlag, Berlin,
1980, p. 531.
24. "Theory of Dense Hydrogen," S. Chakravarty, J. H. Rose, D. Wood, and
N. W. Ashcrofti Phys. Rev. B24, 1624 (1981).
25. "Theory of Inhomogeneous Fluids," R. M. Nieminen and N. W. Ashcroft,
Phys. Rev. A24, 560 .(1981).
26. "Two-Component Fermi-Liquid Theory: Equilibrium Properties of Liquid
Metallic Hydrogen," J. Oliva and N. W. Ashcroft, Phys. Rev. B23, 6399
(1981). ~~
27. "Superconductivity in Liquid Metallic Hydrogen," J. E. Jaffe and N. W.
Ashcroft, Phys. Rev. B23, 6176 (1981).
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Appendix III
CUMULATIVE PUBLICATIONS: NAG-2-159
(Supported by NASA)
28. N. W. Ashcroft, "Ordered Ground States of Metallic Hydrogen and
Deuterium" in Physics of Solids Under High Pressure (J. S. Schilling
and R. N. Skelton, Eds.) (North Holland Publishing Co., 1981), p. 155
29. J. Oliva and N. W. Ashcroft, "Two Component Fermi -Li quid Theory:
Transport Properties of Liquid Metallic Hydrogen," Phys. Rev. B25, 223
(1982). ~~
30. D. M. Wood and N. W. Ashcroft, "Structure and Screening in Molecular
and Metallic Hydrogen at High Pressure," Phys. Rev. B25, 2532 (1982).
31. A. E. Carlsson and N. W. Ashcroft, "Asymptotic Freedom in Solids: A
Theorem," Phys. Rev. B^ 5_, 3474 (1982).
32. J. Jaffe and N. W. Ashcroft, "Critical Fields of Liquid Superconducting
Metallic Hydrogen," Phys. Rev. Bj[7, 5852 (1983). •
33. A. E. Carlsson and N. W. Ashcroft, "ordered Pairing in Liquid Metallic
Hydrogen," Phys. Rev. B^ 7_, 6676 (1983).
34. A. E. Carlsson and N. W. Ashcroft, "Liquid-like States of Dense
Hydrogen," Phys. Rev. B29, 479 (1984).
35. J. Oliva and N. W. Ashcroft, "Comment on 'Acoustic Plasma Modes',"
Phys. Rev. B29, 1067 (1984).


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