Plutonium Condensed-Matter Physics

Plutonium Condensed-Matter Physics

A survey of theory and experimentA. Michael Boring and James L. Smith

90 Los Alamos ScienceNumber 26 2000

Systems like to be in thelowest-energy state, butplutonium metal has

trouble getting there. It hasmany states close to each otherin energy but dramatically different in structure, and so aportion of a sample can changeits structure and density in response to minor changes in itssurroundings. We probably haveyet to see a sample near roomtemperature that has reached trueequilibrium. This metastabilityand its huge effects are part ofthe story of the strange proper-ties of plutonium metal, alloys,and metallic compounds, andthey are extremely important ifwe want to leave nuclearweapons untouched for decades.

Here, we will put plutoniummetal in perspective by compar-ing it to the other actinides andto other metals in the periodic

table of the elements. Plutoniumhas many unusual properties. Instead of having the cubicstructure found in familiar metals, its ground state has avery low symmetry monoclinicstructure with 16 atoms in the unit cell. Its instability is legendary among metallurgists—plutonium goes through six distinct crystallographic phaseswhen heated to its melting pointunder atmospheric pressure. One of those phases is the face-centered-cubic δ-phase, whichcan be stabilized down to rela-tively low temperatures by alloy-ing it with a tiny amount of gallium metal. The δ-phase is itself a tremendous puzzle, hav-ing an unusually low density, as well as a negative thermal-expansion coefficient; that is,δ-plutonium contracts whenheated. Below room temperature,plutonium continues to display

anomalous properties. In particu-lar, it has an unusually high resistivity and an elevated spe-cific heat, suggesting novel inter-actions and correlations amongits electrons. Because they areprobably the root for much ofplutonium’s unusual behavior,we will bring up these unusualelectron correlations in connec-tion with both high- and low-temperature phenomena.Figure 1 shows plutonium sitting at the crossover of manyproperties. As we survey thoseproperties, we will consider the following questions: Is pluto-nium fundamentally differentfrom other metals? Do we needan entirely new theory to explainits behavior?

Definitive answers must await better and more-completeexperimental data leading to a full theory of plutonium andits compounds. We show, however, that the two underlyingconcepts of modern theories ofmetals, the one-electron “band-structure” approach andthe correlated-electron approach,are relevant to plutonium.

Number 26 2000 Los Alamos Science 91


Moreover, many ground-state propertiesof plutonium can be predicted frommodern one-electron band theory. Otherproblems remain to be solved. As youread this and other papers in this vol-ume, you will learn not only about thesuccesses but also about the ongoingmysteries that place plutonium at afrontier of condensed-matter physics.

The f Electrons and the Role ofNarrow Conduction Bands. All met-als, including plutonium, are held to-gether by the electronic, or chemical,bonding between the conduction elec-trons and the positively charged ioncores that constitute the crystal lattice.Conduction electrons are not localizedat individual lattice sites. Instead, theyare itinerant and travel almost freelythrough the crystal. They are the “glue”that binds the ions together. We do notthink of glue as moving around, sometals are a bit tougher conceptuallythan other solids. Nevertheless, it ispossible to calculate the specific bandsof energy levels that are occupied bythe conduction electrons. The structureof those bands determines many proper-ties of metals.

In pure plutonium and other lightactinides, the conduction electrons include not only the s, p, and d valenceelectrons, as in the transition metals,but also the valence electrons unique tothe actinides, namely, those in the 5fvalence shell. Each plutonium atom hasfive 5f electrons to contribute to bond-ing. However, the roles of those 5felectrons in the various solid phases ofpure plutonium metal and in its metalliccompounds and alloys seem to vary.The 5f electrons can be localized (orbound) at lattice sites, in which casethey do not contribute to the bonding,or they can occupy a narrow conduc-tion band and contribute to the glue.Pinning down the exact interactions andcorrelations among the electrons thatlead to this variability between localiza-tion and itinerancy is currently the sub-ject of intense studies.

For decades, scientists thought thatthe pointed shape (angular variation) of

f-electron atomic orbitals and the likeli-hood that those orbitals would form di-rectional bonds were the source of theanomalies in plutonium metal. Thatconcept was used for solids because itworked so well for molecules and mol-ecular complexes. In fact, when youread the articles on actinide chemistry,you will see the importance of theshape of f orbitals for those molecularsystems. In this article, however, weexplain how modern band-structure cal-culations of plutonium in its low-sym-metry ground state (the α-phase) haveled to a less atomic-like view of at leastsome of its properties. Those calcula-tions demonstrate that, for plutonium,as for other metals, it is the energybands that determine such ground-state(T = 0) properties as the cohesive ener-gy, the stability of the crystal structure,and the elastic properties. Furthermore,plutonium’s very low symmetry crystalstructure in the ground state can betraced to a very particular feature of itsenergy bands—its dominant conductionband, the one that contributes the mostto holding the metal together, is therather narrow f electron band.

In the latter half of the article, weexamine the phase instabilities in pureplutonium, as the metal is heated, andtheir possible origin in f-electron narrow-band behavior. We then intro-duce the low-temperature properties thatplace plutonium among the correlated-electron materials. Finally, we discussthe exotic “heavy-fermion behavior” ofcerium and light actinide compoundsbecause, if we can understand these extremely narrow band materials, wemay understand plutonium. In general,the low-temperature behavior of correlated-electron materials includingplutonium appears to be dominated byas yet unexplained interactions involv-ing their narrow-band electrons. For thatreason, the ground states, which in moretypical materials are either magnetic orsuperconducting, are often not well determined and certainly not understoodfor these materials.

Every few years, condensed-matterphysicists find a material with a new

ground state that challenges acceptedparadigms, and the community turns itsattention to this new challenge. But thereis also a compelling need to keep work-ing on plutonium: This metal presentssome of the most puzzling behaviors ofall the elements, and those behaviorsbear on the national security mission ofthe Los Alamos National Laboratory.Although theory can be done anywhere,plutonium cannot be measured at mostlaboratories, and so we, at Los Alamos,are working on experiment and theorywith renewed intensity.

The ideas in our survey of plutoniumrange in acceptance from firm scienceto outright speculation. We will try tomake clear which is which, but we include both in order to cover our deepest understanding of this complexelement efficiently and, we hope, with more interest.

Basic Properties of Metals

For the most part, metals form in acrystalline state. Unlike amorphous orglassy materials, the atoms in a crystalare arranged in a periodic (repeating)array of identical structural units knownas unit cells. The repetition in spacemeans that metals have translationalsymmetry, and this symmetry underliesall metallic behaviors. Most theoreticalmodels of metals are tractable becausethey exploit the translational symmetry.For example, even with modern com-puters, one cannot calculate the elec-tronic structure of a piece of wood because it has no underlying symmetry.The translational symmetry of crys-talline solids (not just metals) leads toelectron wave functions (or Blochstates) that have that same translationalsymmetry up to a phase factor. Thesewave functions are macroscopic, extending over the entire crystal lattice,and they serve as the solid-state equiva-lents of molecular orbitals. That is, justas electrons in molecular orbitals arethe glue that bond atoms into a mole-cule, electrons in Bloch states are theglue that bond atoms into a crystal.



The big difference between molecular orbitals and Bloch states lies in theirnumbers: In a molecule, there are onlya few molecular orbitals, but there areon the order of 1023 Bloch states.Therefore, one does not focus on individual Bloch states (it is difficult to choose a particular one) but on aver-ages over these states, such as the density-of-states functions discussed in the next section.

Formation of Energy Bands. Whenlight metallic elements, such as lithiumor sodium, condense into the solidstate, they typically have cubic struc-

tures at room temperature, and the elec-trons from their atomic valence shellsbecome conduction electrons travelingalmost freely through the lattice. Thatis, these valence electrons occupy one-electron Bloch states, and they aretherefore responsible for bonding thesolid. The allowed energies of thoseBloch states form a broad band of energy levels. In a metal, this energyband is a conduction band because it is only partially filled. Because manyempty states are available, the conduc-tion electrons with the highest energiesrespond to low-energy thermal andelectrical excitations as if they were

a gas of free particles. Figure 2 illustrates band formation

and the formation of Bloch states insodium. The top of the figure showsthat, when two sodium atoms arebrought together, their 3s-electron wavefunctions (orbitals) overlap, and the valence electrons feel a strong electrosta-tic pull from both atoms (depicted as thedouble-well electrostatic potential). The atomic orbitals combine to formmolecular orbitals that may bind the twoatoms into a diatomic molecule. The sin-gle atomic energy level splits into two:one lower in energy, or bonding, and theother higher in energy, or antibonding.



Al

Cu

Pd

K Cu

Cu

K

Fe

Gd

Ce

Gd

UBe13

UPt3

Itinerant

magnetism

Localize

d

electrons

Local-m

oment

magnetism

Free-electron behavior

Heavy-fermion behavior

Direction of increasing electron correlations in dominant band

Low-energyexcitationsat both ends

Direction of decreasing widthof dominant band

Itinerant

electrons

Simple

superco

nductivit

y

d or f bonding

Higher-densit

y metal

Pu

Correlated

ElectronsLow electrical resistivityLow density of statesLow electronic-heat capacity

Low electrical resistivityLow density of statesLow electronic-heat capacity

High electrical resistivityHigh density of states

High electronic-heat capacity

Heavy Fermions

Figure 1. Plutonium at a Crossover in Electronic Properties This figure summarizes some of the unusual electronic properties of plutonium, stemming from the dominant role of its narrow

5f band. Along one diagonal, plutonium stands midway between simple metals, whose conduction electrons are essentially uncorre-

lated and exhibit free-electron behavior, and heavy-fermion materials, whose conduction electrons exhibit very strong correlati ons

leading to extremely high effective masses. Along the other diagonal, plutonium stands at the crossover between materials whose

itinerant broad-band electrons form superconducting ground states and magnetic materials, whose fully localized electrons (infin itely

narrow-band) form local moments and magnetic ground states. Along the horizontal line, plutonium and other correlated-electron

narrow-band materials are distinguished from the elements on either side through their high resistivity and specific heat, high

density of states at the Fermi energy, and enhanced electronic mass.

The energy difference between these twolevels is proportional to the amount ofoverlap of the two s-electron atomic orbitals, and the molecular orbitals(wave functions) corresponding to the bonding and antibonding energy levels are sum and difference, respec-tively, of the atomic orbitals.

Similarly, when N atoms are broughtclose together to form a perfect crystal(bottom of Figure 2), a single valence

electron sees the periodic electrostaticpotential due to all N atoms. Its wavefunction (Bloch state) is now a combi-nation of overlapping 3s wave functionsfrom all the atoms and extends over theentire volume occupied by those atoms.As in the molecular case, that wavefunction can be a bonding state or anantibonding state. The original atomicvalence levels generalize to a band ofvery closely spaced energy levels, half



Na Atom

3s Wave Function

Electrostatic Potential

Energy Levels

s-Band Wave Function

Na Metal

Periodic Potential

s Band Formation

3s

Ene

rgy s Band-

width

EF

EF

Density of states

3s

A

B

3s2p 2p

AB

2s1s

2s1s

φ(3s) φA

φB

Na Molecule

A states

B states

s BandstatesCore

states

Core states

Valence states

A = AntibondingB = Bonding

Figure 2. The Formation of an Energy Band in Sodium The figure shows the transformation of

electronic structure when two sodium

atoms are brought into close proximity

and when numerous atoms condense to

form sodium metal. (a) When two sodium

atoms are brought together to form a

diatomic molecule, the atomic wave func-

tion for the 3s valence electron changes

into two molecular wave functions—one

is bonding ( ϕB) and the other antibonding

(ϕA)—corresponding to the sum and the

difference of the 3s atomic wave func-

tions, respectively. The single potential

well of the isolated sodium atom with

its 3s-valence energy level and its core

energy levels becomes a double-welled

potential, with bonding (B) and antibond-

ing (A) molecular energy levels replacing

the valence energy level. Finally,

the energy level diagram shows the 3s

atomic energy level becoming molecular

energy levels A and B, which correspond

to the molecular wave functions ϕA and

ϕB, respectively. (b) When N atoms are

brought together, the 3s radial wave func-

tion becomes a Bloch state made up of a

3s atomic wave function at each atomic

site modulated by a plane wave. The sin-

gle potential well becomes a periodic

potential well with core level states at

atomic sites and the energy levels of the

Bloch (conduction electron) states above

the potential wells. The energy level dia-

gram shows the original 3s level becom-

ing a band of N very closely spaced

energy levels, whose width is approxi-

mately equal to the energy difference

between levels A and B in the diatomic

molecule. That energy difference or band-

width is proportional to the amount of

overlap between atomic wave functions

from neighboring sites. In sodium metal,

the 3s conduction band is only half full,

and the highest occupied state at

T = 0 is denoted by EF, the Fermi

energy. Also shown is the number of

energy levels per unit energy, or the

density of states, for this s-electron

conduction band.

(a)

(b)

of them bonding and half of them anti-bonding, and the width of the energyband is approximately equal to the energy split between the bonding andantibonding energy levels in the diatomic molecule. This broad bandforms whether the crystal is an insula-tor, a metal, or a semiconductor.

Because in a macroscopic samplethe number of levels in the energy bandis large (approximately 1023) and thespacing between those levels is small,we can consider the electron energies tobe a continuous variable. We describethe number of electron energy levelsper unit energy in terms of a density ofstates that varies with energy. Becauseeach electron must have at least aslightly different energy (the Pauli exclusion principle), electrons fill up

the energy levels one by one, in orderof increasing energy.

A Bloch state, or the three-dimensional extended wave function ofa valence electron in a solid, is repre-sented in Figure 2 in one dimension. In this example, the 3s valence electronwave function of sodium appears atevery atomic site along a line of sodiumatoms, but its amplitude is modulatedby the plane wave e

ik∑r. As we men-

tioned before, this general form for aBloch state in a solid emerges from therequirement of translational invariance.That is, the electron wave function in agiven unit cell must obey the Blochcondition uk(r + Tn) = uk(r), where Tnis a set of vectors connecting equivalentpoints of the repeating unit cells of thesolid. It must therefore be of the form

Ψk(r) = eik·r

uk(r), where a plane wavewith wave vector k modulates the atom-ic wave function in a solid. The wavevector k, or the corresponding crystalmomentum p = hk, is the quantum num-ber characterizing that Bloch state, andthe allowed magnitudes and directionsof k reflect the periodic structure of the lattice. Similar Bloch states exist inall crystalline materials, and their occu-pation by valence electrons is whatbinds the atoms into a single crystal.

The electronic structure gets morecomplicated in metals containing morethan one type of valence electron. Forexample, Figure 3 shows that multipleoverlapping bands are created when theconduction electrons in a solid originatefrom, say, the s, p, d, and f valence orbitals of an atom, as in the light



5f6d7s

7p

a0∞

Isolatedatom

Interatomic distance, a

Ene

rgy

Solid

f band

Bandwidth

EF EF

−5

5

0

Density of states

Ene

rgy

band

wid

th (

eV)

d band

s band

Figure 3. Multiband Formation inthe Light Actinides (a) Illustrated here are the multiple

energy bands that form in going from a

single actinide atom to a solid. Multiple

bands always form when an atom has

more than one valence electron. Note

that the s and p bands are much wider

than the d band, which is much wider

than the f band. Also, because the s, p,

d, and f states overlap in energy, they

can hybridize with each other. That is,

any Bloch state with wave vector k(Ψk)

can be a linear combination of states

from the different bands with that same wave vector. Finally, the density-of-states functions show that the narrow f band domin ates

at the Fermi energy because it is so much narrower than the other bands and therefore has many more states at that energy.

(b) The energy bands, one-electron energies as a function of wave vector k, are shown for cerium. The very narrow f bands at

the Fermi energy are shown in red. The spd bands are broad. States with the same symmetry cannot cross the Fermi energy level.

We therefore show by dashed lines how the spd bands would connect if no f bands of like symmetry were present.

6

4

2

0

–2

–4

Ene

rgy

(eV

)

EF

spd

spd

spd

spd

spdf admixture

f bands

L X

Wave vector in high-symmetry directions (k )

KΓ Γ

(a)

(b)

actinides. The width of each band increases from left to right, as the inter-atomic distance decreases and the over-lap of the wave functions increases.Also, the s and p bands are alwayswider (span a wider energy range) thanthe d band, which in turn is alwayswider than the f band. The overlappingbands in Figure 3 imply that the Blochfunctions with a given quantum number(wave vector) k could be linear combi-nations of states originating from the s,d, p, and f atomic orbitals. In otherwords, the Bloch states could be “hy-bridized” states containing many angu-lar-momentum components, in contrastto atomic orbitals that contain only oneangular-momentum component.Figure 3 also shows the density ofstates D(ε) resulting from this multi-band structure. Note that the f statesoutnumber all the others at the Fermienergy EF, which is defined as thehighest energy level occupied by a con-duction electron at the absolute lowestenergy of the metal (T = 0). Later,when we discuss cohesion, we willshow that these f states dominate thebonding of plutonium in the groundstate (or α-phase), primarily because there are five f electrons peratom and only one d electron per atomoccupying the Bloch states and partici-pating in bonding. (There are, ofcourse, two electrons in s and p bandsbut they contribute little to the bond-ing.) For that reason, we refer to thenarrow f band in plutonium as the dom-inant band. Because narrow bands correspond to small overlaps of wavefunctions, these f band electrons maybe easily pushed toward localization by various effects, in which case theydo not contribute to bonding.

If an energy sub-band is filled (elec-trons occupy all its energy levels), the solid is an insulator. If a band isonly partially filled, the solid is a metal.Thus, in a metal there are many emptystates close in energy to the occupiedstates, and so the electrons can easilychange their motion (energy) in response to small temperature and electromagnetic perturbations.

The Free-Electron Model and Departures from It. Beyond contribut-ing to bonding, conduction electronsare also thermally excited. In simplebroad-band metals such as potassiumand copper, the free-electron model de-scribes these low-energy excitations. Inthat model, the electrons occupying theBloch states in the conduction band aretreated as a gas of identical free parti-cles. That is, the periodic electrostaticpotential seen by the conduction elec-trons and the interactions and correla-tions among the electrons have no ex-plicit role. However, the model doesaccount for the conduction electronsobeying the Pauli exclusion principle,and therefore at T = 0, they fill in theconduction band in order of increasingenergy up to the Fermi energy EF. Ifwe draw the energy states in the three-dimensional space defined by the crys-tal momentum hk, as in Figure 3(b),then EF traces a surface in momentumspace (or k-space) known as the Fermisurface. In the free-electron model,each state corresponds to an electronwith crystal momentum p = hk andwith kinetic energy given by the free-particle formula, ε = (hk)

2/2me.

For a gas of free particles heatedfrom absolute zero to a temperature T,classical statistical mechanics wouldpredict that, on the average, the kineticenergy of each particle would increaseby an amount kBT. But because of theexclusion principle, the electrons re-spond differently. Only those conduc-tion electrons occupying states withinkBT of the Fermi level EF can be heat-ed (by phonon scattering) because onlythey can access states not occupied byother electrons (see Figure 4). Thenumber of electrons that participate inproperties such as electrical conductionand electronic heat capacity decreasesto a fraction T/TF of the total number ofconduction electrons in the metal (here,the temperature at the Fermi surface TFis defined by the relation kBTF = EF).At room temperature, T/TF is about1/200 in most metals. Thus, replacingthe classical Maxwell-Boltzmann statis-tics with the Fermi-Dirac quantum sta-

tistics implied by the exclusion princi-ple has a profound impact on the elec-tronic properties of metals.

The factor T/TF shows up explicitlyin the low-temperature specific heat ofa metal. In general, the specific heat isthe sum of a lattice vibrational term(proportional to T3), which is due to thethermal excitation of the ions, and anelectronic term γT, which is due to thethermal excitation of the electrons. Theclassical coefficient of the electronicterm is γ = NkB, but because of the ex-clusion principle, it becomesγ =NkBT/TF, and only electrons near theFermi energy can be heated. Thus, insimple metals obeying the free-electronmodel, γ is inversely proportional to TF,or equivalently,EF, and therefore proportional to the rest mass of the free electron, me. Later, when we discuss the low-energy excitations incorrelated-electron materials includingplutonium, we show that the conductionelectrons depart from free-particle behavior. They behave more like thestrongly interacting particles of a liquid,more like a Fermi liquid. Because theinteractions slow down the electrons,the effective mass of the electrons appears larger, and it shows up as anincrease in the value of γ over that predicted by the free-electron model.Thus, low-temperature specific-heatmeasurements reveal the strength of the electron-electron correlations in ametal and therefore provide a majortool for identifying unusual metals.

Electrical resistivity at low tempera-tures tells us about the quality of themetal. In a perfect crystal, electrical resistance would be zero at the classicalT = 0 because the noninteracting con-duction electrons, acting as waves,would move through the perfect latticeunimpeded. Above T = 0, the thermalexcitations of lattice vibrations(phonons) make the lattice imperfectand scatter the electrons. The electricalresistance increases linearly with tem-perature, as will be shown later in thisarticle. In general, anything that destroys the perfect translational invari-ance of the crystal lattice will scatter



electrons. Foreign atoms, lattice vacan-cies, more-complicated defects such asstacking faults, and finally, magneticmoments in an array without the fullsymmetry of the lattice can scatter elec-trons. Many of these imperfections aretemperature independent and lead to afinite limiting resistance as T = 0 is approached. Hence, this limit is used as a measure of the quality of metal samples, for which the lowest residualresistance signifies the most perfectsample. We will show that correlated-electron materials often have anomalously high resistivities and very small or zero magnetic momentsat low temperatures.

Models of Conduction Electrons

We have suggested that the electro-static forces holding the metal togethercan be considered averaged forces between the ions and conduction elec-trons and that these forces can bemodeled by a periodic electrostatic

potential. On the other hand, once ametal is formed, its conduction elec-trons (approximately 1023 per cubiccentimeter) can act collectively or in acorrelated manner, giving rise to whatis called quasiparticle or free-electronbehavior (not determined by averagedelectrostatic forces) and to collectivephenomena such as superconductivityand magnetism.

These two seemingly opposingviews of conduction electrons and theirbehavior in solids first appeared in thescientific literature in 1937 and 1957.John Slater (1937) proposed calculatingthe electronic states—the energy bandsin Figures 2 and 3(b)—of solids by thesame self-consistent method that hadbeen applied so successfully to describ-ing the electronic states of atoms andmolecules. In this method, one treatselectrons as independent particles andcalculates the average Coulomb forceson a single electron. The other electronsand all the ions in the solid are thesource of these Coulomb forces on oneelectron. This calculation, repeated forall the electrons in the unit cell, leads to

a charge distribution from which theelectrostatic potential seen by the elec-trons can be obtained as a solution ofPoisson’s equation. Using the new elec-trostatic potential, one then repeats thecalculations for each electron until the charge density (distribution of elec-trons) and the crystal potential (forceson the electrons) have converged toself-consistent values. Slater’s approachled to all the modern electronic-band-structure calculations commonly labeledone-electron methods. These one-elec-tron band-structure methods are adapta-tions of the familiar Hartree-Fock meth-ods that work so well for atoms andmolecules. They were put on a morerigorous footing through Walter Kohn’sdevelopment of density functional theo-ry (DFT). For his achievement, Kohnbecame one of the recipients of the1998 Nobel Prize in Chemistry.

Lev Landau (1957) took a differentview and argued that the collective mo-tion of electrons in a solid’s conductionband was very different from the motionof electrons in atomic or molecular orbitals. He pointed out that particles inthe conduction band act as if they werenearly free even though the individualelectrons are subject to strong Coulombforces. Landau’s way out of this para-dox was to argue that the effect of theelectrons’ correlated motions from mutual interactions in the solid was to“clothe” themselves, which screens theircharge. In heuristic terms, a conductionelectron is very much like an onion withmany layers. When the electromagneticforce is weak, the interaction penetratesonly a few layers, and the electron appears to be clothed. When the forcebecomes stronger, however (as in theejection of conduction electrons by pho-tons in photoemission experiments), theinteraction penetrates many more layersuntil the bare electron with its Coulombforce becomes visible, as it does in the one-electron models.

And yet, the conserved quantumnumbers characterizing the single-particle states of the clothed elec-trons—such as spin, momentum, and charge—are unchanged by the



kBT

B

A

B Bonding StatesCrystal structureCompressibilityItinerant magnetism

Excited StatesSpecific heat

Conduction

A

Den

sity

of s

tate

s

Ene

rgy

EF

EF

Energy

(a) (b)

Density of states

Figure 4. Density of States for a Free-Electron Gas(a) The solid line is the density of single-particle states for a free-electron gas plotted

as a function of one-electron energy ε. At T = 0, electrons occupy all the states up to

the Fermi energy EF. The dashed curve shows the density of filled states at a finite

temperature T, where kBT, the average thermal energy per particle, is much less than

the Fermi energy. Only electrons within kBT of the Fermi level can be thermally excited

from states below the Fermi energy (region B) to states above that level (region A).

(b) This plot of the density of states emphasizes that all electrons in the conduction

band participate in bonding, whereas only those with energies near the Fermi energy

contribute to low-energy excitations, such as specific heat, and conduct electricity.

orbital A or B, and so can electron 2. Second, the electrons

have to obey the Pauli exclusion principle, which means that

the total wave function for the two electrons has to be

antisymmetric, and that antisymmetry implies that the Hamil-

tonian must contain an exchange term. This exchange term

is what separates the Hartree-Fock calculations of many-

electron atoms from the original Hartree calculations of

those atoms.

When the exchange term was included in the calculation of

an electron gas, it was found that around each electron, there

is a “hole,” or depression, in the probability of finding another

electron close by. The accompanying figure shows the

“exchange” hole, which is the probability of finding an electron

of the same spin near a given electron. That probability is

one-half the value it would have without the exchange term.

This exchange hole demonstrates that the electron motion is

correlated, in the sense that electrons with the same spin

cannot get close to each other. In the 1930s, Wigner

performed similar calculations for electrons of opposite spins,

which led to a “correlation” hole (very

similar to the exchange hole) for the

probability of finding an electron of

opposite spin near a given electron.

The picture of an exchange hole and

a correlation hole around each

electron is a great visual image of

electron correlations in solids. Modern

one-electron calculations include

these correlations in an average way

because these terms can be calculated from the average

electron density around a given electron.

Having given a physical basis for the need to include elec-

tron correlations, we now refer to the modern usage of

the phrase. Any theory that includes interactions beyond the

one-electron method is now considered a correlated-electron

theory. Likewise, any solid (metal, insulator, and so on) that

exhibits behavior not explained by either the free-electron

model or the one-electron band model is considered a

correlated-electron system. If the properties of a solid devi-

ate strongly from the predictions of free-electron or band

models, that solid is called a strongly correlated system.

Figure 1 in the main text shows examples of such systems.



Electron correlations are always mentioned in discussions of

electronic structure or excitations, but their physical origin is not

always explained. Here, we will give a simple argument for the

need to include electron correlations. We will also give some

examples of such correlations.

Assume that we have a container of electrons (no ions) that

are noninteracting and that we can remove the container and

turn on the interactions. At that point, the electron cloud will

expand indefinitely because of the Coulomb repulsion between

the electrons. No correlations are needed to describe this mo-

tion of free electrons. Note that the term free-electron behavior

as applied to conduction electrons really means that the elec-

trons act like neutral particles (no charge) that obey Fermi-

Dirac statistics—that is, they act like Landau’s quasiparticles.

Now suppose that we have a container with an equal number

of ions and electrons and that the ions are on closely spaced

lattice sites, as in a real solid. Again, we remove the container

and turn on the interactions. We assume that our electrons

are in random positions and their de

Broglie wavelengths are greater than

the ion spacings when the interactions

are turned on. In this case, the elec-

trons’ motion is much more complex

because the electrons are simultane-

ously attracted to the ions and repelled

from each other. To minimize the total

energy of the system, the electrons

must minimize the electron-electron

repulsion while maximizing the electron-ion attraction, and

the way to minimize the Coulomb repulsion is for them to stay

as far from each other as possible. Below, we will demon-

strate how electrons are kept apart in real calculations,

and the reader will thus get a feel for what we mean by

electron correlations.

We will first consider the helium atom, which has two occupied

atomic orbitals—one for each electron. Helium was the first

many-electron system for which a quantum mechanical calcu-

lation was attempted. In developing a model that would predict

the observed spectral lines of helium, physicists discovered

two unanticipated properties of the electrons. First, the two

electrons are indistinguishable—that is, electron 1 can be in

1

0 1 2 3Normalized electron spacing (x)

Pro

babi

lity

dens

ity

4 5 6

0.5

Electron Correlations vs Correlated-Electron Materials

interactions. This argument led Landauto propose a one-to-one correspondencebetween the interacting and noninteract-ing systems. That is, the number of en-ergy levels in the interacting system(labeled by the conserved quantumnumbers of the noninteracting system)and the number of elementary excita-tions (or clothed particles) in the inter-acting system would be the same asthose in the noninteracting system. Theclothed electrons, called quasiparticles,with their physical properties modified,would then interact very weakly andthus have almost-free-particle behavior.

Landau’s original argument was meant to explain the nearly-free-electron behavior seen in the con-duction electrons in simple metals, inthe atoms in liquid helium-3, and in theprotons and neutrons in nuclear matter.Later, scientists realized that the sameconceptual framework could be appliedto small-energy electronic excitations ofsolids in which the conduction electronsdo not exhibit nearly-free-electron behavior but behave more like a liquid.For this reason, the Landau method canbe considered a correlated-electronmethod. Landau’s genius was to recog-nize that these correlations did not haveto be calculated directly. Instead, onecould assume that the correlations werebuilt into the behavior of the quasiparti-cles, which could therefore no longer

be identified as electrons. The small de-viations of those quasiparticles fromfree-particle behavior were the sign ofresidual interactions, and the effects ofthose interactions on the quasiparticlebehavior could be determined directlyfrom experimental data. The downsideof this theory is that the measurable pa-rameters are not easily derived fromfirst-principles calculations, and so theirmeaning is not always clear.

Although the one-electron band theories and the correlated-electron theories seem incompatible, we havelearned that both are correct and thattheir relevance depends on the energyand time scale used to view the conduc-tion electrons. We also know that alltheories of conduction electrons mustinclude clothing, or correlations. Theone-electron methods include averagedcorrelations, which produce an aver-aged exchange hole and correlationhole around each electron (these termsare defined in the box “Standard Elec-tron Correlations vs Correlated-ElectronMaterials”). For higher energies andlonger times, such as those involved inbonding, these one-electron methodswork well because the averaged corre-lations dominate the behavior of theelectrons. For low-energy excitations,spin and charge fluctuations becomemore important than averaged values,and the Landau method, which incorpo-

rates these fluctuations in the behaviorof the quasiparticles, becomes moreuseful.

Models for Correlated-ElectronMaterials. Until 1980, we had thesetwo general methods for calculatingphenomena involving higher and lowerenergies, and they worked quite well—except for some isolated systems. But,in the early 1980s, we began findingnew metallic materials, the heavy-fermion and mixed-valence materials,whose behavior was as anomalous asthat of the light actinides. Resistivities,specific heats, magnetic susceptibilities,electron masses, and other low-energyexcitations were different in differentmaterials of the same class and hadstrange temperature dependencies.Some of these anomalies had been seenbefore, but they seemed isolated. Almost every new material appeared to exhibit odd behaviors that could notbe explained by one-electron models or the Landau treatment.

We now recognize that the centralfeature of all these materials, includingplutonium, is the presence of a domi-nant, narrow conduction band. We alsorecognize that the exotic behavior ofthese materials has a common source:the spin and charge fluctuations associ-ated with the low-energy excitations inthose narrow bands. As indicated

Table I. Solid-State Excitations

Excitation Excitation Energy Typical Theoretical Method

Photoemission 20–1500 eV One-electron/many-electron

Band formation 2–10 eV One-electron

Cohesion 2–10 eV One-electron

Phase stability 1–25 meV One-electron

Elastic constants 1–25 meV One-electron

Magnetic moments (band) 1–25 meV One-electron

Magnetic moments (local) 1–25 meV One-electron

dHvA signals 0.1 meV One-electron/many-electron

Resistivity 0.1 meV One-electron/many-electron

Specific heat (at low temperature) 0.1 meV One-electron/many-electron

Magnetic ordering 0.1 meV Many-electron

Superconductivity 0.01–0.1 meV Many-electron



before, narrow bands mean that thewave functions from different atoms arebarely overlapping, and the electronsare thus bordering on being localized.These materials are now collectivelycalled correlated-electron materials, andthose with the most extreme behaviorare the heavy fermions. (In the late1980s, another class of materials wasdiscovered, namely, high-temperaturesuperconductors. Although they arecomplicated and interesting, high-temperature superconductors are notnarrow-band materials. Their exotic behavior is therefore unrelated to thatof plutonium.)

Along with the discovery of corre-lated-electron materials came a newclass of many-electron models to describe the exotic behaviors of thosematerials. The Kondo, Hubbard, andAnderson models are among them.These models stand between the one-electron methods and the Landaumethod in the sense that they can beused to add electron-electron interac-tions (correlations) to either the semi-clothed electrons of one-electron theory or to the quasiparticles of theLandau theory. Originally, the Kondoand Anderson models were invented tosolve specific mysteries in materialscontaining impurities, and they can begenerically classed as two-electron“impurity” models. That is, they intro-duce interactions between pairs ofelectrons, one localized on an impurityatom and one in a conduction band.

Unfortunately, these impurity mod-els break the translational invariance ofthe crystal lattice. To become applica-ble to correlated-electron systems, thesemodels must allow translational invari-ance to be restored. In the standard approach, the localized (or “almost localized”) f electrons become the im-purity, and one postulates a lattice ofcouplings between conduction electronsand the f electrons, which are either located at every lattice site or distrib-uted randomly among the sites. In prin-ciple, translational symmetry makesthese extended impurity models soluble,but in practice, the models present

enormous calculational difficulties, andprogress in solving them has been slow.

Energy Scales of Electronic Phenomena. Table I lists solid-stateexcitations or effects, along with theenergy scale and theory most applicablefor each effect. Effects for relativelyhigh energies—2 to 10 electron-volts(eV)—include the ground-state crystalstructures, as well as the particularbonding energies and stability of thosestructures. Those properties are mod-eled with one-electron methods as arethe elastic constants, which are amongthe effects for moderate energies—1 to25 milli-electron-volts (meV). Low-energy (0.1 meV) excitations and phe-nomena such as resistivity, magnetic ordering, and specific heat can be described by either one-electron ormany-electron models, whereas the verylow energy (0.1 to 0.01 meV), collectiveelectronic ground states such as super-conductivity and magnetism can only be

described by many-electron models. Not all phenomena in solids,

however, can be fully treated by a singleapproach. For example, the de Haas–van Alphen (dHvA) effect1, as seen inheavy-fermion materials, requires both.The oscillations can be explained byone-electron theories, whereas the veryheavy electron masses are explained onlyby correlated-electron theories. Anotherexample is provided by the spectra obtained from photoemission experi-ments, which are performed at relativelyhigh energies but for short times. Pho-toemission spectra measure the one-electron density of states predicted fromone-electron methods, but because of theshort time, spin and charge fluctuationsmay add small features to the overallspectra (see the article “PhotoelectronSpectroscopy of α- and δ-Plutonium” on



Figure 5. The Metallic Radii of the Actinide Elements in the Ground State The metallic radius is half the average distance between the atoms in the solid. The

line follows the smoothly varying metallic radii of the simple trivalent actinide metals,

whose f electrons are localized and therefore nonbonding. The metallic radii of the

light actinides—thorium through plutonium—fall on a parabolic curve below the triva-

lent line, showing the contribution of the f electrons in the bonding, that is, in pulling

the atoms closer together.

1.9

1.8

1.7

1.6

1.5

Ac Pa NpElement

Met

allic

rad

ius

(Å)

Am Cm Bk CfTh U Pu

1The dHvA effect is the oscillation in magneticsusceptibility at low temperatures, as an appliedmagnetic field changes.

). Clearly, we need both theo-retical techniques to explain the proper-ties of metallic plutonium and its com-pounds.

To convey an overall picture of theactinide metals, we first discuss theirbonding properties, for which we useone-electron methods. Later in the arti-cle, when we look at phase instabilitiesand other lower-energy phenomena, wewill need arguments and concepts fromcorrelated-electron theories.

Cohesion in the Light Actinides—Similarity with

the Transition Metals

Our discussion of the bonding prop-erties in the actinides focuses on therole of the 5f valence electrons. The 5felectrons in the light actinides (thoriumthrough plutonium) are itinerant, justlike the 5d electrons in the transitionmetals, participating in the bonding ofthe solid and affecting most of thehigh-energy properties such as cohe-sion, crystal structure, and elastic prop-erties. In contrast, the heavy-actinidemetals (americium through lawrencium)have localized (atomic-like as opposedto itinerant) 5f electrons, and they arethus the true counterpart to the rare-earth metals. (The presence of localizedmagnetic moments in both the lan-thanides and the heavy actinides is a direct sign that f electrons are localized.)

Figure 5 is a plot of the metallicradii of the actinides, and it gives explicit evidence for the bonding behavior of the 5f electron series in the ground state. Notice that the lightactinides have smaller metallic radii(therefore, higher densities) than theheavy actinides. The reason is that the f electrons in the light actinidescontribute to the bonding. Also, the metallic radius of the light actinidesgets smaller with increasing atomicnumber (Z) because each additionalf electron per atom pulls the atomscloser together. This trend stops atamericium. The metallic radii of theheavy actinides are all about the same,

and that value is larger than that for the radii of the light actinides becausethe localized 5f electrons have no effecton bonding.

Ironically, before the ManhattanProject, when thorium and uraniumwere the only actinides whose physicalproperties were known, these metalswere thought to belong to a 6d metalseries to be placed below hafnium andtungsten in the periodic table of the ele-

ments. The similarities between thoriumand uranium and the 5d transition met-als were the cause for this misconcep-tion. After McMillan and Abelson discovered neptunium in 1940 and afterthe discoveries of plutonium, americium,and curium during the Manhattan Pro-ject, measurements of atomic spectrashowed that the valence electrons inthis ever-lengthening series were fillinga 5f shell and not a 6d shell of atomic



0

–1

–2

–3

–4

–5

–6

–7

–8

–91

Th2

Pa3U

4Np

5Pu

Number of d(p) or f electrons

Coh

esiv

e en

ergi

es (

eV)

6Am

7 8 9 10

Actinides

3d

4d

5d

Figure 6. One-Electron and Friedel Model Results for the Cohesive Energies of the d and f SeriesThe calculated cohesive energies per atom of the 3d metals (dots) are equal to the cal-

culated difference between the binding energy of an isolated atom (excluding atomic

valence-electron coupling) and the binding energy of an atom in its solid. For peda-

gogical reasons, in these LDA one-electron calculations, the metals were assumed to

be in the fcc crystal structure. (The cohesion of the other structures differs by up to

10% from that of the fcc phase.) The Friedel model predicts a parabolic curve for the

3d metals. This simplified model of bonding assumes that the density of states in the

d band of the transition metals is constant over its width. The model also assumes

that the d electrons fill the energy band in order of increasing energy: First the bond-

ing d states are filled, which increase the binding, and then the antibonding d states,

which decrease the binding. Thus, maximum stability is reached when the band is half

full. (Refer to the article “Actinide Ground-State Properties” on page 128 for a more

detailed discussion of this model.) The LDA one-electron results agree with the predic-

tions of the Friedel model. We show only the parabolic curves for the 3d, 4d, and 5d

elements and for the light actinides. The cohesive energies of the light actinides look

similar to those of the transition metals—that is, they show increased bonding as the

f shell is being filled.

orbitals. These 5f electrons were expected to be localized in the solidstate, like the 4f electrons of the rareearths. Long after the Manhattan Project, the pendulum swung back, asscientists realized that the similaritiesbetween the light actinides and the 5dtransition-metal elements meant that the5f electrons in those early actinides didindeed form a conduction band.

The cohesive energy per atom hold-ing a crystal together is defined as thedifference between the electrostaticbinding energy per isolated atom (ignor-ing coupling among electrons) and thetotal internal (electrostatic binding) energy per atom in a crystal. Both bind-ing energies, and thus the cohesive energy, are calculated self-consistentlyby one-electron methods. Figure 6shows the one-electron predictions forthe cohesive energy per atom of 3dtransition metals in a hypothetical face-centered-cubic (fcc) structure (dots) andthe Friedel model predictions for the 3d,4d, 5d, and 5f elements (paraboliccurves). The Friedel Model, a simplifiedmodel of bonding, assumes that thed electrons are conducting and are fill-ing an energy band in order of increas-ing energy: first, the bonding d states,which increase the amount of binding,and then the antibonding d states, which decrease that amount. Note thatthe Friedel model predictions for the 3d elements fit the results from quan-tum mechanical one-electron calcula-tions quite well. And the bonding in the 4d and 5d metals is greater than thatin the 3d metals because 4d and 5delectronic states have a greater radialextent (more overlap) than 3d states (a feature not obtained from the simpleFriedel model). The cohesive energiesof the light actinides look similar. That is, they show increased bondingwith the filling of the beginning of the 5f shell. However, the parabolictrend in the early actinides ends atamericium because the 5f electrons areno longer contributing to bonding.

Similarly, the lattice constants(length of the edge of the basic cubesin a cubic crystal structure) tend to

decrease as one goes across the firsthalf of all the d and the 5f series, again indicating that each additional electronincreases the bonding or cohesion. Recall that metallic radii also decreaseparabolically across the early part ofthe 5f series, as revealed in Figure 5.

Plutonium and the UniversalBonding Curve of Metals

The parabolic trend in cohesion andlattice constant extends to the bulkmodulus, which is the average elastic(or spring) constant of the solid. Thebulk modulus thus gives the strength of

the restoring forces, or interatomicforces, that bind the solid together asthe ion cores vibrate about their equi-librium positions. For most metals, thebulk modulus, the cohesive energy, andthe lattice constant can be related toeach other through the “universal bond-ing curve.” We will show next that plu-tonium fits this curve as well.

First developed as a parameterizedequation, the bonding curve is a plot ofpressure (cohesive force per unit area)on an atom vs the distance between theatoms. Figure 7 shows the bondingcurve for potassium metal calculatedfrom one-electron band-structure calcu-lations in the atomic-sphere approxima-



Pre

ssur

e ×

volu

me

(eV

)

0Logarithm of interatomic spacing (x)

0.80.4

0

–4

4

–3

3

–2

2

–1

1

a0 = Equilibrium lattice constant

Ecohesive = ( )dxdεdx

∞

0

Figure 7. The Bonding Curve for PotassiumThe plot shows LDA one-electron calculations of the bonding curve for potassium in

the fcc structure, which is the change in cohesive energy per atom, dε/dx , as isolated

potassium atoms at infinite separation are brought together to form a solid. We plot

this quantity as PΩ (pressure times volume or force times length) vs the separation

between atoms (or lattice constant a). The shaded area under the bonding curve is

the cohesive energy to bond the atom in the solid. To compress the bonding curve,

the horizontal axis is a logarithmic scale x = ln(a/a0). Equilibrium occurs when

dε/dx = P = F/A = 0, at which point a = a0 and x = 0 (the dashed line in the figure).

The bulk modulus (average elastic constant) is given by the slope of the curve at

equilibrium (that is, at x = 0 and dε/dx = 0). At the minimum in the curve, the attractive

forces between the atoms are the greatest. Those forces weaken as the atoms move

together until they vanish at the equilibrium separation.

tion (the electrostatic potential aroundeach lattice site is assumed to be spher-ical). Here, the bonding forces are foratoms arranged in an fcc structure, andthey are plotted as a function of x, thenatural log of the lattice spacing a nor-malized by a0. At very large interatom-ic distances, the atoms are isolated, andthe pressure on any atom is zero. Asthe atoms approach each other, theirelectronic wave functions overlap, andthere is an attraction, or pressure, thatpulls them together. Finally, they reachequilibrium at the lattice constant a0,where again the pressure on the atomsis zero. In Figure 7, the cohesive ener-gy is the area between the bondingcurve and the line indicating zero pres-sure; the equilibrium lattice spacing isat x = 0 (or a = a0), the value at whichthe bonding curve crosses the zeropressure line; and the bulk modulus isthe slope of the bonding curve at thisequilibrium spacing. Remember thatthis curve represents the forces on anatom in the solid. Therefore, the mini-mum in the curve is the lattice constantat which the attractive force betweenthe atoms is the greatest. As the atomsmove closer together, the attractiveforce becomes weaker and vanishes atthe equilibrium lattice constant a0.

We will next illustrate that there isonly one bonding curve for all metals,including plutonium. We first calculatethe bonding curve for molybdenum because, among metals, it has thelargest cohesion. We then plot all othermetals on this curve, by drawing thezero pressure line for each metal so thatthe area between that line and the bond-ing curve is equal to its cohesive ener-gy. The results, shown in Figure 8 forseveral metals, reveal the following relationships: First, if the cohesion ofone metal is greater than that of another,then its lattice constant is smaller. Next,the bonding curve becomes steeper asthe interatomic distances get smaller(see the part of the curve to the left ofthe maximum force point). This meansthat the tangent to the curve (bulk mod-ulus) becomes larger as one moves upthe curve. So, if the cohesion of one

metal is larger than that of another, itsbulk modulus will be larger. This uni-versal bonding applies to all metals inthe periodic table. When consideringcohesion and bulk modulus, we can seethat plutonium and the other light actinides fit the universal curve just like any other metal. We note that plu-tonium and the light actinides have relatively small bulk moduli becausethey have large lattice parameters,which, as we shall see, are due to theelectrons in the s and p bands and areconsistent with the universal curve.

More exact calculations of bondingin plutonium are now available and arepresented in the article “Actinide

Ground-State Properties” (page 128),but a few general remarks are in order.Those modern calculations, as well asthe ones we presented above, are basedupon the local density approximation(LDA) of the DFT approach to band-structure calculations. But in contrast toour simplified, pedagogical approach,Wills and Eriksson have performedfull-potential calculations (that is, noconstraints are put on the form of thecrystal potential), so that all types ofcrystal structures can be handled. Theirmodern calculations are fully relativistic(as were our pedagogical calculations),which is necessary for crystals withhigh Z-number atoms. For example, a



10

0

Pre

ssur

e ×

volu

me

(eV

)

0 0.2 0.4

Logarithm of interatomic spacing

0.6 0.8 1

−10

−20

P = 0

P = 0P = 0

P = 0

Mo

Pu

Universal curve

Cu

K

Ecohesive

a0 a0

a0

a0

a0 = Equilibrium lattice constant

Figure 8. A Universal Bonding Curve LDA one-electron calculations of the bonding curves for potassium, copper, plutoni-

um, and molybdenum in fcc structures are overlaid on the same curve. The horizontal

axis for each metal is placed at zero pressure for that metal, and the area between

that axis and the curve gives the correct cohesive energy per atom. (The dotted curve

for copper shows that, in this construction, we are ignoring the large distance tails of

the bonding curve for each element.) A number of relationships are apparent in this

figure. As the cohesive energy increases from one metal to the next, the P = 0 axis

moves up, and the equilibrium lattice constant a0 decreases (moves to the left). At the

same time that a0 moves to the left, the force curve at a0 becomes steeper, and the

bulk modulus, which is the tangent to the curve at a0, increases. So, if the cohesion of

one metal is larger than that of another, its lattice constant will be smaller and its bulk

modulus will be larger.

nonrelativistic calculation of the pluto-nium atom gives a wrong ordering ofthe s, p, d, and f valence levels.

One of the expected connections between electronic-structure calcula-tions based on one-electron methodsand experiment has been provided bythe values for the energy levels of theone-electron states, which can be mea-sured by photoemission experiments. Inthe atomic case, for example, it has beenproved that the energy eigenvalues ofthe one-electron Hartree-Fock equations(see the box on page 98) are equal to theelectron-removal, or ionization, energies(Koopman’s theorem), which are directly measurable quantities. Unfortu-nately, when those atomic eigenvaluesare calculated with the LDA, they do nothave the same physical meaning. Instead, they are equal to the removalenergies plus a small calculable correc-tion, and this correction has not beenshown to vanish for atomic energy lev-els nor has it been calculated for theone-electron Bloch (periodic) states of a solid. Therefore, we have to rely on acomparison with experimental data todetermine how close the LDA-calculatedenergies are to the electron removal energies measured from photoemissionexperiments. Any disagreements do notmean that the theory is wrong.

Finally, we remind the reader that all one-electron methods automaticallypredict the hybridization between the s, p, d, and f states when those statesoverlap. The LDA results we reportedhere agree with experimental data at the 90 percent level, whereas full-potential calculations, such as those inthe article “Actinide Ground-StateProperties” on page 128, are often inbetter agreement. However, it is notonly the calculated values that are important but also the deeper under-standing that LDA calculations enable.Because these calculations are efficient,we can repeat them for many variationsof the parameters, thereby uncoveringdetailed bonding features that determinethe structures and bulk properties of all the metals in the periodic table.

Low-Symmetry Structuresfrom Narrow Bands

One of the striking anomalies ofplutonium metal is its very low symme-try crystal structure. In fact, the lightactinides exhibit the lowest-symmetryground states of all elemental metals:from orthorhombic for α-uranium tomonoclinic for α-plutonium. Figure 9

shows the room-temperature monocliniccrystal structure of α-plutonium and itsdeparture from a hexagonal structure.

In contrast, the transition metals, despite their relatively complex behavior,form high-symmetry cubic ground statessuch as bcc (body-centered cubic), fcc,or hcp (a hexagonal close-packed varia-tion of fcc). These high-symmetry struc-tures look like spheres stacked in an



2

Figure 9. The Crystal Structure of α-PlutoniumThe α-phase, the equilibrium phase of pure plutonium at room temperature and below,

has a monoclinic crystal structure. The parallelogram outlining the 16-atom unit cell

shows 2 flat layers (or planes) of atoms, and 8 distinct atomic sites. The larger num-

bers label the distinct sites in the top plane of atoms, and the smaller numbers label

the equivalent sites in the next layer down into the crystal. The lines between the

atoms show that the layers are somewhat similar to a hexagonal structure. For atoms

numbered 2 through 7, one side of each atom has only short lengths to its nearest

neighbors (2.57–2.78 Å), and the other side has only longer lengths (3.19–3.71 Å). This

pattern could be viewed as a strange packing of individual “half dimers,” which is what

a three-dimensional Peierls distortion might be.

efficient space-filling manner with no directional bonding (increased electrondensity) between the atoms. Once metal-lurgists know the stacking pattern (orcrystal structure), they can predict manyof the metallurgical properties. To illus-trate the connection between electronic

structure and crystal structure, we showhow LDA electronic-structure calcula-tions correctly predict the sequence ofground-state crystal structures for all the transition metals.

The LDA calculations predict thatthe one-electron density-of-states func-

tions for the fcc and hcp transition-metal structures are very similar. Forthis discussion, we will treat those crys-tal structures as equivalent and comparethe one-electron energy contributions ofthe bcc structures with those of the fcc(or hcp) structures. The one-electron



Figure 10. Stability of bcc, fcc, andhcp 4d Transition Metal StructuresDetermined from the Density ofStates (Top) The LDA fcc (hcp) and bcc elec-

tronic density of states for the d conduc-

tion band of the 4d transition metals are

plotted as a function of energy and dis-

play their unique signatures. For peda-

gogical purposes, we ignore the relatively

small differences between the fcc and

hcp density of states. The d-band density

of states is the same for all 4d elements,

but the Fermi level, or highest occupied

level, for each element (dashed vertical

lines shown for Zr and Mo) increases

with the number of 4d electrons. (Bottom)

The total one-electron energy contribu-

tions for each element in both the fcc

(hcp) and bcc structures were calculated

from the first moment, ∫ εD(ε)dε for that

crystal structure, where each ε is a one-

electron energy eigenvalue from LDA cal-

culations. The fcc (hcp) results are the

reference line, the bcc results are plotted

relative to that reference, and the lower

value of the two is the prediction of the

stable structure for that element. Note

that the predicted sequence of structures

matches the observed sequence (listed

below each element) if we ignore the dif-

ference between fcc and hcp. The lines

through the top and bottom figures mark

the level of band filling ( EF) at which the

crystal stability changes from fcc (hcp) to

bcc or vice versa. In Region I, the number

of 4d electrons per atom contributing to

the band increases from 1 to 3. For each

element in Region I, the fcc centroid

(average value of ε) for the filled states is

always lower in energy (farther away from

the highest filled level at EF ) than the bcc centroid. Therefore, for those elements, there is more bonding in the fcc (hcp) phase.

In Region II, the situation is reversed because the EF for these elements goes above the first bcc density-of-states peak, and there-

fore the bcc centroid goes below the fcc (hcp) centroid. Finally, for the elements in Region III, EF reaches the second bcc peak,

the bcc centroid shifts closer to EF than the fcc centroid, and so the fcc (hcp) phases become more stable again. Thus, the d band

filling and the unique signatures of the electronic density of states determine the crystal structures of the 4d transition meta ls.

1Srfcc

0.5

0

−0.5

−1.0

−1.5

−2.02Y

hcp

3Zr

hcp

4Nbbcc

5Mobcc

6Tc

hcp

Valence electrons

Tota

l one

-ele

ctro

n en

ergy

Den

sity

of s

tate

s (a

rbitr

ary

units

)

7Ruhcp

8Rhfcc

8

6

4

2

9Pdfcc

10Agfcc

bcc

fcc

fccreference line

bcc curve

Region Ifcc/hcp

Region IIIfcc/hcp

Region IIbcc

energy contribution for a given elementis equal to the one-electron energy εaveraged over the portion of the one-electron density-of-states function D(ε)that is filled for that particular element(∫εD(ε)dε). The upper half of Figure 10shows the approximate forms of D(ε)for the 4d transition-metal series in thebcc and fcc (or hcp) crystal structures.The lower half of that figure presentsthe one-electron energy contribution forthe 4d series, with the fcc (hcp) resultsplotted as the horizontal reference lineand the bcc results plotted relative tothat line. The structure that gives thelower-energy value for each element isthe LDA prediction for that element’sground-state structure. In all cases, theLDA prediction agrees with the ob-served structure. The same methodsyield similar agreement between pre-dicted and observed crystal structuresfor the 3d and 5d metals.

We remind the reader that the totalinternal energy is equal to this one-elec-tron contribution plus other terms (dou-ble counting, exchange, and correlationterms). Our results suggest, however,that the one-electron contribution is thedominant factor in determining the crys-tal structure. The physical reason for itsdominance is that the states at the bot-tom of the conduction band are the mostbonding, so the crystal structure whoseaverage one-electron energy is lowest,or closest to the bottom of the band,should be the most stable.

Figure 10 shows that one-electrontheory predicts the right crystal struc-tures for the transition metals, and italso illustrates why crystal structures inthe 4d series occur in the sequences ob-served. In principle, we could performthe same calculations for the low-sym-metry structures of the actinide series,but the density-of-states functions for

those elements are more complicated,so those calculations would not be aseasy to interpret.

Because electronic structure deter-mines crystal structure, we will considerhow the electronic structure of the lightactinides, of plutonium in particular, differs from that of the transition metals.An obvious difference is the angularcharacter and symmetry of the orbitalsassociated with their dominant electronbands—that is, d orbitals with evensymmetry for the transition metals vs f orbitals with odd symmetry for thelight actinides. Perhaps the odd symme-try of the f orbitals causes the groundstates to have low symmetry, as do p-bonded metals such as indium, tin, antimony, and tellurium. In these metals,the odd-symmetry p orbitals seem toproduce directional covalent-like bondsand low-symmetry noncubic structures.With no way to check this conjecture



4 8Bandwidth (eV)

12 164 8Bandwidth (eV)

2 6

1.0

0.5

-0.5

0

0.2

0.1

-0.1

0

Ene

rgy

(eV

)

Ene

rgy

(eV

)

Nb U

Weq

Weq

bct – bcc

ort – bcc

bct – bccort – bccfcc – bcc

Figure 11. Internal Ground-State Energies as a Function of Bandwidth DFT results for bonding energy vs bandwidth demonstrate that narrow bands produce low-symmetry ground states and wide bands

produce high-symmetry ground states. Results are shown here for niobium (a d-bonded metal), and uranium (an f-bonded metal).

The atomic number, crystal structure, and lattice constant (or volume) are inputs to these calculations, and the bandwidth and ener-

gy are outputs. The input structures are bcc, bct, and orthorhombic, and in the case of uranium, fcc. The results for the bcc s truc-

ture are plotted as the horizontal reference line, and those for the other structures are plotted relative to the bcc results.

Thus, a structure is more stable than bcc when its bonding energy appears below the reference line. Note that for niobium, the low-

symmetry orthorhombic structure becomes stable when we force the d band to be narrower than 3 eV by decreasing the input

volume. Likewise, when we force the f band in uranium to be broader than 7 eV by increasing the input volume, the high-symmetry

bcc structure becomes stable. The calculated bandwidth at the experimental equilibrium volume is labeled Weq. The results in

the figure suggest that transition metals have broad bandwidths and symmetric structures, whereas the light actinides have narro w

bandwidths and low-symmetry structures. (This figure was reproduced courtesy of Nature .)

for plutonium, metallurgists and physi-cists alike long held on to the notionthat directional bonding plays the domi-nant role in explaining the low symme-try of the light actinides.

Recent LDA electronic-structure cal-culations of ground-state propertieshave demonstrated that the narrowwidth (approximately 2 to 4 eV) of thef bands leads to the low-symmetryground states of the actinides. The dbands of the transition metals are muchbroader—approximately 5 to 10 eV.From our discussion of band formation,one intuitively knows that bandwidthsare a function of volume. They becomenarrower for large volumes (the wavefunctions barely overlap between latticesites), they turn into a single energyvalue for fully separated atoms, andthey become broader for small vol-umes. Using the LDA, one can calcu-late the total bonding energy of a givenelement in any crystal structure andacross a range of volumes (with theirconcomitant bandwidths) and therebydemonstrate the dependence of energyon bandwidth.

The results of such calculations areplotted in Figure 11. The ground-state(or internal) energy is plotted as a func-tion of bandwidth for niobium and uranium in several structures: the bct(body-centered tetragonal) and ort (orthorhombic) structures for niobiumand the bct, ort, and fcc structures foruranium. In both plots, the ground-stateenergy of the bcc (body-centered cubic)phase is the horizontal reference line. Astructure is more stable than bcc whenits internal energy goes below the refer-ence line. Figure 11 shows that, whenthe d band is forced to be narrower than3 eV because an expanded volume isused as input to the calculation, the low-symmetry orthorhombic structure becomes stable. Likewise, when the f band in uranium is forced to be broad-er than 7 eV, the high-symmetry bccstructure becomes stable. This is simplya demonstration that narrow bands favorlower-symmetry structures, not that nio-bium would form in the orthorhombicstructure of α-uranium. That particular

structure is used only for convenience inthe calculations, and we expect thatsome other low-symmetry structurewould be even more stable. Figure 11also indicates that the true equilibriumbandwidths (Weq) are broad (small volumes) for the transition metals and narrow (larger volumes) for thelight actinides.

Lowering the Electronic Energythrough a Peierls-like Distortion. Inmetals, narrow bandwidths (and theirhigh density-of-states functions) lead to low-symmetry structures through aPeierls-like distortion. The originalPeierls-distortion model occurs in a one-dimensional lattice: A row of perfectlyspaced atoms can lower the total energyby forming pairs (or dimers). The lowerperiodicity causes the otherwise degen-erate electronic-energy levels to split,some becoming lower and others becoming higher in energy. The loweredlevels are occupied by electrons, andtherefore the distortion increases thebonding and lowers the total energy ofthe system. In this one-dimensional sys-tem, the distortion opens an energy gapat the Fermi level and makes the systeman insulator. In the higher dimensionalsystems, which we will discuss next, the material remains a metal after thedistortion because there are other Blochstates that fill this gap.

In real three-dimensional lattices, theenergy levels are degenerate alonghigh-symmetry directions. If those lev-els lie close to EF, a crystal structuredistortion (Peierls-like) would increasethe one-electron contribution to bond-ing, again because the degenerate levelsin the band would split, pushing somelevels above EF and others below EF.This mechanism is very effective ifthere are many degenerate levels nearEF—that is, if the energy bands are nar-row and therefore the density of statesis large. Materials with broad bands(wider than 4 eV), gain less energyfrom level splitting because there arefewer levels near EF, and thereforesymmetry-lowering distortions are rarein these materials.

Stabilizing Wide Bands into High-Symmetry Structures. Besides theone-electron contribution, there is another important contribution to theground-state energy, namely, the elec-trostatic Madelung energy. This term isgenerated because the conduction elec-trons do not completely shield the ionson the lattice sites. The unshielded pos-itive charge leads to a long-range attractive force on the conduction electrons that lowers their energy. Thisnegative Madelung energy is largest forhigh-symmetry crystals, opposing thePeierls-like term. So, for the broad s-p2

and d band metals with few degeneratelevels near EF, the Madelung termdominates the effects of the Peierls-likedistortion, and these metals are stabi-lized in high-symmetry structures. Although the Madelung term stabilizesthese high-symmetry structures, remem-ber from our discussion of crystal struc-tures that the one-electron contributiondetermines which structure (fcc, hcp, or bcc) will be stable. For the light actinides with their narrow 5f bandsand a very high density of states nearEF, the Peierls distortion wins out, andthese metals form low-symmetry struc-tures. A closer look at Figure 9, for example, shows that six of the eightdifferent atoms in the α-plutonium unitcell have neighboring atoms that fallinto near distances on one side and fardistances on the other. It is as if onehemisphere around the atom weresmaller and the other larger, and it isdifficult to imagine how to pack suchobjects efficiently. But this type ofproblem is just what one might expectto solve with the three-dimensional version of the Peierls distortion.

Why Are the 5f Bands Narrow?By considering the contribution of eachenergy band (s, p, d, and f) to the bond-ing curve of the universal bonding pic-ture, we can understand why the 5fbands are narrow at the ground-stateequilibria of the light actinides.



2The broad s and p bands often mix strongly andare therefore referred to as the s-p band.

We used the atomic-sphere approxima-tion to calculate the contributions fromindividual bands, and for simplicity, weperformed those calculations in the fccphase. Figure 12 shows the results forplutonium. To understand this figure,consider that, if the metal had only oneenergy band, the lattice constant in theground state would be given by thepoint at which the bonding curve forthat band crossed the zero pressure line.From Figure 12, you can see that, ifplutonium had only an f band contribu-tion, its equilibrium lattice constant

would be smaller than it actually is, itsf band would be wider, and this metalwould stabilize in a high-symmetrycrystal structure. In reality, the contri-bution from the s-p band (a repulsiveterm at true equilibrium) helps to stabi-lize plutonium at a larger volume; the f band is narrow at that larger volume,and the narrowness leads to the low-symmetry crystal structure. This argu-ment is universal for multiband metals.In the transition metals, the s-p band isrepulsive at equilibrium and leads toslightly larger volumes than would be

the case if these metals had only d bands.

We conclude that the width of thedominant band determines the symme-try of the ground state, and one doesnot need to invoke directional bondingto obtain low-symmetry structures.When Willie Zachariasen and FinleyEllinger first identified the crystal struc-ture of α-plutonium in 1957, theycalled it a covalent structure, suggestiveof directional bonds, but they never repeated that name. (We note that thiswas one of the most difficult structureidentifications ever done because singlecrystals were not available, the x-raypattern had many lines, other phases inthe samples contributed to the pattern,and the x-ray lines were terribly broad-ened by strain.) In recent years, JohnWills and collaborators (see the article“Actinide Ground-State Properties” onpage 128) have been able to calculate the charge density for several actinidesusing the full-potential DFT method.They find no dominant directional 5f bonds and, most important, nocharge buildup between atoms. Instead,the calculated ground-state properties of α-plutonium indicate that this metalis quite average.

We already have hinted, however,that plutonium may not be so average.The strong competition between the repulsive s-p band contribution and theattractive f band term in Figure 12 is the first sign of instability near theground state. The second sign is the factthat the density-of-states functions fordifferent low-symmetry crystal struc-tures are very similar. Thus, the totalenergies for different low-symmetrycrystal structures are likely to be veryclose to each other—that is, many stateslie close to the ground state.

The Landscape of ActinidePhases

Knowing that the ground states ofthe light actinides are in a very shallowminimum of energy, we might expectthese metals to change phase as they



Repulsion

Equilibrium distance

Attraction

−0.4 −0.2

−10

−5

0

For

ce p

er a

tom

(eV

)

5

10

0 0.2 0.4 0.6

Expansion


Total

sp

d

f

Compression

Figure 12. Contributions of Different Electrons to δ-Pu Bonding LDA predictions for the bonding curves of δ-Pu (fcc structure) are plotted as a function

of the interatomic spacing x = ln(a/a0). The figure includes the curve for the total cohe-

sive energy per atom, as well as the individual contributions from the s, p, d, and f

energy bands. Note that, if the bonding came from the f band alone, the equilibrium lat-

tice constant of Pu would correspond to the value of x at which the f band contribution

crosses the horizontal axis; that is, the lattice constant would be well below a0. At this

smaller volume, the f band would be wider, and Pu would stabilize in a high-symmetry

crystal structure. In the true equilibrium represented by the vertical line at x = 0, the

s-p bands contribute a repulsive term and therefore help stabilize Pu at the larger vol-

ume. The f band is narrow at that larger volume, and the narrowness leads to the low-

symmetry ground-state crystal structure of δ-Pu. It may seem strange that the s-p bands

are not bonding at equilibrium, but we must remember that they are also not bonding

in d-electron transition metals, which form the bulk of metals in the periodic table.

are perturbed by heating or alloying.An atomic argument leads to the sameconclusion. In the actinides, two orthree different shells of electrons in anatom are partially filled, producingmore states to compete with the lowest-energy ground state. Also, the heavierthe atom (or the higher the value for Z),the smaller the energy difference between the last few valence electronsin different shells, and hence the greaterthe chance for metamorphosis.

A composite phase diagram for theentire actinide series is drawn in Figure13. This plot was constructed from individual-alloy phase diagrams for the actinide neighboring-element pairs.Each original diagram showed the

change in phase as the temperature andthe binary composition were varied. In Figure 13, these diagrams have beensimplified and drawn side by side. Thegray areas represent guesses for uniden-tified details. The resulting “landscape”of the actinides has been reprintedwidely since Smith and Kmetko drew itin 1980 because anything with such apretty pattern was something the ac-tinide community needed to understand.

Phase Instability and the f-f Inter-action. One striking feature of Figure13 is the large number of allotropes, orsolid-state phases. In fact, the actinideshave the largest number of allotropes ofany series in the periodic table. Note

also that most of the allotropes occur atneptunium and plutonium, the elementshaving the highest number of bonding f electrons. Figure 14, a replotting ofthe actinide metallic radii in Figure 5that now includes some of the allotropes, emphasizes the effects of thef electrons on bonding. The f electronbonding begins at thorium, with a frac-tion of an f electron in the energybands, and increases all the way to plu-tonium, which has the highest phase instability. Thus, in the ground-statephases of the early actinides, each addi-tional f electron increases the bondingand decreases the interatomic distance.At americium, the f electrons localizecompletely and become nonbonding



1500

Tem

pera

ture

(°C

)

Liquid

bcc

bcc

Orthorhombic

fccfcc

dhcp

Ac Th Pa U NpElement

Pu Am Cm

1000

500

Com

plexC

ubic

Ortho.

Ortho.

Tetragonal

Monoclinic

Monoclinic

Tetragonal

Figure 13. Schematic Phase Diagram of the Light Actinides This phase diagram connects individual binary alloy diagrams of the light actinide series. The black areas are two-phase region s; the

brown are regions where the details are unknown. Whereas the trend in phases is rather ordinary at either end of this series, t he phase

variation in the middle (at neptunium and plutonium) is very complex (also revealed by the opening illustration to “Atomic Vibr ations and

Melting in Plutonium” on page 190). In these two elements, the 5f electrons play a major role in bonding because they are so nu merous.

This widely reproduced and compelling figure shows nature in action, and we have yet to match it with an equally appealing theor y.

because Coulomb forces have at last become strong enough to pull the f electrons inside the valence shell, leaving only the three (or sometimestwo) valence electrons in the s-p and d bands to form the glue that holds the atoms together for the rest of the actinides. The metallic radii increase at this crossover to localization, and as shown in Figure 14, the low-temperature phases of the heavier actinides (beginning with americium)form close-packed or bcc structures.

The upper (high-temperature) part ofFigure 13 shows the liquid phase, andthe melting curve separating the liquidfrom the solid state goes through a min-imum between neptunium and plutoni-um, at which point the crystal structuresare least stable. In its broadest features,the pattern shows the light actinidesforming low-symmetry ground states

but melting out of high-symmetry struc-tures as the heavy actinides and mostother metals do. This observation isstriking, but the really difficult aspect toexplain is that the transition from lowsymmetry to high symmetry occurs overan extremely small temperature range,small enough that the f band remainsnarrow. What property of narrow bands(or, specifically, narrow f bands) enablesthem to stabilize high-symmetry struc-tures at modest temperatures?

We know from standard Wigner-Seitz rules that, for electrons in narrowbands, the radial portion of the elec-tronic wave function measured from agiven lattice site must change rapidly asa function of energy. These rules alsotell us that the electrons in the upperhalf of a band are less bonding, so wecan use the molecular concept of bond-ing and antibonding orbitals to under-

stand the Bloch states of solids.3 Also,the electronic wave functions of atomson different lattice sites overlap less fornarrow bands than for wide bands.Cerium, the only 4f metal with a nar-row f band, is a good example. The f-electron wave functions overlaponly the nearest-neighbor sites, whereasthe s-p and d wave functions, which are associated with wide bands, overlapsites that are far from a given site. In other words, narrow f bands imply ashort-range, rapidly changing interac-tion among the f electrons. This meansless overlap between f electrons on dif-ferent sites and also weaker hybridiza-tion of electrons in f orbitals with elec-trons in s, p, and d orbitals.

The many phase transformations inthe center of the actinide landscape sug-gest that the f-f interaction varies dra-matically with very small changes in theinteratomic distances, such as those pro-duced by lattice vibrations or heating.Also, electron-phonon coupling appearsto be very strong in these narrow bands.We would even suggest that the varia-tion with temperature in the f-f interac-tion (and f-spd interactions) might causesome of the f orbitals to become local-ized as plutonium transforms from theα- to the δ-phase. The situation is com-plicated by several different interatomicdistances in the low-symmetry structuresof the light actinides (see the crystalstructure of plutonium in Figure 9). Unlike the phase transformations in the transition metals, which follow sim-ple, well-understood routes, such as theBain path (fcc↔bcc) or the fcc↔hcppath, the individual transformations inplutonium (say, α↔β or β↔γ) are diffi-cult to characterize. And the whole pathfrom the α- to the δ-phase is even moredifficult to characterize. We need moredetailed analysis and many more calcu-lations to work out the details of thesephase transformations. But we do know



1.9

1.8

1.7

1.6

1.5

Ac Pa NpElement

Met

allic

rad

ius

(Å)

Am Cm Bk CfTh U Pu

βα α

β

βα

γ, ε

γ

δ, δ′

Figure 14. Metallic Radii of Actinide AllotropesHere, we have replotted Figure 5 to include additional allotropes (red) of the actinides.

As in Figure 5, the metallic radius is half the average distance between the atoms in

the solid, and the line represents simple trivalent metals with nonbonding f electrons.

The points falling below the line show the involvement of the f electrons in the bond-

ing, which pulls the atoms closer together. Notice that the various allotropes of

uranium and neptunium have similar metallic radii. In contrast, the radii of the phases

from α- to δ-plutonium increase monotonically. The δ-phase still departs from the triva-

lent line but is much larger than the α-phase, indicating that the f electrons are not

completely localized and therefore still contribute to bonding.

3According to the Wigner-Seitz rules, the radialpart of the wave function at the top of the band is zero at the Wigner-Seitz radius (a radius that is roughly half the interatomic distance). At the bottom of the band, the derivative of that radialpart is zero at that same radius.

that the short-range nature of the f-f interaction in the light actinides andthe partial transition from bonding (itin-erancy) to localization (rather than di-rectional bonding) are key ingredientsfor a better understanding of plutonium.

The Role of Entropy in Phase Sta-bility. To understand fully the stabilityof these different phases as a functionof temperature, we must go beyond the one-electron methods. Rememberthat those methods determine only the internal energies of solids (they describe the system at T = 0), whereasphase stability depends on the free energy, which is the internal energyplus the entropy of the system. So far,theorists have not been able to calculatethe free energy from first principles.Nevertheless, we can make some gener-al comments about the role of entropy.

For the low-symmetry structures ofthe light actinides, which have similardensity-of-states functions and thereforesimilar internal energies, the internal-energy path between different low-symmetry phases is very shallow. Thatis, the lower-energy phase sits in a rela-tively flat potential well connecting it to the higher-energy phase. For thatreason, it often has very soft (low-frequency) vibrational modes in itsphonon spectrum. Low-frequencymodes imply large numbers of phononsper mode and therefore lots of entropy.For example, entropy is a driver in thefirst three solid-to-solid phase transfor-mations in plutonium (α to β, β to γ,and γ to δ), in which low-symmetrystructures are involved. (A recent measurement on phonons in uraniumgave the entropies for uranium. See the article “Vibrational Softening in α-Uranium” on page 202).

The δ- to ε-phase transformation in plutonium is between two high-symmetry structures, following thestandard Bain’s path (fcc↔bcc) seen in most transition metals. Typically, the internal-energy path between cubicstructures is not shallow. However, experimental data indicate that this pathis shallow in plutonium. In Figure 15,

we show the Bain’s path for a few typi-cal transition metals and the postulatedpath for the δ- to ε-phase in plutonium.When the internal-energy path betweenphases is shallow and the entropy islarge, the range of stability of a phasecan be small. And that is exactly whatis seen in the δ- to ε-phase transforma-tion. The δ-phase remains stable for

only 160°C, whereas in a typical transi-tion metal, the fcc phase remains stablefor around 1000°C before becomingbcc. (See the article “Elasticity, Entropy, and the Phase Stability of Plutonium” on page 208 for a discus-sion of how to estimate the vibrationalentropy through measurements of elastic constants.)

The actinides are not unique in hav-ing shallow internal-energy paths between them. The same is true ofsome transition metals (see Figure 15)and even simple metals such as lithium. In all these metals, entropy plays a biga role. We emphasize, however, that in all the actinides, and particularly inplutonium, the initial instabilities (lead-ing to high entropy and frequent phasechanges) are due to the role of the f electrons and the existence of manyinternal-energy states whose energy values are very close.

Low Melting Temperatures andElectron-Phonon Coupling. The lastremarkable feature of Figure 13 that wewill comment on is the pronounced dipin the melting curve at neptunium andplutonium. We suggest that two effectsmay lead to this reduction in meltingtemperature: the strong electron-phononcoupling in narrow-band materials(which in a one-electron theory wouldbe seen as the large variation of the f-finteraction with temperature) and the instability of the phases due to the largeentropy term. Strong electron-phononcoupling implies that small temperaturechanges lead to larger-than-usual elec-tronic changes. As a result, the effectivetemperature is larger than the actualone, and therefore the material melts ata relatively low temperature. Evidencefor strong electron-phonon coupling is seen in the strong temperature depen-dence of the elastic constants of gallium-stabilized δ-plutonium deducedfrom neutron diffraction measurements(see the article “Atomic Vibrations andMelting in Plutonium” on page 190).Strong electron-phonon coupling is alsoimplied by the high resistivity of pureplutonium just below room temperature.



4

6

2

0

1.0 1.1 1.2

c/a Ratio1.3 1.4 1.5

Ene

rgy

(mR

y)

Os

Ir

Re

Pu and Pt

bccfccBain’s path (bct)

Figure 15. The Bain Paths for Typical Transition Metals and forthe δ- to ε-Plutonium Transformation We performed these calculations by

varying the c/a ratio in the bct structure

to go from the fcc to the bcc phase.

The calculated internal energy as a

function of the c/a ratio yields the path

the solid must follow to get from the fcc

to the bcc phase. As shown, the Bain’s

paths for typical transition metals (5d),

such as osmium and iridium, are steep.

The platinum path is very shallow,

and we use it as our model for the plu-

tonium path (red dotted lines). The

shallow well in the fcc phase for plat-

inum and plutonium leads to soft modes

and large entropy. The observation that

plutonium goes through the δ′-phase

(bct) in going from δ- to ε-phase implies

that the Bain’s path for plutonium is flat

enough to make the δ′-phase metastable

at least at some value of c/a.

The large entropies (soft vibrationalmodes) in these materials also imply that relatively small temperature changesproduce relatively large changes in freeenergy. As a result, the average tempera-ture change between phases in plutoni-um is 85°C, whereas in typical transitionmetals, it is around 1000°C. In otherwords, a free-energy change largeenough to cause melting requires only asmall temperature change—that is, themelting temperature is relatively low.These general insights notwithstanding,Migliori is right to emphasize that muchmore work is needed to quantify the vibrational energy and entropy contribu-tions to the free energy and phase stabil-ity. We now return to the problem ofhow electronic structure affects thephase stability of plutonium, especiallyits δ-phase.

Delta Plutonium, f Electron Localization, and Plutonium

Metallurgy

Delta-phase plutonium is of specialinterest because its fcc structure allowsplutonium to be formed into shapes aseasily as aluminum. Used in buildingnuclear weapons, this phase is most im-portant to understand from first princi-ples. Ironically, although electronic-structure calculations of plutonium haveimproved, the δ-phase has been thetoughest one to get right in spite of itssimple cubic structure because its f elec-tron behavior does not fit the usual cate-gories. The plot of metallic radii in Fig-ure 14 indicates the dramatic increase inthe metallic radius (decrease in density)of the δ-phase, suggesting that f electronbonding decreases markedly from the α- to the δ-phase. On the other hand, f electron localization is not completebecause the interatomic distance for δ-plutonium is still below the trivalentline defined by actinium and the heavieractinides, whose f electrons are fully localized. (The side a0 of the cube of theunit cell is 4.637 angstroms in δ-plutonium and 4.894 angstroms incubic americium.) Thus, the f electrons

in δ-plutonium are in limbo between localization and itinerancy, a state thathas yet to be modeled in any convincingway. Plutonium can easily change itscrystal structure and density in responseto its surroundings, and the δ-phase isthe most sensitive.

The only other transformation thatcomes close to producing the 26 percentvolume expansion seen in the transfor-mation fromα- to δ-plutonium is theisostructural expansion from the fcc α-phase to the fcc γ-phase of cerium. Thistransition produces a 20 percent volumeexpansion. Before f bands were known

to exist, the reverse transition in ceri-um—the collapse from the fcc γ-phaseto the denser fcc α-phase near roomtemperature and under pressure—was attributed to a change of an f electron(localized) to a d electron (itinerant),which would explain the volume changeand the magnetic properties. However,accumulating data showed no change in the number of f electrons through thechange in volume. In 1974, Börje Johansson suggested that the collapse incerium involved a Mott transition. Motthad described how the localization ofconducting electrons turned conductors



Slip planes 111 Slip planes

Figure 16. Stacking Close-Packed Crystal Structures The red and blue layers of the fcc and hcp crystal structures are stacked identically.

In the fcc structure, the third layer (yellow) is in a new position; in the hcp structure, it is

perfectly aligned with the first layer (both are red). The ability of one plane to slide (slip)

over another leads to the ductility of fcc metals. It takes a bit of practice to visualize

the close-packed planes (shaded) within the fcc unit cell. The many slip systems in

fcc metals are discussed in the box “Atomic Packing and Slip Systems in Metals” on

page 308.

fcc structure hcp structure

into insulators, and delocalization did thereverse. Johansson saw that cerium’s fccisostructural expansion involved local-ization and its collapse involved delocal-ization just as in the Mott metal-insulator transition, but for cerium, onlythe f electrons were involved. The s-pand d electrons remain conducting inboth fcc phases, and thus this Mott-liketransformation is between two metallicphases of cerium. Many theorists believethat a similar localization mechanism is probably responsible for the series of phase transitions from α- to δ-plutonium, except that the net resultfalls short of the more complete transfor-mation to localization seen in the trans-formation from α- to γ-cerium becausefive, instead of one, f electrons are involved (see the article “A PossibleModel for δ-Plutonium” on page 154).

Despite its weird electronic state, δ-plutonium signals the return of the actinides to the typical metallic close-packed crystal structures. That is, itforms the tip of the “iceberg” of the fccphases at the right in Figure 13. The fccstructure is what makes metals such ascopper, aluminum, stainless steel, and δ-plutonium very useful. It is a high-symmetry, close-packed structure thatone can easily construct by stackingmarbles, as in Figure 16. The first layerof marbles (red) forms hexagons madeof triangles. The second layer (blue)falls naturally into a set of first-layer depressions. The marbles in the thirdlayer can be either directly above thosein the first layer (red) or in a third set ofpositions not found in the layers below(yellow). The hcp structure is formed ifthe third layer is on top of the first one;the fcc structure, if the third layer is inthe third position. Notice that, in allcases, it would be fairly easy to slideone layer across the next one becausethe marbles do not have to move up anddown too much in order to land into thenext set of holes. Sliding them acrosssome other plane would not work aswell because there are fewer atoms in allother nonequivalent planes and the wellsare deeper. It is precisely because oneplane can easily slide across the next

one that fcc metals are ductile and there-fore easy to shape.

Ductility, a remarkable feature ofmetals, is not necessarily long lasting.As a plane of atoms slides, some of theatoms may not move uniformly. And asmore and more planes “misslide” in thisway, the misfits accumulate, and themetal gets stronger—that is, it takesgreater force to deform it or to make itfail. This process is called work harden-ing. Alpha plutonium is an example of anaturally brittle and strong material.

From the α-plutonium structure shownin Figure 9, it is easy to see that slidingplanes across each other in any directionwould be extremely difficult because theatoms would no longer fit. In contrast,the planes of δ-plutonium slide very eas-ily, like those of aluminum.

On the other hand, new single crys-tals of very pure α-uranium were shownto bend easily when squeezed by hand.Single crystals are expected to be easierto bend because there are no grainboundaries and dislocations to pin



8

7

6

5

4

3

2

1

0 200 400

Temperature (°C)

Leng

th c

hang

e (%

)

600 700100 300 500

α

β

γ

δ

ε

L

Iron

Pureplutonium

δ′

Crystal structure Density (g/cm3)

α Simple monoclinic 19.86

β Face-centered monoclinic 17.70

γ Face-centered orthorhombic 17.14

δ Face-centered cubic 15.92

δ′ Body-centered tetragonal 16.00

ε Body-centered cubic 16.51

L Liquid 16.65

Figure 17. Thermal Expansion of Plutonium The percentages by which the lengths of plutonium and iron have changed are

shown as a function of temperature for 0°C to 660°C. Abrupt changes in the curve

for plutonium denote phase transitions, and each of the smooth segments is in

the labeled structures. The crystal structures and densities of each phase are also

listed. Note that plutonium’s thermal expansion is very large, the material goes through

six solid-state phases before melting, the δ-phase contracts as it is heated, and pluto-

nium contracts as it melts. These features are in stark contrast to the small linear

expansion of iron and most metals.

the sliding. However, these uraniumcrystals seem softer than expected. Wedo know that uranium deforms by mak-ing twins, which are structures that looklike mirror images of each other sepa-rated by planar boundaries. In general,low-symmetry structures have moreplanes across which twins can form, andthis may mean that, if we had singlecrystals of very pure α-plutonium, theywould be far softer than any crystals wehave ever seen. For a more completedescription of these metallurgical prop-erties, see the article “Plutonium and Its Alloys” on page 290.

Unfortunately, δ-plutonium is notvery stable. It takes only about 1 kilobarof pressure to transform δ-plutoniuminto a lower-symmetry phase such as α-plutonium. A fascinating reversal of thistransformation was seen during the frac-ture of α-plutonium and is described in“Plutonium and Its Alloys.” The surfaceof fractured typical-purity α-plutoniumappears jagged and broken, as would beexpected for a brittle monoclinic metal.However, under a scanning electron microscope, the surface looks coveredby ductile dimples, like a peanut buttersandwich that has been pulled apart.Such dimples are never seen in brittlematerials because the plastic flow needed to make them does not occur.So, where did the dimples come from?Although the following explanation hasnot been proved so far, it is believed thata region of hydrostatic expansion preced-ing the tip of the crack is created duringthe fracture of α-plutonium. The expan-sion, or effective negative pressure, inthis region instantaneously transformsthe α- into the δ-phase. After the frac-ture has propagated through the region,the material relaxes and returns to the α-phase. These fracture results show thatplutonium easily moves back and forthfrom the α- to the δ-phase and thatspreading out the atoms (during hydro-static expansion) stabilizes the fcc phase,as one would guess. Yet, this quicktransformation to the δ-phase increasesthe volume by 26 percent.

Figure 17 shows the complex patternof length changes produced by heating

plutonium metal. Plutonium goesthrough six solid phases before it reach-es its melting point, with the δ-phasehaving the lowest density (and the most-localized f electrons) of all the phasesincluding the liquid state. These changesin volume are not only challenging forphysicists to explain but also extremelytroublesome to metallurgists. For exam-ple, when castings are made from pluto-nium in the liquid state, the metal expands as it solidifies (see the box“Plutonium in the Liquid State”) andthus helps fill a mold. During subsequentcooling, however, the many changes involume lead to the formation of voids,which reduce the structural integrity ofcast plutonium.

Another surprising feature is that δ-plutonium contracts when heated, as re-vealed in Figure 17. Remember thatthermal expansion is an anharmonic ef-fect and therefore not predicted by theusual harmonic models of a solid. Inharmonic models, a solid is described asa set of masses held together by springs.That is, the potential energy as a func-tion of distance between neighboringatoms is a parabola, and the forces arelinear in the displacement from equilibri-um, as they are for springs. In thesemodels, the coefficient of thermal expan-sion is always zero, and this is true forall even powers of the potential energy

as a function of interatomic spacing. So, the common belief that atoms

vibrate more at higher temperatures andtherefore take up more room is notcorrect. Instead, when atomic excursionsbecome large enough to run into the“hard core” of the filled-shell electrons ofthe neighboring ions, the interatomic potential becomes steeper than a parabola, and this anharmonicity, or increase in pressure, leads to an expan-sion in volume with increasing tempera-ture. Conversely, a potential that increas-es more slowly than a parabola(including higher even powers) at largeatomic excursions yields a contraction involume with increasing temperature andreflects a change in electronic structure.We will return to this topic when we dis-cuss magnetism. For the moment, we cansay that a negative thermal-expansion coefficient may be due to a conversion ofmagnetic into bonding energy as in Invar(see the box “Itinerant Magnetism, Invar,and δ-Plutonium” on page 120).

New Models for Delta Plutonium

As yet, there has been no estab-lished model for the unusual features ofthe fcc δ-phase. Here, we suggest thatboth the unusual thermal properties and



Plutonium in the Liquid State

Plutonium must be happier (have lower energy) in the liquid because it has a higher

density than the phase from which it melts. It seems likely that plutonium forms clus-

ters in the liquid (as water does) because it has the highest viscosity of any molten

metal and because it forms complexes, as discussed in the article “The Complex

Chemistry of Plutonium” on page 364. We note that the bcc crystal structure has its

second-nearest neighbor atoms almost as close as the closest neighbor atoms and has

little shear strength; that is, it is the most liquidlike crystal structure and the highest-

temperature solid phase for most metals. That is why bcc sodium cuts with a butter

knife. Because water and plutonium are both good solvents, it may be fair to draw a

comparison between the polar molecules of water and the atomic misfits of plutonium.

To find the nature of liquid plutonium, we therefore need to use neutrons or some other

probe. We remind the reader that, if plutonium were not such a good solvent, we would

have more usable containers in which to put it while we make measurements at

high temperatures.

the absence of measured magnetism inδ-plutonium may be related to the deli-cate balance of forces between the repul-sive s-p bands and the attractive f band(and the sensitivity of the f-f interac-tions) shown in Figure 12. First princi-ples ground-state calculations haveshown that all five 5f electrons are bond-ing in the α-phase. Because it has neverbeen demonstrated that plutonium hasless than five 5f electrons in its atoms,molecules, and solids, we must assumethat the δ-phase also has five.

In the δ-phase, some or all of theseelectrons may be localized, a fact consis-tent with Figure 12. Localization wouldreduce the contribution of the five 5felectrons to pressure, and the latticewould expand. If their number were reduced from five to, say, one, then the three electrons in the spd broadbands would become more important in determining the crystal structure (see Figure 18). And because broadbands favor cubic structures, the δ-phaseis a reasonable outcome of localization.The real problem, however, is to deter-mine a mechanism for such localizationin the f shell.

Wills and Eriksson (see the article“Actinide Ground-State Properties” onpage 128) have used one-electron cal-culations to show that localizing four ofthe five 5f electrons leads to a correctprediction of the δ-phase volume.Through a separate atomic calculation,they show that the energy gain from lo-calizing those four electrons yields thelowest-energy state at the δ-phase vol-ume. However, those calculations pro-vide no mechanism for the localization.

In “A Possible Model for δ-Plutonium” (page 154), Cooper pre-sents the self-induced Anderson localiza-tion model, a two-electron impurity-likemodel. According to this model, all the5f electrons on some plutonium sites become localized. The strong scatteringfrom these “impurity sites” can disruptthe coherence of the 5f band states anddrive the whole system toward localized5f states. The Anderson Hamiltonian, as used by Cooper, has both the Hubbardrepulsive Coulomb term and a Kondo-

like two-electron term (originally invent-ed to treat the spin-spin interaction between an electron on an impurity atomand a conduction electron). The Hubbardterm keeps the 5f electrons localized,and the two-electron term leads to a par-tial localization of the conduction elec-trons. This model has therefore all theright ingredients for yielding localizationin going from the α- to the δ-phase, butit may take years before it could be usedfor realistic calculations. Moreover, tostart the calculation, one must have someimpurity sites. In effect, this prerequisitehelps avoid the question of how the five5f electrons become localized.

Can we obtain a translationally invariant solution for fcc plutonium inwhich the five 5f electrons are localizedby starting from a calculable Hamilton-

ian? The usual one-electron calculation(translationally invariant) at the δ-phasedensity predicts itinerant 5f electrons.So, we must modify the Hamiltonian to predict localization. In the Andersonmodel, the Hubbard term prevents the f electrons on other sites from hoppingonto a given site. This is the impurityview, but as we have pointed out, it is almost impossible to obtain transla-tionally invariant solutions from impurity models. Instead, let us find outwhat will keep the 5f electrons fromleaving a site. Clearly, we need to addan attractive Coulomb term. We there-fore propose a model with a small attractive Coulomb term added (in the f states only) to the band Hamiltonian in a one-electron calculation. Such a calculation would have to be performed



Repulsion

Equilibrium distance

Attraction

−0.4 −0.2

−10

−5

0

Pre

ssur

e ×

volu

me

(eV

)

5

10

0 0.2 0.4 0.6

Expansion


Total

sp

df

Compression

Figure 18. Modified Bonding in Plutonium with Fewer 5f Bonding Electrons In this figure, we show how the bonding would change if only one 5f electron is in the

conduction band instead of the usual five, and the other four are localized at lattice

sites. We illustrate the effect of this localization by artificially reducing the f band con-

tribution to bonding by roughly 80% relative to the results plotted in Figure 12. In this

case, the s-p repulsion term is not balanced by the f bonding term, and the lattice must

expand to reach a new equilibrium.

self-consistently so that the hybridizationof the f states with the non-f states couldchange in value as the 5f electrons be-come more localized. We believe such acalculation is feasible within the standardcodes of electronic-structure calculations.Having such a solution, we could thenapply the two-electron model to obtainthe dynamics of partial localization.

A feasible approach closer to the im-purity models would be an electronic-structure calculation of an alloy, inwhich there are two plutonium sites: Onehas an added attractive Coulomb term,and the other does not. Such a self-con-sistent calculation would demonstratewhether random sites with localized fstates could drive all the f states to local-ization on all sites. The physical basisfor an attractive term may be the poorscreening of the nuclear charge by the 5fstates, as it is by the 4f states in the caseof the lanthanide contraction. As we arelooking at new models for the δ-phase,we must also consider the alloy-stabi-lized δ-phase.

Gallium-Stabilized δ-Plutonium.Although no one understands the elec-tronic structure of δ-plutonium, peoplein the nuclear weapons program workwith this phase all the time. In fact, thesecret to stabilizing this phase down toroom temperature dates back to theManhattan Project. At room tempera-ture, pure plutonium would be in thevery brittle α-phase, but as luck wouldhave it, one of the very first “high-puri-ty” samples made by Manhattan Projectworkers had enough impurities to become ductile. Those pioneers knewtherefore from the start that they coulddeform the new metal into a requiredshape. They soon figured out that it wasthe addition of a few percent of a triva-lent metal at high temperature that heldplutonium in the δ-phase down to roomtemperature.

But why should the addition of thosesmall atoms stabilize a phase with a dra-matically low density and the desiredmalleability? The answer is not knownyet, so we can only make suggestions.The impurity picture suggests that atoms

at random lattice sites create strong scat-tering, thereby blurring the periodicity ofthe lattice and killing the coherence ofband states. Applying this mechanism toplutonium, we would say that the addi-tion of impurity atoms containing no felectrons would destroy the coherence ofthe f band. Without its narrow f band,plutonium could no longer lower its en-ergy by lowering its symmetry through aPeierls-like distortion into the α-phase; itwould therefore remain in the δ-phase.Another view is that plutonium atomsrelax and move toward the smaller non-fatoms, thereby reducing the f-f interac-tions that stabilize the α-phase. So, thesolid remains stable in the high-symme-try fcc phase even as the temperature islowered. In this cubic structure, only 3atomic percent aluminum or gallium dis-solved in plutonium (the replacement ofone plutonium atom by a gallium atomin every three-atom-sided cube) isenough to hold the fcc structure down toroom temperature. Roughly, any atomsthat are soluble in pure δ-plutonium atits elevated temperatures and that haveno f electrons should stabilize the δ-phase to lower temperatures. (Like plu-tonium, cerium and americium have f electrons, but they also stabilize the δ-plutonium. Understanding howtheir more-localized f electrons interactwith the itinerant f electrons in plutonium, however, is best left to bandor alloy calculations that can includesome of these effects.)

Our discussion of the plutonium δ-phase and other phases highlights thecomplexity of the 5f bonding in plutoni-um and its impact on metallurgical prop-erties. We believe that improved one-electron methods, along with newmeasurements conducted with moderntechniques, will clarify our understand-ing of bonding in all plutonium phases.One might also have to go beyond theaverage electron correlations inherent inone-electron methods to include specificspin interactions and specific interactionsbetween more-localized and less-localized electrons, as in the two-electrondynamics described in the article by Cooper.

We now leave the topic of phase instabilities, the least understood aspectof the light actinides, and turn to thelow-temperature properties of the lightactinides. Recently, these properties have been recognized as similar to the low-temperature properties of othercorrelated-electron materials.

A Revised View of the Periodic Table

Plutonium is not alone in having itselectrons precariously balanced betweenbonding and localized states although itexhibits this property better than anyother element. If the periodic table isarranged to show only the d and f elec-tron series, and the f series are put ontop and squeezed together to fit as inFigure 19, then the elements highlightedby the white diagonal stripe are therough dividing line between localized(local moment magnetism) and itinerant(superconductivity) long-range collectivebehavior. This is a good indication thatthe f and d electrons of the diagonal ele-ments may be balanced between beinglocalized and itinerant, and their behav-ior is therefore interesting. We will dis-cuss the “normal metals” off the diago-nal before we consider the diagonalelements.

Normal Behavior off the Diagonal:Magnetism vs Superconductivity. Fig-ure 19 indicates that the f and d electronmetals away from the diagonal divideneatly into two categories. In one catego-ry, the f and d electrons are itinerant andfully bonding, and they tend to form asuperconducting ground state at very lowtemperatures. In the other category, the fand d electrons are fully localized, usual-ly forming local magnetic moments andordering into a magnetic ground state atsome temperature.

This association of localized electronswith local magnetic moments and ofshared electrons with bonding is familiarto chemists. When chemists make anew insulator or organic compound con-taining an f or d electron atom, they can



deduce the valence state of that atom bymeasuring the magnetic moment of thecompound. The measured value tellsthem the number of electrons contribut-ing to the moment and therefore thenumber of electrons localized on theatom. The remaining f or d electrons areparticipating in bonding, and their countgives the valence. Usually, metals arenot so simple, but it is useful to think of electrons as magnetic or bonding.

So, how do we know if there arelocal magnetic moments in a material?Magnetic susceptibility χ measures theinternal response of a material to an applied magnetic field. That is, M = χB,where M is the magnetization of the ma-terial and B is the magnetic field intensi-ty. In 1845, Michael Faraday showedthat some materials were drawn to thehigh-field region of his magnet and oth-ers were repelled. He called the formerbehavior paramagnetic with a positivesusceptibility and the latter diamagneticwith a negative susceptibility. Most com-mon metals have no magnetic momentsand are weakly paramagnetic or diamag-

netic. As shown in Figure 20, these non-magnetic metals have low susceptibili-ties, which are roughly temperature inde-pendent. Magnetic materials havepositive, much larger susceptibilities, andbecause thermal motion counters the ten-dency for moments to align with the ap-plied field, this positive susceptibility isinversely proportional to temperature, asshown in Figure 21. We can extract thesize of the magnetic moment in a mag-netic material from the slope of χ vs T.

In the solid state of metals or insula-tors, local moments at lattice sites tendto line up at some temperature as thematerial is cooled, producing a state oflong-range order called magnetism. (Fol-lowing Hund’s rules, the local magneticmoments at each lattice site are propor-tional to a vector sum of the total spinand total orbital angular momentum ofthe localized electrons.) In particular,each magnetic material has a criticaltemperature, at which the free energy ofthe system is lowered through an align-ment of the local moments. If the mo-ments line up parallel to each other,

forming a ferromagnetic state, the transi-tion temperature is called the Curie pointTC. If the moments line up antiparallel,forming an antiferromagnetic state, thetemperature is called the Néel tempera-ture TN. Magnetic transition temperaturesrange from close to absolute zero toabout 1000 kelvins, and a ferromagneticmaterial whose TC is above room tem-perature is called a permanent magnet.Figure 21, a generic plot of the inversesusceptibility vs temperature, shows howthe temperature intercept can indicatewhether a material will be a ferromagnetor an antiferromagnet at lower tempera-tures. If the intercept is at a positive temperature, the material should be fer-romagnetic, and if it is at a negative temperature, antiferromagnetic.

Local moments in one of these mag-netic ground states in a metal arearranged following the symmetry of thelattice, and they do not scatter the con-duction electrons well. As a crystal isheated above TC or TN, the magnetic ordering disappears, the local momentspoint in random directions, and those



La Ce Pr Nd Pm Sm Eu Gd

Partially filled shell

Tb Dy Ho Er Tm Yb Lu

Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn

Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd

Ba

4f

Magnetism

Superconductivity

5f

3d

4d

5d Lu Hf Ta W Re Os Ir Pt Au Hg

Figure 19. A Revised Periodic Table of the f and d Series The periodic table is rearranged with the rare-earth, or lanthanide, elements in the top row, the actinides in the second row, and the

d-electron transition elements below them. Most metals have predictable ground states and become superconducting (blue) or mag-

netic (red) as the temperature is lowered. But the low-temperature metallic properties of the elements along the diagonal are d ifficult

to explain because, in the solid-state, their f or d valence electrons are poised between localization and itinerancy.



0.007

0.006

0.00055

0.00050

200 400 60000.005

0.004

0.003

0.002

0.001

0 200 400

Temperature (K)

T (°C)

Mag

netic

sus

cept

ibili

ty (

emu/

mol

e)

x (e

mu/

mol

e)

600 800 1000

UBe13

Pu inset

α-Pu δ-Pu

Most metals—Nb, Mo, Ta, Hf,Pt, Na, K, Rb

α β γ δ εδ′

Figure 20. Magnetic Susceptibilityof Plutonium and Other Materials The susceptibility of most metals is close

to zero on this plot, but that of UBe 13 is

very large as though this heavy-fermion

compound had local moments. It was

therefore conjectured that UBe 13 would

become an antiferromagnet at low tem-

peratures. Instead, however, UBe 13

becomes a superconductor at low tem-

peratures, and the reason for its high

susceptibility is not well understood.

The susceptibilities of plutonium’s vari-

ous phases are also higher than those

of most metals, but they are lower than

those of materials with local moments.

The blowup of the plutonium curve

(see inset) displays more clearly the

variation in susceptibility as plutonium

changes phase. Although essentially tem-

perature independent, the susceptibility

of plutonium appears to increase slightly

as the temperature decreases. It is fair

to say that plutonium behaves as if it

were almost magnetic.

χ−1

Temperature

TN

Antif

erro

mag

net

Inde

pend

ent m

omen

tsFe

rrom

agne

t

Tc

Figure 21. Curie-Weiss Plots of Inverse Magnetic Susceptibility The magnetic susceptibility due to local

moments follows the Curie-Weiss law: It is

inversely proportional to temperature

because thermal motion tends to wash out

the natural alignment of local moments as

temperature increases. The inverse sus-

ceptibility is plotted for several cases.

If there is no interaction among local

moments, the inverse susceptibility extrap-

olates to zero at T = 0. If there is a ferro-

magnetic interaction among the local

moments, it extrapolates to zero at a posi-

tive temperature intercept, and if there is

an antiferromagnetic interaction, it extrapo-

lates to a negative intercept. The absolute

value of the intercept tends to correlate

with the magnetic ordering temperatures.

moments can now scatter the conduc-tion electrons. The scattering occursthrough a magnetic interaction that canflip the spins of the conduction elec-trons. We shall see that plutonium scatters the conduction electrons verywell although it does not have local moments.

Superconductivity is a state of trulyperfect conductivity that can exist in an electrical conductor at low tempera-tures. In general, materials with magnetic moments do not become superconductors at low temperaturesbecause magnetism in any form is inimical to superconductivity. The typi-cal superconducting state forms at tem-peratures of a few kelvins and consistsof pairs of conduction electrons withopposite spins and momenta that arebound together through their interactionwith the crystal lattice. These pairs actlike a Bose condensate and travelthrough the lattice with no resistancewhatsoever. This low-energy collectivestate was explained in the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity (1957). However, if amaterial has ordered moments, a netmagnetization, or spins from randomimpurities, the magnetic field resultingfrom any of these sources would tendto flip one of the two opposing spins inthe pair and prevent the pairs fromforming. Thus, elements with localizedd or f electrons typically have magneticground states, and those with conduct-ing d or f electrons typically have superconducting ground states.

Anomalous Properties of Elements on the Diagonal. Now, weturn our attention to the elements on thediagonal in Figure 19. On the atomiclevel, these elements are like plutoniumbecause they have two or more incom-pletely filled atomic shells that areclose in energy. The electrons cantherefore shift between shells relativelyeasily. As a result, these elements showgiant resonances in their atomic opticalspectra, reflecting their many close-lying energy levels.

These energy levels in the atom

translate into a multiband system withone relatively narrow band and a highdensity of states at the Fermi energy inthe metal. This description is true of thetransition metals on the diagonal, eventhough their d bands are not quite asnarrow as the f bands in elements alongthe diagonal. Narrow bands tend to mixor hybridize with other close-lyingbands (a phenomenon that is predictedby one-electron band calculations).Moreover, weakening or strengtheningthe hybridization can push the d or felectrons in these narrow bands towardgreater localization or greater itineran-cy, respectively. Thus, the electrons inthese narrow bands are highly sensitiveto small perturbations. This sensitivity

can lead to many allotropic crystalstructures in the same element and to catalytic activity. It can also lead tothe metal’s ability to absorb hydrogeneasily and to spark when struck. Cat-alytic activity and hydrogen absorptionoccur when perturbations can modifythe chemical state, and the diagonal isthe place to look for those behaviors.

First, let us consider a metal’s abilityto make sparks. All metals shower offburning pieces when put to a grindingwheel because the wheel heats the piecesas they break off, and most hot, finely divided metals burn. However, steel, cerium, uranium, neptunium, and plutoni-um make sparks under far more gentleconditions. For example, it is the cerium



100

150

50

200

0 200 400

Temperature (K)

K

Pu

UBe13

Res

istiv

ity (

µΩ c

m)

600 800

α β γ δ εδ′

Figure 22. Resistivity of UBe 13, Plutonium, and Potassium Electrical resistance in most metals increases linearly with temperature because

electrons are scattered by phonons, as shown for potassium. Imperfections such as

foreign atoms, lattice vacancies, and more-complicated defects, which are temperature

independent, also scatter electrons and lead to a finite limiting resistance as T = 0.

Plutonium has much higher resistivity values at all temperatures, showing enhanced

scattering of the conduction electrons. The electron scattering comes from electron

correlations. The compound UBe 13 yields one of the most extreme examples of

electron scattering in a metal.

in a lighter flint that makes the sparks forlighting fires. We argue without proofthat the high density of states near theFermi surface causes the transfer of elec-trons to or from a tiny piece of highly de-formed metal as it breaks off from a largepiece, no matter how gently the breakingoccurs. The resulting voltage differenceproduces a tiny electrical discharge across

the gap, which then ignites the shard.Lanthanum, which is mechanically simi-lar to cerium and is far more chemicallyactive, does not make sparks because ithas no f electrons and its density of statesis low. The uranium-manganese com-pound UMn2, which is made of two nar-row-band elements from the diagonal inFigure 19, sparks so readily that its pow-

der burns as soon as it is poured in air. The most-common anomalous prop-

erty, and the simplest one to measure, isthe departure from the usual linear rela-tion between electrical resistivity andtemperature that occurs below room tem-perature. In Figure 22, we compare theresistivities of potassium and plutonium.Over the temperature range shown,



In some magnetic materials such as iron, conduction electrons

rather than localized electrons produce ferromagnetic ground

states. In these “itinerant magnets,” the conduction band splits

into two distinct bands, one composed of spin-up states and the

other of spin-down states. Although those two sets of states exist

in all materials, in most they remain degenerate, except in the

presence of an applied magnetic field. In iron, for example, the

somewhat narrow d band splits spontaneously (without an applied

magnetic field) into spin-up and spin-down bands. As in the

Peierls distortion described in the main text, the separation of

degenerate states into two bands lowers the total energy of the

filled states. (Note that itinerant magnetism does not follow

Hund’s rules because the orbital angular momentum of conduc-

tion electrons is quenched.) Itinerant magnets are less robust

than local-moment magnets because the split conduction bands

remain close to each other in energy and various perturbations

can easily overwhelm the energy advantage from the split. The

ferromagnetic state of iron, for example, disappears when iron is

alloyed with other elements to make stainless steel.

We note the little-appreciated correlation between corrosion and

magnetism. Most stainless steels are nonmagnetic and, of course,

do not corrode. After welding or deformation, they sometimes rust

in exactly those spots that have converted to a magnetic form of

iron, martensite. Why? We suggest that any electrons that are not

fully participating in bonding, including localized electrons or con-

ducting electrons that are magnetically ordered, are more available

for chemical reaction and thus for corrosion. We also note that, if

the electrons in an itinerant magnet become less bonding and

more magnetic, the lattice becomes bigger and vice versa. It is this

effect that made Invar possible. Invar, as well as being an itinerant

magnet, is an alloy with a zero coefficient of thermal expansion

around room temperature. Accurate pocket watches were made of

Invar a century ago. As an itinerant magnet is heated, thermal mo-

tion interferes with the spin alignment of electrons in the narrow

bands, and this decrease in magnetic ordering makes those elec-

trons a little more bonding (van Schilfgaarde et al. 1999), thus

countering the normal thermal expansion. By choosing the correct

combination of elements, one can make an alloy that maintains its

size over some temperature range because these competing

effects exactly balance out over that temperature range. It is now

clear that Invar watches had to be kept out of magnetic fields

because those fields could have bent the parts. We will not dis-

cuss the Invar effect in detail but will give a simple example that

illustrates the electronic-structure changes leading to this effect.

The energy contained in the magnetic field of a permanent mag-

net can become noticeable on the scale of the cohesive energy.

When dropped, modern neodymium-iron-boron permanent mag-

nets break not because the metal is weak but because the huge

magnetic energy decreases dramatically .

Consider the itinerant magnets iron, cobalt, and nickel. By one-

electron calculations, we can determine the volumes of the true

ferromagnetic state and of the hypothetical paramagnetic state.

We calculate the ferromagnetic state by including spin, and thus

we have spin-up and spin-down bands as mentioned before. In

the paramagnetic calculation, we do not include spin, and the

spin-up and spin-down bands are degenerate. In all cases, the

calculated volume of the paramagnetic state is smaller than that

of the ferromagnetic state. In other words, if the itinerant electrons

contribute to magnetism, their contribution to bonding is reduced.

Can this tradeoff be related to the contraction upon heating that is

seen in the δ-phase of pure plutonium?

Although plutonium has no local magnetic moments, one might

ask whether its conduction electrons could make it an itinerant

magnet, like iron. There is little evidence now for magnetism in

pure plutonium, but many plutonium compounds are magnetic and

tend to be itinerant magnets. Simply dissolving hydrogen in pluto-

nium is enough to make the electrons localize and the system

ferromagnetic. Also, a comparison of the light actinides with the

transition metals indicates that the light actinides should be super-

conductors unless they have local moments. So, the fact that

plutonium is not a superconductor may indicate that plutonium

is an incipient, weak itinerant magnet and that the loss of

magnetic ordering with heating plays a role in the contraction

of the δ-phase.

Itinerant Magnetism, Invar, and δ-Plutonium

potassium’s resistivity increases linearlywith temperature, as expected from theinteraction of conduction electrons withlattice vibrations (phonons). Instead offollowing this simple linear increase,plutonium’s resistivity bulges high abovelinearity at temperatures below roomtemperature and remains extremely highabove room temperature, where it showsoff its many phase transitions. The resis-tivity of plutonium at the bulge is of theorder of 150 micro-ohm centimeters (µΩ cm), which means, microscopically,that an electron is scattered by roughlyevery atom in the lattice. This type ofscattering is considered the highest possible simple resistance that can beseen in a metal and is called the unitarylimit. Many correlated-electron metalsshow higher resistivities, which can onlybe explained by correlated-electron mod-els such as the Kondo and Andersonmodels for extended systems. Thesemodels add electron interactions and correlations beyond those included inone-electron methods.

This puzzling enhanced electrical resistivity upon cooling began receivingattention thirty-five years ago, when itwas noticed that a small amount of irondissolved in gold showed an increasingelectrical resistance as it was cooled attemperatures of a few kelvins. At thesame time, the iron atoms had no mag-netic moments. (The increase in the re-sistance of plutonium from room temper-ature down had already been measured,but this curve was neither as repro-ducible as the curve for the iron-goldalloy nor as easy for everyone to check.)

Jun Kondo was the first scientist tohave come up with an explanation forthe increase in the gold alloy’s resistivi-ty: The missing magnetic moment iniron was responsible for it. Kondo thenmade a model in which the spins of con-duction electrons could flip the spins ofthe electrons on the iron atoms andthereby cancel the magnetic momentwithin the small volume around eachiron atom impurity. This model predictedthe desired increase in resistivity withdecreasing temperature. In time, theorieswith a characteristic Kondo temperature

were used for explaining not only the increase in resistivity upon cooling butalso the leveling off in magnetic suscep-tibility upon further cooling, once a highvalue had been reached. They could alsoexplain why magnetic moments weremissing in so many materials. So, theKondo impurity model was expanded to a Kondo lattice model, an extendedsystem, in which crystallographically ordered moments could again be made to disappear, even though there seemednot to be enough conduction electrons todo the job. The most extreme exampleare the “Kondo insulators,” in which noconduction electrons are present, but their spins are assumed to makemagnetic moments vanish. We believethat these are examples of overworking amodel simply because it fits the data.Such situations would not occur if wehad a comprehensive theory for thesephenomena. We note that the disappear-ance of the iron moment in the goldalloy should have come as no surprisebecause that moment is also known todisappear when iron is alloyed to makestainless steel. Indeed, it has been wellknown since the 1930s that the magnet-ism of iron is not robust (see the box“Itinerant Magnetism, Invar, and δ-Plutonium” on page 120). However,we are so used to calling iron a perma-nent magnet that even today many peo-ple believe iron is an archetypal exampleof a magnetic atom or metal.

Correlated-Electron Materials. De-spite exaggerated claims, the Kondomodel and all its offshoots have shownthe condensed-matter physics communitythat there are very interesting anomaliesgoing on in metallic systems, especiallythe metals on the diagonal in Figure 19,and this group includes plutonium. Although we do not have a simple pre-dictive model to explain the origin ofhigh resistivities, the materials exhibitingthese conundrums are now lumped underthe single category of correlated-electronmaterials. As shown in Figure 1, apartfrom high resistivities, these materialsexhibit high magnetic susceptibilities,and they have no local moments at low

temperatures. Other anomalies includetiny magnetic moments (as much as1000 times smaller than expected), high-er-than-expected magnetic ordering tem-peratures, failure to follow the quadratictemperature dependence of resistivitypredicted by the Fermi-liquid model, andhigh specific heats indicating enhancedelectron masses. All the signatures ofcorrelated-electron materials are believedto result from strong electron-electroncorrelations involving spin and chargeinteractions.

Plutonium is probably the clearest example of a pure element in the corre-lated-electron category. Figure 22 showsits anomalous resistivity. Plutonium alsohas the most enhanced low-temperaturespecific heat of any pure element. Thespecific heat shown in Figure 23 indi-cates that the electron effective mass ofplutonium is 10 times larger than that ofmost metals. Finally, plutonium has arelatively large magnetic susceptibility inall its solid-state phases, as seen in theinset on Figure 20.

The relatively large paramagnetic sus-ceptibilities of the various phases of plu-tonium suggest that phases with magneticmoments or itinerant magnetism musthave similar internal energies. Perhaps allthe phases in pure plutonium above theα-phase have magnetic moments, but because the temperature range of eachphase is so narrow, we have not beenable to demonstrate them by magneticsusceptibility or neutron diffraction mea-surements. Correlated-electron materialsusually show increases in susceptibilityas they cool, but that susceptibility levelsout within a few kelvins of absolutezero. Our best chance of finding outwhether plutonium has local moments isto do synchrotron x-ray and neutron-scattering experiments on single crystals.Those measurements would be adequatelysensitive to magnetic moments. We notethat Méot-Raymond and Fournier (1996)claim to have found the signature of localmoments on top of a huge paramagneticbackground in δ-plutonium. Although the presence of local moments is consis-tent with the model of δ-plutonium described in Cooper’s article, we



remain somewhat skeptical of this datainterpretation because so many correlat-ed-electron systems show the same typeof susceptibility even when they do nothave local moments.

Magnetism and local moments do be-come manifest in plutonium if we spreadout the plutonium atoms. As mentionedin the box “Itinerant Magnetism, Invar,and δ-Plutonium,” plutonium hydride isferromagnetic because the increased sep-aration between plutonium atoms causesthe f electrons to localize and pushestheir behavior across the diagonal limit in

Figure 19 and into the magnetic regime.This last statement leads us into a discus-sion of actinide compounds that exhibitlocalized 5f states or whose ground statesbecome superconducting.

Hill Plots and Actinide Compounds

The crossover from itinerant to localized electrons can clearly beachieved if the atoms are spread out. Before 1970, investigators could not

guess or understand whether ceriumand uranium compounds would be superconducting or magnetic. In 1970,at Los Alamos, Hunter Hill explainedunder what conditions metallic com-pounds containing f electron elementswould be superconducting or magnetic.Until then, the light-actinide and lan-thanide compounds seemed to showthese two collective ground states in anunpredictable manner.

Hill realized that in cerium and light-actinide compounds, the distance between the f electron elements deter-mined whether their ground states weresuperconducting or magnetic, usuallyquite independently of which atoms con-taining non-f electrons were separatingthem. Figure 24 shows what is nowcalled a Hill plot for uranium com-pounds. Superconducting or magnetictransition temperatures are plotted verti-cally, and the spacing between the f elec-tron elements is plotted horizontally. InHill’s original plot, the known behaviorsfell into two of the four quadrants—largespacing correlated with magnetism andshort spacing with superconductivity.

At the time, it was quite strange tothink that superconductivity did not depend on the intervening atoms contain-ing non-f electrons. Energy band calcula-tions were just beginning to show thatthe f electrons would be in bands, butthose calculations were limited to thesimplest structures. Hill conjectured thatthe f electrons could hybridize only with f electrons at other sites and that the intervening non-f electron atoms werejust spacers to change the degree of over-lap between the f-electron wave func-tions. In this way, Hill’s plot became amajor step toward understanding the lightactinides. This picture is far more generalin Figure 19, where we see that elementson or near the diagonal can have theirproperties easily modified.

Figure 24 shows many more materialsthan were plotted in Hill’s original version, including two superconductingcompounds that, based on the relativelylarge distance between two f-electronatoms, should have been magnetic. Thesetwo, namely, the uranium-platinum and



1

400

300

200

100

0 20 40

T2 (K2)

T (K)C

/T (

mJ/

g-at

om K

2 )

60 80 100

2 3 4 5 6 7 8 9 10

UBe13

Ga-stabilized δ-Pu

Pure α-Pu

Cu

Figure 23. The Specific Heat of Plutonium and Other Metals The low-temperature specific heat of a metal is the sum of a lattice term proportional

to T3 and an electronic term γT; that is, CV = γ T + AT3. Hence, when we plot CV divid-

ed by T vs T2, we get a straight line that has an intercept equal to γ at T = 0 and a

slope equal to A. That slope yields the Debye temperature ΘD via the relation ΘD3 =

1944 A–1 J·mol –1 ·K–1. As we explained in the section on the free-electron model, the

specific-heat coefficient γ is proportional to T/TF, and TF is inversely proportional to

the mass of the electron. So, γ is proportional to the electron mass. In this figure, the

line for copper represents the behavior of most metals whereas the lines for α- and δ-

plutonium have the highest values of γ (intercept values) of any pure element, indicat-

ing that conduction electrons have an enhanced effective mass. The compound UBe 13

has an extremely high electronic specific heat, which continues to increase until it is

cut off by the compound’s transition to superconductivity just below 1 K (not shown).

The superconductivity of UBe 13 proves that its large heat capacity must be associated

with the conduction electrons rather than the local moments and that the anomalously

large values of γ are enhanced electron masses, or heavy-fermion masses, m*.

uranium-beryllium compounds UPt3 andUBe13, are examples of heavy-fermionsuperconductors.

Heavy-Fermion Materials

Many compounds and alloys contain-ing elements near the diagonal in Figure19 are called heavy-fermion materials,which means that their conduction elec-trons behave as if they had extremelyheavy masses. The very high heat capac-ity of UBe13 shown in Figure 23 indi-cates that the electrons have an effective

mass hundreds of times larger than thatof conduction electrons in normal metals.Heavy-fermion materials also show verystrange magnetic and superconductingground states. All these materials are narrow-band metallic systems. So, theirunusual collective ground states arisefrom very strong electron correlations involving the electrons in their narrowbands. Also, their low-energy excitationsare associated with the spin and chargefluctuations in that narrow band. One dif-ference between these materials and thelight actinides is that the former do notexhibit crystal structure instabilities

because their narrow bands are not thedominant bonding bands. This featurecan be seen in UPt3, whose wide d bandhas roughly 27 electrons (9 electrons for each platinum atom) and its narrow f band has only 3 electrons. So, it is the low-energy excitations(mostly at low temperatures) that makethe heavy-fermion materials resemble the light actinides. We will discuss the discovery of the heavy-fermion materials and the significant role playedby the Los Alamos scientists, as well as the exotic properties of these materials.



U2Ti

UB4USi2

URh3

UCo2UNi2

UCo

UB2

U-18 Mo (γ–U)

β–UH3

URe2

UAl2U6Ni

U6Mn

U6Co

U6Fe

UC2

UO2

U2PtC2

U2C3

UC UPt3 UGa3

UGe2Si2

U3P4

UTeUP UAs

UOSeUPt

UN

USeUS

UCu2

U2Zn17UCu5

USb2UP2

USb

Ul3UAl3

UPt5

UNi5

UBe13

UB12

1

2

3

4

0

50

100

150

200

2503.0 3.5 4.0

U-U spacing (Å)

Magnetism

Sup

erco

nduc

tivity

4.5 5.0

TS (

K)

TN o

r T

C (K

)

TNéel

TCurie

No ordering

TSuperconducting

α,β–U

Figure 24. Hill Plot for Uranium CompoundsThe Hill plot shows the superconducting or magnetic transition temperatures vs interatomic spacing separating the f electron at oms.

We augmented the original Hill plot for uranium compounds to include more data—in particular, the transition temperatures of th e

two superconducting heavy-fermion compounds UPt 3 and UBe 13. Hill conjectured that the overlap of the f-electron wave functions

between the uranium atoms determines whether the f electrons are localized (magnetic) or itinerant (superconducting) independen t

of the intervening atoms. Most compounds behaved as Hill expected. The superconducting compounds occurred at short

f-electron spacings (blue quadrant), and the magnetic compounds at large f-electron spacings. The heavy-fermion superconducting

compounds are exceptions. Although the spacing between the uranium atoms in those compounds is fairly large, the f electrons ar e

still not fully localized and can condense into a superconducting state.

In the late 1970s, in Cologne, Germany, Frank Steglich and cowork-ers were the first scientists to observe an anomalous superconducting groundstate. They were measuring the low-temperature properties of the cerium-copper-silicon compound CeCu2Si2.The behavior of this compound at tem-peratures above 100 kelvins suggestedthat, at very low temperatures,CeCu2Si2 would become an antiferro-magnet. That is, this compound showedthe standard inverse temperature depen-dence of the magnetic susceptibility, asshown in Figure 21, with a negative intercept on the temperature axis. It alsohad a huge heat capacity at low temper-

atures, of the order of joules per mole-kelvin, the size associated with localizedmagnetic moments.4 On the other hand,the magnetic susceptibility of CeCu2Si2became temperature independent at lowtemperatures, implying that magnetic ordering never took place. The real sur-prise was a large jump in the specificheat, as the compound went supercon-

ducting just below 1 kelvin. The specif-ic heat of a superconductor at its transi-tion point is shown in Figure 25. The opening of a large gap in the midstof the huge heat capacity of CeCu2Si2meant that the electrons that presum-ably were localized and associated with large magnetic moments had become superconducting. But that conclusion simply did not make sense.Localized electrons cannot even con-duct electricity, much less be supercon-ducting. These properties were considered so unlikely that they werediscounted as artifacts of bad samples,but Steglich defended his results.

Then, in the early 1980s, Los Alam-os scientists found that the compoundsUBe13 and UPt3 have the same proper-ties: very large heat capacities and a superconducting transition at low tem-peratures. The Los Alamos resultsmade clear that a new ground state ofmatter existed, one in which electroninteractions were so highly correlatedthat only an extreme quasiparticle pic-ture could cause that behavior. That is,the electrons were so heavily clothedby interactions that they acted as iftheir masses were 1000 times largerthan the mass of a free electron. A heatcapacity that should typically belong tomagnetic moments belonged, in thiscase, to the electrons moving throughthe metal.

The existence of this new groundstate blurred forever the distinction between localized and itinerant elec-trons. Fully localized electrons, withtheir definite energy levels, can bethought of as belonging to an infinitelynarrow band. Because no energy levelsare near by, those electrons act as if theyhad infinite mass—that is, they cannotmove at all. By analogy, the f electronsin heavy-fermion materials occupy sucha narrow energy band that the electronsact as if they had a huge mass—that is,as if they were almost localized.

Zachary Fisk, Hans Ott, Al Giorgi,Greg Stewart, Joe Thompson, JeffWillis, and other Los Alamos scientistswent on to identify more compoundsthat had the same heavy-fermion, or



0

320

280

240

200

160

120

80

40

1 2 3 4 5 6 7 8 9 10 11

Superconducting

Normal

Temperature (K)

Cv

(mJ/

mol

. K)

Tc = 8.7 K

Figure 25. Specific Heat Signature of the Superconducting Transition in Niobium Common superconductors, such as niobium, show a drop in specific heat as the

superconducting transition temperature is reached because a gap opens in the spec-

trum of possible electronic energies. It is this energy gap that makes the resistance

of the material vanish. That is, once in the superconducting state, conduction electrons

cannot be scattered as they move through the material because the scattering would

cause them to have energies within the energy gap. If an external magnetic field is

applied to the material, the superconducting transition will be suppressed, and the spe-

cific heat will vary with temperature as in a normal metal. Note that this plot is not

a straight line because C, rather than C/T, is plotted on the vertical axis. The same

specific-heat signatures of superconducting transitions occur in heavy-fermion com-

pounds such as CeCu 2Si2 and UBe 13 but at temperatures that are 10 times lower

and with changes in specific heat that are 10 times larger.

4The electronic heat capacity of a magnetic ma-terial is much larger than that of a normal metalbecause the moments on all the atoms contributeto the former whereas only T/TF of the conduc-tion electrons contribute to the latter. The low-temperature electronic specific heat of normalmetals is of the order of 1–10 millijoules permole-kelvin.

enhanced-mass, behavior but did notbecome superconductors. Some ofthese heavy-fermion materials becameantiferromagnetic and others, as mea-sured then, had neither magnetic order-ing nor superconducting ground states.Heavy-fermion compounds had beenseen earlier, but until some were seento be superconducting, there was noway to distinguish them from the mag-netic materials, which have large heatcapacities. Indeed, in the mid 1970s,Jim Trainor5 and coworkers noticedthat the antiferromagnetism of the nep-tunium-tin compound NpSn3 showed,at its magnetic ordering temperature, aheat capacity anomaly in the shape of a superconducting anomaly, similar to the one shown in Figure 25 for a superconductor. Because they could fitthe shape of a magnetic anomaly withthat of a superconducting one, the sci-entists knew that something quite unusual was occurring. However, noone was listening then, and the puz-zling result went unnoticed.

Several many-body theory groupshave tried to understand the unusualmetallic properties of the heavy-fermioncompounds by developing Fermi-liquiddescriptions. In the mid 1990s at LosAlamos, Kevin Bedell and his cowork-ers Gerry Brown, David Meltzer, Chris Pethick, David Pines, KhandkerQuader, and Carlos Sanchez-Castro carried out the most detailed theoreticalstudy of this kind. They developed verysimplified one- and two-band Fermi-liquid models of heavy-fermion com-pounds such as UPt3. The surprise wasthat these simplified models yieldedquantitative results in agreement withthe low-energy and low-temperaturephysics of these materials. Again, likethe Kondo model, the Fermi-liquidquasiparticle turns out to be a far moreuseful model than originally expected.Lev Landau’s principle of one-to-onecorrespondence between electron andquasiparticle states continues to be correct and to yield profound insightsinto systems of strongly interactingparticles.

The series of discoveries on heavy-fermion materials established LosAlamos as an institution for leading-

edge research into condensed-matterphysics and turned the attention of theo-rists and experimentalists worldwide tothe actinide elements. High interest inthese materials continues today.

High Magnetic Fields for Measur-ing Enhanced Masses.In the early1930s, W. J. de Haas and P. M. vanAlphen were measuring the magneticsusceptibility of a bismuth crystal andobserved oscillations in susceptibility as they varied the field. The period ofthe oscillations varied as the inverse of the applied magnetic field. The explanation is now simple. Placing ametal in a large magnetic field at lowtemperatures triggers a new quantiza-tion condition. The continuum of energy levels in the conduction bandbecomes a new set of discrete Landaulevels with a splitting between levelsgiven by heH/2πmc. For most experi-ments, the number of these levelsranges from 1 to 1010. This new set ofavailable states affects all the propertiesof the metal, provided the applied fieldis very large and the temperature verylow. As the magnetic field increases,one discrete level at a time rises abovethe Fermi energy and causes the fixednumber of electrons (or quasiparticles)to be redistributed on the remaininglevels. This electron redistribution isseen as an oscillation in all metallicproperties. By measuring the periodicityof the oscillations in different crystallo-graphic directions, we can map theshape of a metal’s Fermi surface. And by measuring the temperature dependence of the oscillations, we candetermine the effective masses of the electrons on the Fermi surface. Figure 26 shows the dHvA oscillationsin ThBe13.

This dHvA effect was used in the 1950s and 1960s to show that Fermisurfaces existed and to measure theirshape for simple metals. Later, however, scientists stopped using thistechnique because most pure metals had been measured. And then, in the late 1980s, the dHvA effect becamefashionable again because verifying



Figure 26. The de Haas–van Alphen (dHvA) Effect At low temperature and in a high magnetic field, the susceptibility of a metal oscillates

with a period that is inversely proportional to the field. This oscillation occurs because

the field imposes a quantization condition on the allowed levels of the electrons.

The Fermi surface can be extracted in this way. Illustrated here are the oscillations

for the ThBe 13 compound, which is a noncorrelated-electron counterpart of UBe.

2

0

Osc

illat

ory

mag

netiz

atio

n (χ

H)

–2

0 5 10

Applied magnetic field (T)

15

ThBe13 <100>

5At that time, Trainor was a postdoctoral fellow at Argonne National Laboratory.

the heavy-fermion masses in cerium anduranium compounds was of interest andthe higher magnetic fields necessary forthat work became available at the Na-tional High Magnetic Field Laboratoryin Tallahassee, Gainesville, and at LosAlamos. It was thus immediately clearthat dHvA methods would allow mea-suring the heavy-electron masses inheavy-fermion compounds. In the early1990s, at the Cavendish Laboratory inCambridge, England, Gil Lonzarich,Stephen Julian, and Louis Taillefer res-urrected the 20-year-old technique andused high magnetic fields (up to 20tesla) and the very low temperatures of adilution refrigerator (down to 10 mil-likelvins). After much work, they saweffective electron masses as high as 210in UPt3. This mass is so large that thehighest oscillations visible at a tempera-ture of 10 millikelvins were gone by 30millikelvins. This means that the highestfields, lowest temperatures, and highest-quality samples are a must. Althoughmany groups have looked for similar os-cillations in UBe13, such oscillationshave never been reported in crediblework because UBe13 masses are muchlarger than UPt3 masses, and the oscilla-tions are already gone at the lowest tem-perature attainable at this time.

In 1998, Jason Detwiler, GeorgeSchmeideshoff, and Neil Harrison observed dHvA oscillations in thepraseodymium-beryllium and thorium-beryllium compounds PrBe13 andThBe13 by working with a 70-teslapulsed magnet at the National High-Magnet Laboratory at Los Alamos.These compounds do not have the heavyelectrons of UBe13, and so the scientistsfocused on measuring the Fermi surfaceand showed that their results agreed withone-electron calculations. Then, to tryand sneak up on UBe13, they next triedthe thorium-uranium-beryllium com-pound Th0.95U0.05Be13 and saw nothing.They did, however, see oscillations inTh0.995U0.005Be13, but much work liesahead before the masses of UBe13 canbe seen. We will have to create the fol-lowing mandatory conditions: very lowtemperatures, very high fields, and sam-

ples of even higher quality. Because the one-electron behavior of PrBe13 andThBe13 has already been confirmed, the natural next step is to measure theclothed masses in UBe13.

Future Steps

If researchers think of narrow-bandphenomena when thinking of plutonium,we have succeeded in making our point.The narrow 5f bands lead to all the un-usual behaviors of plutonium at all ener-gy scales. At higher energies, the narrowbands predicted by one-electron DFTcalculations lead to low-symmetry crystalstructures but normal elastic properties.At low temperatures and energies, strongspin and charge fluctuations connected tothe excitations from those narrow bandslead to all the anomalous heavy-fermion-like behaviors. The strong electron-phonon coupling of narrow-band electrons may lead to higher effectivetemperatures, which form a depressedmelting curve in the light actinides. The instability of these band states (due to the small radial overlap of the f-electron wave functions and the manystates with similar energies) leads to themany-phase transitions in these materials.This instability most likely leads to thepartial localization of these 5f states sus-pected to exist in δ-plutonium.

For the past 40 years, we focused on the difference between the angularcharacter of d and f wave functions asthe key to understanding the light ac-tinides although, for most of that time,we knew that the 5f electrons go frombeing itinerant in plutonium to being localized in americium. That change tolocalization occurs only because of achange in the radial part of the wavefunction. We have also known for almost10 years that the low-symmetry crystalstructures are less related to the angularcharacter than to the narrow bondingbands in these metals. Therefore, weneed to study other features, such as theshort-range nature of the f-f interactionsin Bloch states. How can we constructstates with partial localization, that is,

with two kinds of 5f electrons (localizedand itinerant) in the same metal?

Over the next few years, improvedone-electron theories will better describethe short-range nature of the 5f interac-tions. To determine the range of the f-finteractions, these theories might involvesome Slater-Koster fitting of the f bands.Such a scheme could also be used indetermining the variation of the f-f inter-action as a function of volume. Somecalculations of this nature have been per-formed, but details of the f-f interactionshave not yet been extracted. Also, themodern electronic-structure calculationshave only demonstrated that the angularcharacter of the single-atom orbitals isnot a main factor in determining crystalstructures. Indeed, it appears to bewashed out when a large number ofstates build up to form bands, and forthat reason little or no charge buildup isseen between atoms in the actinide met-als. However, improved calculations mayshow some small charge buildup, andthen we would have to reconsider thisfeature. Models of localization of the 5f states need to be developed.

Many low-energy properties are stillso poorly understood that existing correlated-electron theories need to beimproved and new theories might have tobe developed. Theorists are still unableto predict the collective ground states inmany of these materials. Also, we cannotexplain why neptunium and plutoniumare the only f electron elements that donot develop a collective magnetic or superconducting state. After examiningheavy-fermion materials more closely,we now believe that all may have a mag-netic ground state. By using the most-modern experimental tools, we now cantry and measure magnetic moments thatare 1000 times smaller than they shouldbe. From this work, we hope that new insights into plutonium will emerge.

But right now, the most-modern experimental techniques cannot be applied to plutonium because they require large single crystals, and thoseare not yet available for plutonium.Learning how to grow such crystals in arepeatable fashion would be a real break-



through in plutonium studies. One couldthen use the crystals in neutron-scatteringexperiments to measure the entirephonon spectrum of plutonium and inphotoemission experiments to measureits energy bands as a function of crystalmomentum. Photoemission (photon inand electron out) is now sufficiently accurate to determine the widths andstructure of the narrow 5f bands in mate-rials such as plutonium. In all the earlierphotoemission experiments, the instru-ment resolution for x-rays was so poorand the surface contamination for ultravi-olet rays so high that the result wasmerely featureless spectra. Now, withsome of the new photoemission machines and improved surface-cleaningtechniques, we should obtain actinidespectra that show the structure predictedby the electronic-structure calculations.At Los Alamos, we are currently makingtiny single crystals of plutonium and related materials with techniques that had been used decades ago at RockyFlats and Argonne National Laboratory.As reported in this volume, we are measuring these tiny samples with ultrasound, neutrons, and x-rays.

Although we may eventually understand the electronic structure ofplutonium single crystals, the landscapeof close-lying but different states cannotbe removed to give homogeneity whenwe make a large plutonium casting. And because plutonium is radioactive, itsatoms will always be converted into impurities, and this process damages the lattice. Nevertheless, if we know howsingle crystals behave, it is much simplerto model a collection of them in a largecasting than to reverse-engineer a largepiece to see its components. This is whywe are looking forward to more progressin these areas.

We must study plutonium and itsneighbors, alloys, and compounds because we deal with them all inweapons. We may never know what idealplutonium is. However, the gift from nature is that this most-complicated element teaches us about all the elementsand shows us that there still is new science to discover.

Further Reading

Fisk, Z., D. W. Hess, C. J. Pethick, D. Pines, J. L. Smith, J. D. Thompson, and J. O. Willis. 1988. Science239: 33.

Harrison, N., A. L. Cornelius, H. Harima, K. Takegahara, J. A. Detwiler, G. M.Schmiedeshoff, and J. C. Cooley. 2000. Phys. Rev. B 61: 1779.

Hess, D.W., P. S. Riseborough, and J. L. Smith. 1993. In Encyclopedia of Applied PhysicsVol. 7, p. 435. Edited by G. L. Trigg. New York: VCH Publishers.

Johansson, B. 1974. Phil. Mag. 30: 469.

Lander, G. H., E. S. Fisher, and S. D. Bader. 1994.Adv. Phys. 43: 1.

Méot-Reymond, S., and J. M. Fournier. 1996. J. Alloys and Compounds232: 119.

Ott, H. R., H. Rudigier, Z. Fisk, and J. L. Smith. Phys. Rev. Lett. 50: 1595.

Smith, J. L., and R. G. Haire. 1978.Science200: 535.

Smith, J. L., and E. A. Kmetko. 1983. J. Less-Common Metals90: 83.

Söderlind, P. 1998. Adv. Phys. 47: 959.

Söderlind, P., O. Eriksson, B. Johansson, J. M. Wills, and A. M. Boring. 1995. Nature374 (6522): 524.

Stewart, G. R., Z. Fisk, J. O. Willis, and J. L. Smith. 1984.Phys. Rev. Lett. 52: 679.

van Schilfgaarde, M., I. A. Abrikosov, and B. Johansson. 1999. Nature400: 46.

A. Michael Boring received a B.S. in mathematics and physics, an M.S. in experimental nuclear physics, and a Ph.D. in theoretical solid-state physics from the University of Florida. Mike came to LosAlamos as a staffmember directlyfrom the Universityof Florida. Heworked on solidstate and neutrontransport problemsas well as theoreti-cal chemistry andatmospheric chemi-cal reactions. Mikeorganized a re-search effort into the electronic structure of solids(energy band theory) for actinide materials andwent on to become Deputy Director of the Centerfor Materials Science at Los Alamos. For most ofhis career at Los Alamos, his research focused onthe quantum theory of atoms, molecules, andsolids. In 1993, he retired from the Laboratoryafter 25 years of service. Since retiring, he hasmaintained his connection with the Laboratory asan affiliate working with staff members and post-doctoral fellows on condensed-matter physicsproblems.



James L. Smithreceived his B.S. in Physicsfrom Wayne State University and his Ph.D. inPhysics from Brown University. He is currentlya Los Alamos Laboratory Fellow in the Materi-als Science andTechnology Divi-sion. Jim is also anadjunct professorat several universi-ties, editor ofPhilosophicalMagazine B, andmember on theboard of theBrown UniversityAlumni Associa-tion. His researchinterests include superconductivity, magnetism,actinides, high magnetic fields, electronic prop-erties of metals, experimental physics, and newmaterials. During his career, Jim published morethan 350 papers. For his work, he received theE. O. Lawrence Award from the Department of Energy in 1986 and the American PhysicalSociety International Prize for New Materials in 1990.

Plutonium Condensed-Matter Physics

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