Comprehensive nuclear materials 1 10 interatomic potential development

Comprehensive nuclear materials 1 10 interatomic potential development Comprehensive nuclear materials 1 10 interatomic potential development Comprehensive nuclear materials 1 10 interatomic potential development Comprehensive nuclear materials 1 10 interatomic potential development Comprehensive nuclear materials 1 10 interatomic potential development Comprehensive nuclear materials 1 10 interatomic potential development Comprehensive nuclear materials 1 10 interatomic potential development Comprehensive nuclear materials 1 10 interatomic potential development

G J Ackland

University of Edinburgh, Edinburgh, UK

ß 2012 Elsevier Ltd All rights reserved.

1.10.5.3 Embedded Atom Methods and Density Functional Theory 273

1.10.12.1 Effective Pair Potentials and EAM Gauge Transformation 288

267

1.10.13 Analyzing a Million Coordinates 289

DFT Density functional theory

DOS Density of electronic states

EAM Embedded atom method

fcc Face-centered cubic

FS Finnis–Sinclair

GGA Generalized gradient approximation

for exchange and correlation

hcp Hexagonal closed packed

LDA Local density approximation for

exchange and correlation

LJ Lennard Jones

MD Molecular dynamics

MEAM Modified embedded atom method

NFE Nearly free electron

Nuclear materials are subject to irradiation, and their

behavior is therefore not that of thermodynamic

equilibrium To describe the behavior that leads to

radiation damage at a fundamental level, one must

follow the trajectories of the atoms Since millions of

atoms may be involved in a single event, this must be

done by numerical simulation, either molecular

dynamics (MD) or kinetic Monte Carlo For either

of these, a description of the energy is needed: this is

the interatomic potential

Now that accurate quantum mechanical force

calcu-lations are available, one might ask whether there is still

a role for atomistic potentials In practice, ab initio

cal-culations are currently limited to a few hundred atoms

and a few picoseconds, or a few thousands at T¼ 0 K

With MD and interatomic potentials, one can run

cal-culations of millions of atoms for nanoseconds With

kinetic Monte Carlo calculation, timescales extend foryears These methods still provide the only method ofatomistic simulation in these regimes The issue is notwhether they are still necessary; rather it is which oftheir predictions are correct

Before embarking on any simulation, it is essential

to consider whether the potential contains enough ofthe right physics to describe the problem at hand Forexample, in studying copper, one might ask about thefollowing:

 Color (total electronic structure)

 Conductivity (electronic structure around Fermilevel)

 Crystal structure (ground state electronic structure)

 Freezing mechanism (bond formation and crystalstructure)

 Dislocation dynamics (stacking fault energy)

 Surface structures (bond breaking energies)

 Interaction with primary radiation (short-rangeatom–atom interactions)

 Phonon spectrum (curvature of potential)

 Corrosion (chemical reaction)

In each case, a totally different aspect of physics

is required An essential issue for empirical potentials

is transferability: can potentials fitted to one set ofproperties describe another? In general they cannot,

so one has to be careful to use a potential in the type

of application that it was intended for

For many physical problems such as dislocationdynamics, swelling, fracture, segregation, and phasetransitions, much of the physics is dominated bygeometry Here, finite size effects are more importantthan accurate energetics and empirical potentials findtheir home

1.10.2 BasicsMaterials relevant to reactors are held together byelectrons An interatomic potential expresses energy

in terms of atomic positions: the electronic and netic degrees of freedom are integrated out

mag-Most empirical potentials are derived on the basis ofsome approximation to quantum mechanical energies

If they are subsequently used in MD of solids, then what

is actually used are forces: the derivative of the energy.

For near-harmonic solids, it is actually the second

derivative of the energy that governs the behavior

A dilemma: Does the primary term covering

ener-getics also dominate the second derivative of the

energy? To take an extreme example, an equation

which calculates the energy of a solid exactly for all

configurations to 0.1% is: E¼ mc2 : most of the

energy is in the rest mass of the atoms But this is

patently useless for calculating condensed matter

properties We encounter the same problem in a less

extreme form in metals: should we concentrate the

energy gained in delocalizing the electrons to form

the metal, or is treating perturbations around the

metallic state more useful? In general, the issue is

‘What is the reference state.’ Most potentials

implic-itly assume that the free atom is the reference state

1.10.3 Hard Spheres and Binary

Collision Approximation

The simplest atomistic model is the binary collision

approximation (BCA), which simulates cascades as a

series of atomic collisions The ‘potential’ is then

basically a hard sphere, with collisions either elastic

or inelastic and the possibility of adding friction to

describe an ion’s progress through an electron field

There is no binding energy, so the condensed phase is

stabilized due to confinement by the boundary

con-ditions This is a poor approximation, since the

struc-tures and energies of the equilibrium crystal and

defects do not correspond to real material However,

it allows for quick calculation and the scaling of defect

production with energy of the primary knock-on atom

The best-known code for this type of calculation is

probably Oak Ridge’s MARLOWE.1

1.10.4 Pair Potentials

Pairwise potentials are the next level in complexity

beyond BCA They allow soft interactions between

particles and simultaneous interaction between many

atoms Pair potentials always have some parameters

that relate to a particular material, requiring fitting to

experimental data This immediately introduces the

question of whether the reference state should be free

atoms (e.g., in argon), free ions (e.g., NaCl), or ions

embedded in an electron gas (e.g., metals)

The two classic pair potentials used for modeling

are the Lennard Jones (LJ) and Morse potential Each

consists of a short-range repulsion and a long-rangeattraction and has two adjustable parameters

ad hoc

Because they have only two parameters, all lations using just LJ or Morse are equivalent, thevalues of e, s, or a , which simply rescale energyand length Hence, these potentials cannot be fitted

simu-to other properties of particular materials

Both LJ and Morse potentials stabilize packed crystal structures, and both have unphysicallylow basal stacking fault energies Equivalently, theenergy difference between fcc and hcp is smallerthan for any real material For materials modeling,this introduces a problem that the (110) dislocationstructure is split into two essentially independentpartials For radiation damage, this means that con-figurations such as stacking fault tetrahedra are over-stabilized, and unreasonably large numbers ofstacking faults can be generated in cascades, fracture,

close-or defclose-ormation If the energy scale is set by the sive energy of a transition metal, then the vacancy andinterstitial formation energy tend to be far too large; ifthe vacancy energy is fitted, then the cohesive energy

cohe-is too small

For binary systems, these potentials can stabilize ahuge range of crystal structures, even without explicittemperature effects; some progress has been made

to delineate these, but it is far from complete.2Onceone moves to N-species systems, there are e and sparameters for each combination of particles, that is,N(Nþ 1) parameters Now it is possible to fit to prop-erties of different materials, and the rapid increase

in parameters illustrates the combinatorial problem

in defining potentials for multicomponent systems.1.10.4.1 LJ Phase Diagram

In practice, pair potentials are cut off at a certainrange, which can have a surprising effect on stability

as shown inFigure 1While the LJ fluid is very well studied, the finitetemperature crystal structure has only recently beenresolved The problem is that the fcc and hcp structuresare extremely close in energy (seeFigure 1(b)), so theentropy must be calculated extremely accurately

This has been done by Jackson using

‘lattice-switch Monte Carlo3(seeFigure 2) The equivalent

phase diagram for the Morse potential remains

unsolved The LJ potential has been used extensively

for fcc materials, and it still comes as a surprise to

many researchers that fcc is not the ground state

1.10.4.2 Necessary Results with Pair

Potentials

Apart from the specific difficulties with Morse and LJ

potentials, there are other general difficulties that are

common to all pair potentials, which make them

unsuitable for radiation damage studies

Expanding the energy as a sum of pairwise

inter-actions introduces some constraints on what data can

be fitted, even in principle It is important to

distin-guish this problem from a situation where a particular

parameterization does not reproduce a feature of a

material There are many features of real materials

that cannot be reproduced by pair potential whatever

the functional form or parameterization used

1.10.4.2.1 Outward surface relaxation

For a single-minimum pair potential, the nearest

neigh-bors repel one another, while longer ranged neighneigh-bors

attract When a surface is formed, more long-range

bonds are cut than short-range bonds, so there is an

overall additional repulsion Hence, the surface layer is

pushed outward But in almost all metals, the surface

atoms relax toward the bulk, because the bonds at

the surface are strengthened Similarly, pair potentials

give too large a ratio of surface to cohesive energy,again consistent with the failure to describe thestrengthening of the surface bonds

1.10.4.2.2 Melting pointsWith LJ, the relation between cohesive energy andmelting is Ec=kBTm 13, other pair potentials beingsimilar Real metals are relatively easier to melt, withvalues around 30 One can fit the numerical value ofthe e parameter to the melting point, and accept thediscrepancy as a poor description of the free atom.1.10.4.2.3 Vacancy formation energyFor a pair-potential, removing an atom from thelattice involves breaking bonds The cohesive energy

of a lattice comes from adding the energies of thosebonds Hence, the cohesive energy is equal to thevacancy formation energy, aside from a small differ-ence from relaxation of the atoms around the vacancy

hcp

1 0 0.1

0.2

0.3 0.4

Salsburg and Huckaby

for the crystalline region of the Lennard-Jones system in reduced units where p is pressure and r is density The equilibrium density is at rs 3 ¼ 1:0915 Filled squares are the harmonic free energy integrated to the thermodynamic limit from Salsburg, Z W.; Huckaby, D A J Comput Phys.

1971, 7, 489–502 All other points are from lattice-switch Monte Carlo simulations with N atoms, lines showing the phase boundary deduced from the Clausius–Clapeyron equation, from Jackson, A N Ph D Thesis, University of Edinburgh, 2001; Jackson, A N.; Bruce, A D.; Ackland, G J Phys Rev E 2002, 65, 036710.

rc/s

Figure 1 Energy difference between hcp and fcc for the

Lennard-Jones potential at 0 K, as a function of cutoff (r c )

with either simple truncation or with the potential shifted

to remove the energy discontinuity at the cutoff Without

truncation the difference is 0.0008695 e, with hcp more stable.

In real metals, the vacancy formation energy is

typi-cally one-third of the cohesive energy, the discrepancy

coming yet again from the strengthening of bonds to

undercoordinated atoms

1.10.4.2.4 Cauchy pressure

Pairwise potentials constrain possible values of the

elastic constants Most notably, it is the ‘Cauchy’

relation which relates C12–C66 In a pairwise

poten-tial, these are given by the second derivative of the

energy with respect to strain, which are most easily

treated by regarding the potential as a function of r2

rather than r ; whence for a pair potential Vðr2Þ, it

of the system

In metals, this relation is strongly violated (e.g., in

gold, C12¼ 157GPa; C44¼ C66¼ 42GPa)

1.10.4.2.5 High-pressure phases

Many materials change their coordination on

pres-surization (e.g., iron from bcc (8) to hcp (12)) and

some on heating (e.g., tin, from fourfold to sixfold)

This suggests that the energy is relatively insensitive

to coordination – for pair potentials, it is

propor-tional These problems suggest that a potential has

to address the fact that electrons in solids are not

uniquely associated with one particular atom, whether

the bonding be covalent or metallic Ultimately,

bond-ing comes from lowerbond-ing the energy of the electrons,

and the number of electrons per atom does not change

even if the coordination does

1.10.4.2.6 Short ranged

It is worth noting that some properties that are

claimed to be deficiencies of pair potentials are

actu-ally associated with short range So, for example, the

diamond structure cannot be stabilized by

near-neighbor potentials, but a longer ranged interaction

can stabilize this, and the other complex crystal

struc-tures observed in sp-bonded elements.4

1.10.5 From Quantum Theory to

Potentials

To understand how best to write the functional form

for an interatomic potential, we need to go back to

quantum mechanics, extract the dominant features,

and simplify Quantum mechanics can be expressed

in any basis set, so there are several possible startingpoints for such a theory Thus, a picture based onatomic orbitals (i.e., tight binding) or plane waves (i.e.,free electrons) can be equally valid: for potentialdevelopment, the important aspect is whether thesemethods allow for intuitive simplification

When a potential form is deduced from quantumtheory, approximations are made along the way Anaspect often overlooked is that the effects of termsneglected by those approximations are not absent inthe final fitted potential Rather they are incorporated

in an averaged (and usually wrong) way, as a tion of the remaining terms Thus, it is not sensible toadd the missing physics back in without reparameter-izing the whole potential

distor-1.10.5.1 Free Electron TheoryFor a free electron gas with Fermi wavevector kF, theenergy U of volume O is5

U ¼
h2kF5

10p2me

OThis contribution to the energy of the condensed phasegenerates no interatomic force since U is independent

of the atomic positions However, its contribution issignificant: metallic cohesive energy and bulk moduliare correct to within an order of magnitude Consider-ation of this term gives some justification for ignoringthe cohesive energy and bulk modulus in fitting apotential, and fitting shear moduli, vacancy, or surfaceenergies instead The discrepancy is absorbed by aputative free electron contribution which does notcontribute to the interatomic atomic forces in a con-stant volume ensemble calculation

1.10.5.2 Nearly Free Electron Theory

In nearly free electron (NFE) theory, the effects ofthe atoms are included via a weak ‘pseudopotential.’The interatomic forces arise from the response of theelectron gas to this perturbation To examine theappropriate form for an interatomic potential, weconsider a simple weak, local pseudopotential V0ðrÞ.The total potential actually seen at ridue to atoms at

rj will be as follows:

VðriÞ ¼X

j

V0ðrijÞ þ W ðrÞwhere WðrÞ describes how the electrons interactwith one another Given the dielectric constant,

we can estimate W in reciprocal space using linear

where a0 is the Bohr radius A more accurate

approach due to Lindhard:

1 xj

where x¼ q=2kFaccounts for the reduced screening

at high q, and r0is the mean electron density

From this screened interaction, it is possible to

obtain volume-dependent real space potentials.6

The contributions to the total energy are as

In this model, interatomic pair potential terms arise

only from the band structure and the electrostatic

energy (the difference between the Ewald sum and a

jellium model) and give a minor contribution to the

total cohesive energy However, these terms are

totally responsible for the crystal structure

A key concept emerging from representing the

Lindhard screening in real space is the idea of a

‘Friedel oscillation’ in the long-range potential:

VðrÞ /cos 2kFr

ð2kFrÞ3

This arises from the singularity in the Lindhard

function at q¼ 2kF: physically, periodic lattice

per-turbations at twice the Fermi vector have the largest

perturbative effect on the energy The effect of

Friedel oscillations is to favor structures where the

atoms are arranged with this preferred wavelength It

gives rise to numerous effects

 Kohn anomalies in the phonon spectrum are

par-ticular phonons with anomalously low frequency

The wavevector of these phonons is such as to

match the Friedel oscillation

 Soft phonon instabilities are an extreme case of theKohn anomaly They arise when the lowering ofenergy is so large that the phonon excitation hasnegative energy In this case, the phonon ‘freezesin,’ and the material undergoes a phase transfor-mation to a lower symmetry phase

 Quasicrystals are an example where the atomsarrange themselves to fit the Friedel oscillation.This gives well-defined Bragg Peaks for scattering

in reciprocal space, and includes those at 2kF but

no periodic repetition in real space

 Charge density waves refer to the buildup ofcharge at the periodicity of the Friedel oscillation

 ‘Brillouin Zone–Fermi surface interaction’ is yetanother name for essentially the same phenome-non, a tendency for free materials from structureswhich respect the preferred 2kFperiodicity for theions – which puts 2kFat the surface

 ‘Fermi surface nesting’ is yet another example ofthe phenomenon It occurs for complicated crystalstructures and/or many electron metals Here,structures that have two planes of Fermi surfaceseparated by 2kFare favored, and the wavevector q

is said to be ‘nested’ between the two

 Hume-Rothery phases are alloys that have idealcomposition to allow atoms to exploit the Friedeloscillation

NFE pseudopotentials enabled the successful tion of the crystal structures of the sp3elements It istempting to use this model for ‘empirical’ potentialsimulation, using the effective pseudopotential coreradius and the electron density as fitting parameters;indeed such linear-response pair potentials do anexcellent job of describing the crystal structures of

predic-sp elements

For MD, however, there are difficulties: the tron density cannot be assumed constant across a freesurface and the elastic constants (which depend onthe bulk term) do not correspond to long-wavelengthphonons (which do not depend on the bulk term).Since most MD calculations of interest in radiationdamage involve defects (voids, surfaces), phonons,and long-range elastic strains, NFE pseudopotentialshave not seen much use in this area They may beappropriate for future work on liquid metals (sodium,potassium, NaK alloys)

elec-The key results from NFE theory are the following:

 The cohesive energy of a NFE system comes marily from a volume-dependent free electron gasand depends only mildly on the interatomic pairpotential

pri- The pair potential is density dependent: structures

at the same density must be compared to

deter-mine the minimum energy structure

 The pair potential has a long-ranged, oscillatory

tail

 These potentials work well for understanding

crystal structure stability, but not for simulating

defects where there is a big change in electron

density

 The reference state is a free electron gas:

descrip-tion of free atoms is totally inadequate

1.10.5.3 Embedded Atom Methods and

Density Functional Theory

In the density functional theory (DFT), the

elec-tronic energy of a system can be written as a

func-tional of its electron density:

U ¼ F½rðrÞ

The embedded atom model (EAM)7postulates that

in a metal, where electrostatic screening is good, one

might approximate this nonlocal functional by a local

function And furthermore, that the change in energy

due to adding a proton to the system could be treated

by perturbation theory (i.e., no change in r) Hence,

the energy associated with the hydrogen atom would

depend only on the electron density that would exist

at that pointr in the absence of the hydrogen

UHðrÞ¼ FHðrðrÞÞThe idea can be extended further, where one con-

siders the energy of any atom ‘embedded’ in the

effective medium of all the others.8Now, the energy

of each (ith) atom in the system is written in the

same form,

Ui ¼ FiðrðriÞÞ

To this is added the interionic potential energy,

which in the presence of screening, they took as a

short-ranged pairwise interaction This gives an

expression for the total energy of a metallic system:

To make the model practicable, it is assumed that r

can be evaluated as a sum of atomic densities fðrÞ, that

is, rðriÞ ¼PjfðrijÞ and that F and V are unknown

functions which could be fitted to empirical data The

‘modified’ EAM incorporates screening of f and

addi-tional contributions to r from many-body terms

1.10.6 Many-Body Potentials and Tight-Binding Theory

1.10.6.1 Energy of a Part-Filled Band

An alternate starting point to defining potentials istight-binding theory As this already has localizedorbitals, it gives a more intuitive path from quantummechanics to potentials Consider a band with a den-sity of electronic states (DOS) n(E) from which thecohesive energy becomes

U ¼

ðEf

ðE  E0ÞnðEÞ dEwhere E0is the energy of the free atom, which to afirst approximation lies at the center of the band.For example, a rectangular d-band describing bothspin states and containing N electrons, width W hasnðEÞ ¼ 10=W , and EF¼ W ðN  5Þ=10 þ E0whence(Figure 3),

U¼ Nð10  NÞ

This gives parabolic behavior for a range of related properties across the transition metal group,such as melting point, bulk modulus, and Wigner–Seitz radius For a single material, the cohesion isproportional to the bandwidth Even for more com-plex band shapes, the width is the key factor indetermining the energy

energy-The width of the band can be related to its secondmoment9here:

Figure 3 Density of states for a simple rectangular band model.

S ¼ hijVijiihðrijÞ ¼ hijVijjiThe electron eigenenergies come from diagonalizing

this matrix (there are, of course, cleverer ways to

do this than brute force) Typically, we can use

them to create a density of states, n(E), which can be

used to determine cohesive energy (as above)

The width of this band depends on the off-diagonal

terms (in the limit of h¼ 0, the band is a delta

function) One can proceed by fitting S and h, or

move to a further level of abstraction

1.10.6.2 The Moments Theorem

A remarkable result by Ducastelle and

Cyrot-Lackmann10 relates the tight-binding local density

of states to the local topology If we describe the

density of states in terms of its moments where the

basis of the eigenvectors But, the trace of a matrix is

invariant with respect to a unitary transformation,

that is, change of basis vectors to atomic orbitals i

By counting the number of such chains, we can build

up the local density of states

Unfortunately, algorithms for rebuilding DOS anddeducing the energy using higher moments tend toconverge rather slowly, the best being the recursionmethod.11

The zeroth moment simply tells us how manystates there are

The first moment tells us where the band center is.Taking the band center as the zero of energy, thesecond moment is as follows:

tight-h h

0 0

h s h

0 0

h h s h h

0 0

h s h

0 0

h h s

Figure 4 Matrix of onsite and hopping integrals for a

planar five-atom cluster – in tight binding this gives five

eigenstates, each of which contributes one level to the

‘density of states’: five delta functions In an infinite solid,

the matrix and number of eigenstates become infinite, so

the density of states becomes continuous Of course, tricks

then have to be employed to avoid diagonalizing the

matrix directly.

1 hhhhh

2 hhshh

Figure 5 Dashed and dotted lines show two of the chains

of five hops which contribute to the fifth moment of the tight-binding density of states.

This gives the relationship between cohesive

energy, bandwidth, and number of neighborsðziÞ In

the simplest form Wi/

ffiffiffiffiffiffiffi

mðiÞ2q

Ui¼ Nð10  NÞ

20 Wi / pffiffiffiz

½4

that is, the band energy is proportional to the square root of

the number of neighbors

Note that this is only a part of the total energy

due to valence bonding There is also an

electrosta-tic interaction between the ions and an

exclusion-principle repulsion due to nonorthogonality of the

atomic orbitals – it turns out that both of these can be

written as a pairwise potential VðrÞ

The moments principle was laid out in the late

1960s.12 To make a potential, the squared hopping

integral is replaced by an empirical pair potential

fðrijÞ, which also accounts for the prefactor ineqn

[4] and the exact relation between bandwidth and

second moment Once the pairwise potential VðrijÞ

is added, these potentials have come to be known as

j

fðrijÞ

s

½5

where V and f are fitting functions

Further work14 showed that the square root lawheld for bands of any shape provided that there was

no charge transfer between local DOS and that theFermi energy in the system was fixed For bcc, atoms

in the second neighbor shell are fairly close, and arenormally assumed to have a nonzero hopping integral.Notice that the first three moments only containinformation about the distances to the shells of atomswithin the range of the hopping integral Therefore, athird-moment model with near neighbor hoppingcould not differentiate between hcp and fcc (in fact,only the fifth moment differentiates these in a near-neighbor hopping model!) This led Pettifor to con-sider a bond energy rather than a band energy, andrelate it to Coulson’s definition of chemical bondorders in molecules.15Generalizing this concept leads

to a systematic way of going beyond second momentsand generating bond order potentials

One can investigate the second-moment hypothesis

by looking at the density of states of a typical transitionmetal, niobium, calculated by ab initio pseudopotentialplane wave method,Figure 6, and comparing it withthe density of states at extremely high pressure Thesimilarity is striking: as the material is compressed,the band broadens but the structure with five peaksremains unchanged The s-band is displaced slightly

to higher energies at high pressure, but still provides a

Energy (eV) 0

0.5

1 1.5

low, flat background, which extends from slightly

below the d-band to several electron volts above

1.10.6.3 Key Points

 In a second-moment approximation, the cohesive

(bond) energy is proportional to the square root of

the coordination

 Other contributions to the energy can be written as

pairwise potentials

1.10.7 Properties of Glue Models

The embedded atom and FS potentials fall into a

general class of potentials of the form:

with a many-body cohesive part and a two-body

repulsion Both fðrijÞ and V ðrijÞ are short ranged,

so MD with these potentials is at worst only as costly

as a simple pair potential (computer time is

propor-tional to number of particles)

These models are sometimes referred to as glue

potentials,16the many-body F term being thought of

as describing how strongly an atom is held by the

electron ‘glue’ provided by its environment

The pragmatic approach to fitting in all glue

schemes is to regard the pair potential as repulsive

at short-range with long-range Friedel oscillations

Compared with most pair potential approaches, this

is unusual in that the repulsive term is longer ranged

than the cohesive one

1.10.7.1 Crystal Structure

According to the tight-binding theory on which the

FS potentials are based, the relative stability of bcc

and fcc is determined by moments above the second,

which in turn relate to three center and higher hops

These third and higher moments effects are explicitly

absent in second-moment models, and so by

implica-tion, the correct physics of phase stability is not

contained in them There is no such clear result in

the derivation of the EAM; however, since the forms

are so similar, the same problem is implicit

In glue models, energy is lowered by atoms having

as many neighbors as possible; thus, fcc, hcp, and bcc

crystal structures (and their alloy analogs) are

nor-mally stable (seeTable 1); bcc is normally stable in

potentials when the attractive region is broad enough

to include 14 neighbors, fcc/hcp are stable for rower attractive regions in which only the eight near-est bcc-neighbors contribute significantly to thebonding Indeed, without second neighbor interac-tions, bcc is mechanically unstable to Bain-type sheardistortion The fcc–hcp energy difference is related

nar-to the stacking fault energy: it is common nar-to see MDsimulations with too small an hcp–fcc energy differ-ence producing unphysically many stacking faultsand over widely separated partial dislocations.Phase transitions are observed in some potentials

As free energy calculations are complicated andtime consuming,17 it is impractical to use themdirectly in fitting – one would require the differen-tial of the free energy with respect to the potentialparameters, and this could only be obtained numer-ically Consequently, most potentials are only fitted

to reproduce the zero temperature crystal structure,and high-temperature phase stability is unknown forthe majority of published potentials One counter-example is in metals such as Ti and Zr, where thebcc structure is mechanically unstable with respect

to hcp, but becomes dynamically stabilized at hightemperatures Here, the transition temperature isdirectly related to a single analytic quantity: theenergy difference between the phases Althoughabout half of this difference comes from electronicentropy,18which suggests a temperature-dependentpotential, phase transition calculations have beenexplicitly included in some recent fits.19The case

of iron is also anomalous, as the phase transition isrelated to changes in the magnetic structure

1.10.7.2 Surface RelaxationGlue models atoms seek to have as many neighbors aspossible; therefore, when a material is cleaved, thesurface atoms tend to relax inward toward the bulk toincrease cohesion This effect also arises because of

Table 1 Neighbor distances in fcc, hcp (c =a ¼pffiffiffiffiffiffiffiffi8 =3 ), and bcc, in units of the nearest neighbor separation Structure

2 p ð6Þ pffiffiffi3

5 p ð24Þ

ffiffi 4 q

2 p (12)

ffiffiffiffi 11 3

q

3 p (8)

2 p (6)

ffiffi 8 q

3 p (18)

ffiffiffiffi 11 3

q (6) 2(6) The number of neighbors at that distance is given in brackets For fcc the shells fall at ffiffiffiffi

N

p for all integers up to 30 except one As fcc and hcp structures have identical numbers and ranges of first and second neighbors, glue or pair potentials can only distinguish them via long-range interactions.

the longer range of the repulsive part of the potential:

at a surface, the further-away atoms are absent This

is in contrast to pair potentials and in agreement with

real materials

1.10.7.3 Cauchy Relations

The functional form of the glue model places fewer

restrictions on the elastic constants of materials than

pair potentials do; for example, the Cauchy pressure

for a cubic metal is as follows20:

If the ‘embedding function’ F (minus square root in

FS case) has positive curvature, the Cauchy pressure

must be positive, as it is for most metals A minority of

metals have negative Cauchy pressure It is debatable

whether this indicates negative curvature of the

embed-ding function, or a breakdown of the glue model

There are also some Cauchy-style constraints on

the third-order elastic constants But in general, ‘glue’

type models can fit the full anisotropic linear

elastic-ity of a crystal structure

1.10.7.4 Vacancy Formation

In a near-neighbor second-moment model for fcc,

breaking one of twelve bonds reduces the cohesive

energy of each atom adjacent to the vacancy by a

factor ofð1 pffiffiffiffiffiffiffiffiffiffiffiffi11=12Þ ¼ 4:25% Other glue models

give a similar result Meanwhile, the pairwise

(repul-sive) energy is reduced by a full 1=12 ¼ 8:3%

Thus, energy cost to form a vacancy is lower in

glue-type models than in pairwise ones For actual

parameterizations, it tends to be less than half the

cohesive energy

1.10.7.5 Alloys

To make alloy potentials in the glue formalism, one

needs to consider both repulsive and cohesive terms

Thinking of the repulsive part as the NFE pair

potential, it becomes clear that the long-range

behav-ior depends on the Fermi energy This is composition

dependent – the number of valence electrons is

criti-cal, so it cannot be directly related to the individual

elements The short-ranged part should reflect the

core radii and can be taken from the elements

Despite this obvious flaw, in practice, the pairwise

part is usually concentration-independent and is

refitted for the ‘cross’ heterospecies interaction

In the EAM, the function Fidepends on the atom ibeing embedded, while the charge densityP

jfjðrijÞinto which it is embedded depends on the species andposition of neighboring atoms By contrast for FS poten-tials, the function F is a given (square root), while fðrijÞ

is the squared hopping integral, which depends onboth atoms There is no obvious way to relate thisheteroatomic hopping integral to the homoatomicones, but a practical approach is to take a geometricmean21: one might expect this form from consideringoverlap of exponential tails of wavefunctions

1.10.8 Two-Band Potentials

In the second-moment approximation to tight ing, the cohesive energy is proportional to the squareroot of the bandwidth, which can be approximated as

bind-a sum of pbind-airwise potentibind-als representing squbind-aredhopping integrals Assuming atomic charge neutral-ity, this argument can be extended to all band occu-pancies and shapes22(Figure 7)

The computational simplicity of FS and EAM lows from the formal division of the energy into a sum

fol-of energies per atom, which can in turn be evaluatedlocally Within tight binding, we should consider alocal density of states projected onto each atom Thepreceding discussion of FS potentials concentratessolely on the d-electron binding, which dominatestransition metals However, good potentials are diffi-cult to make for elements early in d-series (e.g., Sc, Ti)where the s-band plays a bigger role An extension tothe second-moment model, which keeps the idea of

locality and pairwise functions, is to consider two

separate bands, for example, s and d

This was first considered for the alkali and

alka-line earth metals, where s-electrons dominate These

appear at first glance to be close-packed metals,

forming fcc, hcp, or bcc structures at ambient

pres-sures However, compared with transition metals,

they are easily compressible, and at high pressures

adopt more complex ‘open’ structures (with smaller

interatomic distances) The simple picture of the

physics here is of a transfer of electrons from an s- to

a d-band, the d-band being more compact but higher

in energy Hence, at the price of increasing their

energy (U ), atoms can reduce their volumes (V )

Since the stable structure at 0 K is determined by

minimum enthalpy, H¼ U þ PV at high pressure, this

sd transfer becomes energetically favorable The

net result is a metal–metal phase transformation

characterized by a large reduction in volume and

often also in conductivity, since the s-band is free

electron like while the d-band is more localized

Two-bands potentials capture this transition, which

is driven by electronic effects, even though the crystal

structure itself is not the primary order parameter

Materials such as cerium have isostructural

tran-sitions It was thought for many years that Cs also had

such a transition, but this has recently been shown to

be incorrect,23 and the two-band model was

origi-nally designed with this misapprehension in mind.24

For systems in which electrons change, from an s-type

orbital to a d-type orbital as the sample is pressurized,

one considers two rectangular bands of widths W1and

W2as shown inFigure 8with widths evaluated using

eqn [3] The bond energy of an atom may be written asthe sum of the bond energies of the two bands on thatatom as ineqn [4], and a third term giving the energy ofpromotion from band 1 to band 2 (seeeqn [8]):

For an ion with total charge T, assuming chargeneutrality,

The difference between the energies of the band ters a1and a2is assumed to be fixed The values of acorrespond to the appropriate energy levels in theisolated atom Thus, a2 a1 is the excitation energyfrom one level to another For alkali and alkaline earthmetals, the free atom occupies only s-orbitals; thepromotion energy term is therefore simply

cen-Eprom¼ n2ða2 a1Þ ¼ n2E0 ½8where E0¼ a2 a1

Thus, the band energy can be written as a function

of ni1, ni2, and the bandwidths (evaluated at eachatom as a sum of pair potentials, within the second-moment approximation) Defining,

Figure 8 Schematic picture of density of electronic states in rectangular two-band model Shaded region shows those energy states actually occupied.