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

1.10 Interatomic Potential Development G J Ackland University of Edinburgh, Edinburgh, UK ß 2012 Elsevier Ltd All rights reserved 1.10.1 Introduction 268 1.10.2 1.10.3 1.10.4 1.10.4.1 1.10.4.2 1.10.4.2.1 1.10.4.2.2 1.10.4.2.3 1.10.4.2.4 1.10.4.2.5 1.10.4.2.6 1.10.5 1.10.5.1 1.10.5.2 1.10.5.3 1.10.6 1.10.6.1 1.10.6.2 1.10.6.3 1.10.7 1.10.7.1 1.10.7.2 1.10.7.3 1.10.7.4 1.10.7.5 1.10.8 1.10.8.1 1.10.8.2 1.10.8.3 1.10.8.4 1.10.9 1.10.10 1.10.10.1 1.10.10.2 1.10.10.3 1.10.11 1.10.12 1.10.12.1 1.10.12.2 1.10.12.2.1 1.10.12.2.2 1.10.12.3 Basics Hard Spheres and Binary Collision Approximation Pair Potentials LJ Phase Diagram Necessary Results with Pair Potentials Outward surface relaxation Melting points Vacancy formation energy Cauchy pressure High-pressure phases Short ranged From Quantum Theory to Potentials Free Electron Theory Nearly Free Electron Theory Embedded Atom Methods and Density Functional Theory Many-Body Potentials and Tight-Binding Theory Energy of a Part-Filled Band The Moments Theorem Key Points Properties of Glue Models Crystal Structure Surface Relaxation Cauchy Relations Vacancy Formation Alloys Two-Band Potentials Fitting the s–d Band Model Magnetic Potentials Nonlocal Magnetism Three-Body Interactions Modified Embedded Atom Method Potentials for Nonmetals Covalent Potentials Molecular Force Fields Ionic Potentials Short-Range Interactions Parameterization Effective Pair Potentials and EAM Gauge Transformation Example: Parameterization for Steel FeCr FeC Austenitic Steel 268 269 269 269 270 270 270 270 271 271 271 271 271 271 273 273 273 274 276 276 276 276 277 277 277 277 279 280 282 282 283 284 284 285 285 286 287 288 288 288 289 289 267 268 Interatomic Potential Development 1.10.13 1.10.13.1 1.10.14 References Analyzing a Million Coordinates Useful Concepts Without True Physical Meaning Summary Abbreviations BCA bcc DFT DOS EAM fcc FS GGA hcp LDA LJ MD MEAM NFE Binary collision approximation Body-centered cubic Density functional theory Density of electronic states Embedded atom method Face-centered cubic Finnis–Sinclair Generalized gradient approximation for exchange and correlation Hexagonal closed packed Local density approximation for exchange and correlation Lennard Jones Molecular dynamics Modified embedded atom method Nearly free electron Symbols Ec Cohesive energy KB Boltzmann constant KF Fermi wavevector 1.10.1 Introduction 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 calculations are available, one might ask whether there is still a role for atomistic potentials In practice, ab initio calculations are currently limited to a few hundred atoms and a few picoseconds, or a few thousands at T ¼ K With MD and interatomic potentials, one can run calculations of millions of atoms for nanoseconds With 289 289 290 290 kinetic Monte Carlo calculation, timescales extend for years These methods still provide the only method of atomistic simulation in these regimes The issue is not whether they are still necessary; rather it is which of their predictions are correct Before embarking on any simulation, it is essential to consider whether the potential contains enough of the right physics to describe the problem at hand For example, in studying copper, one might ask about the following:  Color (total electronic structure)  Conductivity (electronic structure around Fermi level)  Crystal structure (ground state electronic structure)  Freezing mechanism (bond formation and crystal structure)  Dislocation dynamics (stacking fault energy)  Surface structures (bond breaking energies)  Interaction with primary radiation (short-range atom–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 of properties 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 dislocation dynamics, swelling, fracture, segregation, and phase transitions, much of the physics is dominated by geometry Here, finite size effects are more important than accurate energetics and empirical potentials find their home 1.10.2 Basics Materials relevant to reactors are held together by electrons An interatomic potential expresses energy in terms of atomic positions: the electronic and magnetic degrees of freedom are integrated out Most empirical potentials are derived on the basis of some approximation to quantum mechanical energies If they are subsequently used in MD of solids, then what Interatomic Potential Development 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 energetics 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: E ¼ mc : 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 implicitly 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 conditions This is a poor approximation, since the structures and energies of the equilibrium crystal and defects 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 269 consists of a short-range repulsion and a long-range attraction and has two adjustable parameters sÀ12 sÀ6 ! À VLJ ¼ e r r VMorse ¼ e½2eÀ2ar À eÀar Š ÀsÁÀ6 cohesion is on the basis of van der Waals The r interactions, while the eÀar is motivated by a screenedCoulomb potential The repulsive terms were invented ad hoc Because they have only two parameters, all simulations using just LJ or Morse are equivalent, the values of e, s, or a , which simply rescale energy and length Hence, these potentials cannot be fitted to other properties of particular materials Both LJ and Morse potentials stabilize closepacked crystal structures, and both have unphysically low basal stacking fault energies Equivalently, the energy difference between fcc and hcp is smaller than for any real material For materials modeling, this introduces a problem that the (110) dislocation structure is split into two essentially independent partials For radiation damage, this means that configurations such as stacking fault tetrahedra are overstabilized, and unreasonably large numbers of stacking faults can be generated in cascades, fracture, or deformation If the energy scale is set by the cohesive energy of a transition metal, then the vacancy and interstitial formation energy tend to be far too large; if the vacancy energy is fitted, then the cohesive energy is too small For binary systems, these potentials can stabilize a huge range of crystal structures, even without explicit temperature effects; some progress has been made to delineate these, but it is far from complete.2 Once one moves to N-species systems, there are e and s parameters for each combination of particles, that is, N(N ỵ 1) parameters Now it is possible to fit to properties 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 certain range, which can have a surprising effect on stability as shown in Figure While the LJ fluid is very well studied, the finite temperature crystal structure has only recently been resolved The problem is that the fcc and hcp structures are extremely close in energy (see Figure 1(b)), so the entropy must be calculated extremely accurately 270 Interatomic Potential Development 400 500 600 700 800 900 1000 300 200 0.5 100 ps 3/e fcc 0.4 kT/e 0.3 hcp Harmonic N = Harmonic N = 123 Harmonic+fit N = 123 LS-NPT N = 63 LS-NVT N = 63 LS-NPT N = 123 Salsburg and Huckaby 0.2 rc/s Figure Energy difference between hcp and fcc for the Lennard-Jones potential at K, as a function of cutoff (rc ) 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 This has been done by Jackson using ‘latticeswitch Monte Carlo3 (see Figure 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 interactions introduces some constraints on what data can be fitted, even in principle It is important to distinguish 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 neighbors repel one another, while longer ranged neighbors 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 0.1 1.5 rs Figure Pressure versus temperature phase diagram 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 rs3 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 give too large a ratio of surface to cohesive energy, again consistent with the failure to describe the strengthening of the surface bonds 1.10.4.2.2 Melting points With LJ, the relation between cohesive energy and melting is Ec =kB Tm % 13, other pair potentials being similar Real metals are relatively easier to melt, with values around 30 One can fit the numerical value of the e parameter to the melting point, and accept the discrepancy as a poor description of the free atom 1.10.4.2.3 Vacancy formation energy For a pair-potential, removing an atom from the lattice involves breaking bonds The cohesive energy of a lattice comes from adding the energies of those bonds Hence, the cohesive energy is equal to the vacancy formation energy, aside from a small difference from relaxation of the atoms around the vacancy Interatomic Potential Development In real metals, the vacancy formation energy is typically 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 potential, 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 r rather than r ; whence for a pair potential V ðr Þ, it follows: X 00 2 V ðrij Þxij yij C12 ¼ C66 ¼ O ij where i; j run over all atoms and O is the volume of the system In metals, this relation is strongly violated (e.g., in gold, C12 ¼ 157GPa; C44 ¼ C66 ¼ 42GPa) and simplify Quantum mechanics can be expressed in any basis set, so there are several possible starting points for such a theory Thus, a picture based on atomic orbitals (i.e., tight binding) or plane waves (i.e., free electrons) can be equally valid: for potential development, the important aspect is whether these methods allow for intuitive simplification When a potential form is deduced from quantum theory, approximations are made along the way An aspect often overlooked is that the effects of terms neglected by those approximations are not absent in the final fitted potential Rather they are incorporated in an averaged (and usually wrong) way, as a distortion of the remaining terms Thus, it is not sensible to add the missing physics back in without reparameterizing the whole potential 1.10.5.1 Free Electron Theory For a free electron gas with Fermi wavevector kF, the energy U of volume O is5 U¼ 1.10.4.2.5 High-pressure phases Many materials change their coordination on pressurization (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 proportional 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, bonding comes from lowering the energy of the electrons, and the number of electrons per atom does not change even if the coordination does 271 h"2 kF5 O 10p2 me This contribution to the energy of the condensed phase generates no interatomic force since U is independent of the atomic positions However, its contribution is significant: metallic cohesive energy and bulk moduli are correct to within an order of magnitude Consideration of this term gives some justification for ignoring the cohesive energy and bulk modulus in fitting a potential, and fitting shear moduli, vacancy, or surface energies instead The discrepancy is absorbed by a putative free electron contribution which does not contribute to the interatomic atomic forces in a constant volume ensemble calculation 1.10.4.2.6 Short ranged It is worth noting that some properties that are claimed to be deficiencies of pair potentials are actually associated with short range So, for example, the diamond structure cannot be stabilized by nearneighbor potentials, but a longer ranged interaction can stabilize this, and the other complex crystal structures 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, 1.10.5.2 Nearly Free Electron Theory In nearly free electron (NFE) theory, the effects of the atoms are included via a weak ‘pseudopotential.’ The interatomic forces arise from the response of the electron gas to this perturbation To examine the appropriate form for an interatomic potential, we consider a simple weak, local pseudopotential V0 ðr Þ The total potential actually seen at ri due to atoms at rj will be as follows: X V0 rij ị ỵ W r ị V ri ị ẳ j where W r Þ describes how the electrons interact with one another Given the dielectric constant, 272 Interatomic Potential Development we can estimate W in reciprocal space using linear response theory: W ðqÞ ¼ V0 ðqÞ=eðqÞ where eðqÞ is the dielectric function In Thomas Fermi theory, eqị ẳ ỵ 4kF =pa0 q Þ where a0 is the Bohr radius A more accurate approach due to Lindhard:    4kF 1 x2 ỵ x ln j ỵ j eqị ẳ ỵ pa0 q 4x 1x where x ¼ q=2kF accounts for the reduced screening at high q, and r0 is 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 follows:  The free electron gas (including exchange and correlation)  The perturbation to the free electron band structure  Electrostatic energy (ion–ion, electron–electron, ion–electron)  Core corrections (from treating the atoms as pseudopotentials)  Soft phonon instabilities are an extreme case of the Kohn anomaly They arise when the lowering of energy is so large that the phonon excitation has negative energy In this case, the phonon ‘freezes in,’ and the material undergoes a phase transformation to a lower symmetry phase  Quasicrystals are an example where the atoms arrange 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 of charge at the periodicity of the Friedel oscillation  ‘Brillouin Zone–Fermi surface interaction’ is yet another name for essentially the same phenomenon, a tendency for free materials from structures which respect the preferred 2kF periodicity for the ions – which puts 2kF at the surface  ‘Fermi surface nesting’ is yet another example of the phenomenon It occurs for complicated crystal structures and/or many electron metals Here, structures that have two planes of Fermi surface separated by 2kF are favored, and the wavevector q is said to be ‘nested’ between the two  Hume-Rothery phases are alloys that have ideal composition to allow atoms to exploit the Friedel oscillation This arises from the singularity in the Lindhard function at q ¼ 2kF : physically, periodic lattice perturbations 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 NFE pseudopotentials enabled the successful prediction of the crystal structures of the sp3 elements It is tempting to use this model for ‘empirical’ potential simulation, using the effective pseudopotential core radius and the electron density as fitting parameters; indeed such linear-response pair potentials an excellent job of describing the crystal structures of sp elements For MD, however, there are difficulties: the electron density cannot be assumed constant across a free surface and the elastic constants (which depend on the bulk term) not correspond to long-wavelength phonons (which not depend on the bulk term) Since most MD calculations of interest in radiation damage involve defects (voids, surfaces), phonons, and long-range elastic strains, NFE pseudopotentials have not seen much use in this area They may be appropriate for future work on liquid metals (sodium, potassium, NaK alloys) The key results from NFE theory are the following:  Kohn anomalies in the phonon spectrum are particular phonons with anomalously low frequency The wavevector of these phonons is such as to match the Friedel oscillation  The cohesive energy of a NFE system comes primarily from a volume-dependent free electron gas and depends only mildly on the interatomic pair potential 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: V ðr Þ / cos 2kF r ð2kF r Þ3 Interatomic Potential Development  The pair potential is density dependent: structures at the same density must be compared to determine 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: description of free atoms is totally inadequate 1.10.5.3 Embedded Atom Methods and Density Functional Theory In the density functional theory (DFT), the electronic energy of a system can be written as a functional of its electron density: U ẳ F ẵrrị The embedded atom model (EAM) postulates 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 point r in the absence of the hydrogen UHrị ẳ FH ðrðrÞÞ The idea can be extended further, where one considers the energy of any atom ‘embedded’ in the effective medium of all the others.8 Now, the energy of each (ith) atom in the system is written in the same form, Ui ẳ Fi rri ịị 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: X X Fi rri ịị ỵ V rij ị Utot ẳ i ij To make the model practicable, it is assumed that r can be evaluated P as a sum of atomic densities fr ị, that is, rri ị ẳ j fðrij Þ and that F and V are unknown functions which could be fitted to empirical data The ‘modified’ EAM incorporates screening of f and additional contributions to r from many-body terms 273 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 is tight-binding theory As this already has localized orbitals, it gives a more intuitive path from quantum mechanics to potentials Consider a band with a density of electronic states (DOS) n(E) from which the cohesive energy becomes ð Ef U ẳ E E0 ịnEị dE where E0 is the energy of the free atom, which to a first approximation lies at the center of the band For example, a rectangular d-band describing both spin states and containing N electrons, width W has nEị ẳ 10=W , and EF ẳ W N 5ị=10 ỵ E0 whence (Figure 3), N 10 N ị W ẵ1 U ẳ 10 This gives parabolic behavior for a range of energyrelated properties across the transition metal group, such as melting point, bulk modulus, and Wigner– Seitz radius For a single material, the cohesion is proportional to the bandwidth Even for more complex band shapes, the width is the key factor in determining the energy The width of the band can be related to its second moment9 here: ð m2 ¼ ðE E0 ị2 nEị dE ẳ E0 ỵW =2 E0 W =2 10E E0 ị2 =W dE ẳ 5W =6 ½2Š To build a band in tight-binding theory, we set up a matrix of onsite and hopping integrals (Figure 4) For a simple s-band ignoring overlap, n(E) W E0 EF E Figure Density of states for a simple rectangular band model 274 Interatomic Potential Development Hopping integral h(r) h h s s Onsite term s s h h 0 h s h 0 h h s h h 0 h s h 0 h h s hhhhh hhshh Figure 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 Figure 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 S ẳ hẩi jVi jẩi i hrij ị ¼ hÈi jVi jÈj i The electron eigenenergies come from diagonalizing this matrix (there are, of course, cleverer ways to 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 h ¼ 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 CyrotLackmann10 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 pth moment is defined by ð1 E p nEị dE mp ẳ and recall that by definition X nEị ẳ dE Ei ị i i i where H is the Hamiltonian matrix written on 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 Therefore, X X ½H p Šii  mðiÞ p i i ðiÞ A sum of local moments of the density of states mp These diagonal terms of H p are given by the sum of all chains of length p of the form Hij Hjk Hkl Hni These in turn can be calculated from the local topology: a prerequisite for an empirical potential They consist of all chains of hops along bonds between atoms which start and finish at i (e.g., see Figure 5) By counting the number of such chains, we can build up the local density of states Unfortunately, algorithms for rebuilding DOS and deducing the energy using higher moments tend to converge rather slowly, the best being the recursion method.11 The zeroth moment simply tells us how many states there are The first moment tells us where the band center is Taking the band center as the zero of energy, the second moment is as follows: X X iị ẵHij Hji ẳ hrij ị2 ½3Š m2 ¼ ½H Šii ¼ j where i labels the eigenvalues, we get ð1 X X p Ep dE Ei ị dE ẳ Ei ẳ TrẵH p mp ẳ mp ẳ TrẵH p ẳ j where h is a two-center hopping integral, which can therefore be written as a pairwise potential This result, that the second moment of the tightbinding density of states can be written as a sum of pair potentials, provides the theoretical underpinning for the Finnis–Sinclair (FS) potentials Referring back to the rectangular band model, we can take the ðiÞ second moment of the local density of states m2 as a measure of the bandwidth Interatomic Potential Development This gives the relationship between cohesive energy, bandwidth, and number of neighbors ðzi Þ In qffiffiffiffiffiffiffi ðiÞ the simplest form Wi / m2 pffiffiffi N ð10 À N Þ Wi / À z ½4Š 20 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 electrostatic interaction between the ions and an exclusionprinciple repulsion due to nonorthogonality of the atomic orbitals – it turns out that both of these can be written as a pairwise potential V ðr Þ 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 fðrij Þ, which also accounts for the prefactor in eqn [4] and the exact relation between bandwidth and second moment Once the pairwise potential V ðrij Þ is added, these potentials have come to be known as FS potentials.13 X X X sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ecoh ẳ V rij ị frij ị ẵ5 Ui ẳ À i ij j where V and f are fitting functions 275 Further work14 showed that the square root law held for bands of any shape provided that there was no charge transfer between local DOS and that the Fermi energy in the system was fixed For bcc, atoms in the second neighbor shell are fairly close, and are normally assumed to have a nonzero hopping integral Notice that the first three moments only contain information about the distances to the shells of atoms within the range of the hopping integral Therefore, a third-moment model with near neighbor hopping could not differentiate between hcp and fcc (in fact, only the fifth moment differentiates these in a nearneighbor hopping model!) This led Pettifor to consider a bond energy rather than a band energy, and relate it to Coulson’s definition of chemical bond orders in molecules.15 Generalizing this concept leads to a systematic way of going beyond second moments and generating bond order potentials One can investigate the second-moment hypothesis by looking at the density of states of a typical transition metal, niobium, calculated by ab initio pseudopotential plane wave method, Figure 6, and comparing it with the density of states at extremely high pressure The similarity is striking: as the material is compressed, the band broadens but the structure with five peaks remains unchanged The s-band is displaced slightly to higher energies at high pressure, but still provides a Density of states (electrons eV–1) 1.5 0.5 -5 10 15 Energy (eV) Figure Density of states for bcc Nb Dotted blue line is at ambient pressure, and solid red line is for 32% reduction in volume The Fermi energy is set to zero in each case 276 Interatomic Potential Development low, flat background, which extends from slightly below the d-band to several electron volts above Table Neighbor distances in fcc, hcp (c=a ¼ and bcc, in units of the nearest neighbor separation pffiffiffiffiffiffiffiffi 8=3), Structure 1.10.6.3 Key Points fcc  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(12) bcc 1(8) hcp 1(12) pffiffiffi 2ð6Þ qffiffi (6) pffiffiffi (6) pffiffiffi (24) pffiffiffi (12) qffiffi (2) 2(12) qffiffiffiffi 11 (24) pffiffiffi (18) pffiffiffi 5ð24Þ pffiffiffi (8) qffiffiffiffi 11 (6) 2(6) The number of neighbors pffiffiffiffi at that distance is given in brackets For fcc the shells fall at N 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 1.10.7 Properties of Glue Models The embedded atom and FS potentials fall into a general class of potentials of the form: " # X X X Ui ẳ Fi frij ị þ V ðrij Þ i j j with a many-body cohesive part and a two-body repulsion Both fðrij Þ and V ðrij Þ are short ranged, so MD with these potentials is at worst only as costly as a simple pair potential (computer time is proportional to number of particles) These models are sometimes referred to as glue potentials,16 the 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 implication, 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 normally stable (see Table 1); bcc is normally stable in potentials when the attractive region is broad enough to include 14 neighbors, fcc/hcp are stable for narrower attractive regions in which only the eight nearest bcc-neighbors contribute significantly to the bonding Indeed, without second neighbor interactions, bcc is mechanically unstable to Bain-type shear distortion The fcc–hcp energy difference is related to the stacking fault energy: it is common to see MD simulations with too small an hcp–fcc energy difference producing unphysically many stacking faults and over widely separated partial dislocations Phase transitions are observed in some potentials As free energy calculations are complicated and time consuming,17 it is impractical to use them directly in fitting – one would require the differential of the free energy with respect to the potential parameters, and this could only be obtained numerically Consequently, most potentials are only fitted to reproduce the zero temperature crystal structure, and high-temperature phase stability is unknown for the majority of published potentials One counterexample is in metals such as Ti and Zr, where the bcc structure is mechanically unstable with respect to hcp, but becomes dynamically stabilized at high temperatures Here, the transition temperature is directly related to a single analytic quantity: the energy difference between the phases Although about half of this difference comes from electronic entropy,18 which suggests a temperature-dependent potential, phase transition calculations have been explicitly included in some recent fits.19 The case of iron is also anomalous, as the phase transition is related to changes in the magnetic structure 1.10.7.2 Surface Relaxation Glue models atoms seek to have as many neighbors as possible; therefore, when a material is cleaved, the surface atoms tend to relax inward toward the bulk to increase cohesion This effect also arises because of Interatomic Potential Development 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: " #2 d2 F ðxÞ X 2 C12 C66 ẳ f r ịxj dx j 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 embedding 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 elasticity 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 ffi adjacent to the vacancy by a p factor of 11=12ị ẳ 4:25% Other glue models give a similar result Meanwhile, the pairwise (repulsive) 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 1.10.7.5 277 the atom i In the EAM, the function Fi depends on P being embedded, while the charge density j fj ðrij Þ into which it is embedded depends on the species and position of neighboring atoms By contrast for FS potentials, the function F is a given (square root), while fðrij Þ is the squared hopping integral, which depends on both atoms There is no obvious way to relate this heteroatomic hopping integral to the homoatomic ones, but a practical approach is to take a geometric mean21: one might expect this form from considering overlap of exponential tails of wavefunctions 1.10.8 Two-Band Potentials In the second-moment approximation to tight binding, the cohesive energy is proportional to the square root of the bandwidth, which can be approximated as a sum of pairwise potentials representing squared hopping integrals Assuming atomic charge neutrality, this argument can be extended to all band occupancies and shapes22 (Figure 7) The computational simplicity of FS and EAM follows from the formal division of the energy into a sum of energies per atom, which can in turn be evaluated locally Within tight binding, we should consider a local density of states projected onto each atom The preceding discussion of FS potentials concentrates solely on the d-electron binding, which dominates transition metals However, good potentials are difficult to make for elements early in d-series (e.g., Sc, Ti) where the s-band plays a bigger role An extension to the second-moment model, which keeps the idea of DOS 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 behavior depends on the Fermi energy This is composition dependent – the number of valence electrons is critical, 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 Ef E DOS Ef E Figure In second-moment tight binding, the band shape is assumed constant at all atoms, the effect of changing environments being a broadening of the band 278 Interatomic Potential Development locality and pairwise functions, is to consider two separate bands, for example, s and d This was first considered for the alkali and alkaline earth metals, where s-electrons dominate These appear at first glance to be close-packed metals, forming fcc, hcp, or bcc structures at ambient pressures 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 K is determined by minimum enthalpy, H ¼ U þ PV at high pressure, this sÀd 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 transitions 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 originally 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 W1 and W2 as shown in Figure with widths evaluated using eqn [3] The bond energy of an atom may be written as the sum of the bond energies of the two bands on that atom as in eqn [4], and a third term giving the energy of promotion from band to band (see eqn [8]): X Wi1 ni1 ðni1 À N1 Þ Ubond ẳ 2N1 i Wi2 ỵ ni2 ni2 N2 ị ỵ Eprom ẵ6 2N2 where N1 and N2 are the capacities of the bands (2 and 10 for s and d respectively) and ni1 and ni2 are the occupation of each band localized on the ith atom For an ion with total charge T, assuming charge neutrality, ni1 ỵ ni2 ẳ T ẵ7 The difference between the energies of the band centers a1 and a2 is assumed to be fixed The values of a correspond to the appropriate energy levels in the isolated atom Thus, a2 À a1 is the excitation energy from one level to another For alkali and alkaline earth metals, the free atom occupies only s-orbitals; the promotion energy term is therefore simply Eprom ¼ n2 a2 a1 ị ẳ n2 E0 ẵ8 where E0 ¼ a2 À a1 Thus, the band energy can be written as a function of ni1 , ni2 , and the bandwidths (evaluated at each atom as a sum of pair potentials, within the secondmoment approximation) Defining, i ¼ ni1 À ni2 ½9Š and using eqn [7], we can write as follows: D(E) N/W1 + N/W2 d-band N/W1 s-band a1 − W1/2 a1 Ef a2 − W2/2 a1 + W1/2 E a2 a + W /2 2 Figure Schematic picture of density of electronic states in rectangular two-band model Shaded region shows those energy states actually occupied Interatomic Potential Development X  T À i ðWi1 À Wi2 ị Wi1 ỵ Wi2 ị 4 i   2  ỵ T Wi1 Wi2 ỵ þ i 8 N1 N2  T Wi1 Wi2 þ i À N1 N2 T À i E0 ½10Š þ Although this expression looks unwieldy, it is computationally efficient, requiring only two sums of pair potentials for Wi and a minimization at each site independently with respect to i , which can be done analytically In addition to the bonding term, a pairwise repulsion between the ions, which is primarily due to the screened ionic charge and orthogonalization of the valence electrons, is added In general, this pair potential should be a function of i and j But to maintain locality, one has to write this pairwise contribution to the energy in the intuitive form, as the sum of two terms, one from each ‘band,’ proportional to the number of electrons in that band: Ubond ¼ V rij ị ẳ ni1 ỵ nj ịV1 rij ị ỵ ni2 ỵ nj ịV2 rij ị ẵ11 We rearrange this to give the energy as a sum over atoms: " # X X X Upair ¼ ni1 V1 rij ị ỵ ni2 V2 rij ị ẵ12 i j 6¼i j 6¼i The total energy is now simply Utot ẳ Upair ỵ Ubond ẵ13 This depends on i , which takes whatever values to minimize the energy, s @Utot ¼0 @ i ½14Š explicitly for i0 independently at each atom, with the constraint that ji j cannot be greater than the total number of electrons T per atom The fixed capacities of the bands (N1 and N2 ) can also prohibit the realization of i0 It is therefore necessary to limit the values which i may have to those where i0 does not imply negative band occupation The expressions for i involves only constants and sums of pair potentials, and can be evaluated independently at each atom at a similar computational cost to a standard many-body type potential 279 The variational property expressed in eqn [14] can be exploited to derive the force on the ith atom: dUtot fi ¼ À dri @Utot @Utot @ ¼À j À @ri  @ dri @Utot ẳ j @ri  ẵ15 Hence, the force is simply the derivative of the energy at fixed  Basically, this is the Hellmann– Feynman theorem25 which arises here because  is essentially a single parameter representation of the electronic structure This result means that, like the energy, the force can be evaluated by summing pairwise potentials Hence, the two-band second-moment model is well suited for large-scale MD The force derivation itself is somewhat tedious, and the reader is referred to the original papers There is no Hellman–Feynman type simplification for the second derivative, so analytic expressions for the elastic constants in two-band models are long ranged and complicated Consequently, elastic constants are best evaluated numerically 1.10.8.1 Fitting the s–d Band Model To make a usable potential, the functional forms of f and V must be chosen Although this is somewhat arbitrary, the physical picture of hopping integral and screened ion–ion potential suggests that both should be short ranged, continuous, and reasonably smooth Popular choices are cubic splines, power series, and Slater orbitals The promotion energy E0 is simply that required to promote an electron from the s level into the d level of an isolated atom The band capacities are Ns ¼ 2, Nd ¼ 10 and the total number of electrons per atom depends on the element, for example, in Cs T ¼ In the first application, parameters were fitted to the energy–volume relations for bcc and fcc cesium and the transition pressure between phases Cs-II and Cs-III Figure shows the energy–volume curves for the fcc and bcc structures calculated using the model At ambient conditions for Cs, there are no d-electrons, so the fitting process is just like a normal FS potential This determines the s-band parameters, and the d-band parameters are then fitted to the high-pressure phase data, where both s- and d-electrons contribute Although an isostructural phase transition is likely to be accompanied by instability of the bulk modulus, there may also be a precursor shear instability Thus, Interatomic Potential Development 1.0 0.5 0.0 -0.5 9.0 fcc lattice parameter (Å)/ cohesive energy (eV per atom) h 280 -1.0 Energy per atom (eV) 1.5 fcc bcc 4.3 GPa 1.0 0.5 0.0 -0.5 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 -1.0 -1.5 50 100 150 Volume per atom (Å3) 200 Figure Top: Variation of i with compression, showing s!d transfer in model cesium (T ¼ 1) Bottom: An energy– volume curve for the two-band potential The minimum lies at –1.3163 eV and 115.2 A˚3 per atom The gradient of the straight dash–dotted line is the experimental Cs-II–Cs-III transition pressure In reality, cesium also has a bcc–fcc phase transition at 2.3 GPa; however, first principles calculations show that these two structures are almost degenerate in energy at K the mechanism may involve shearing rather than isostructural collapse, particularly if a continuous interface between the two phases exists, as in a shockwave With the two-band model, the transition is first order, the volume collapse occurring before the bulk modulus becomes negative in the unstable region Although the shear and tetragonal shear decrease in the unstable region, neither actually goes negative The two-band model is applicable to transition metals, but since the d-band is occupied at all pressures, electron transfer is continuous and there is no phase transition This makes the empirical division of the energy into s and d components challenging However, once appropriately scaled for ionic charge and number of electrons T, in principle, the method could be extended to alloys with noninteger T The results of such an extrapolation are extraordinarily good (Figure 10), considering that there is no fitting to any material other than Cs The extrapolation breaks down at high Z where the amount of sp hybridization is not fully captured in the parameterization As with FS, no information about band shape is included, and so the sequence of crystal structures cannot be reproduced While the extrapolated potentials not represent the optimal parameterization for specific transition Number of sd electrons 10 11 Figure 10 Extrapolation of two-band model fitted to Cs to various {6s5d}-band materials, with the lengths being scaled according to the Fermi vector (Z À1=3 ) and the energies by (Z 1=2 ) Lattice parameters are shown by the dashed line (calculated) and squares (experiment), and the cohesive energies by the solid line (calculated) and circles (fcc at experimental density) metals, the recovery of the trends across the group lends weight to the idea that the two-band model correctly reproduces the physics of this series The s–d two-band approach has also been applied with considerable success by considering the s-band as an alloying band.26 This has been applied to the FeCr system, which we discuss in more detail later 1.10.8.2 Magnetic Potentials The two-band approach can be applied to magnetic materials, where the bands spin up and spin down bands have the same capacity (N ¼ Nd" ¼ Nd# ¼ 5) If in addition we assume that the bands have the same width and shape (see, e.g., Figure 11), there is a remarkable collapse of the model onto the singleband EAM form, with a modified embedding function The formalism here extends to the two-band model, but the physics is analogous to other magnetic potentials.27 For simplicity, consider a rectangular d-band of full width W centered on E0 The bond energy for a single spin-up band relative to the free atom is given by    "  ð Ef ¼ Z=N À W " " Z À W =2 NE=W dE ¼ Z U ¼ N ÀW =2 ½16Š where Z" is the occupation of the band and uparrows denote ‘spin up.’ Interatomic Potential Development 281 2.5 Density of states 1.5 0.5 -8 -6 -4 -2 Energy (eV) Figure 11 bcc iron density of states for majority and minority spin bands from ab initio spin-dependent GGA pseudopotential calculations with 4913 k ¼ points, adjusted so that the Fermi energy lies at the zero of energy The two-band model assumes that the bands have the same shape and width, but are displaced in energy relative to each other: the figure shows this to be reasonable To describe the ferromagnetic case, it is assumed that there are two independent d-bands corresponding to opposite spins, and that these can be projected onto an atom to form a local density of states For a free atom atomic case, Hund’s rules determine a high-spin case (e.g., S ¼ for iron), and there is an energy U x associated with transferring an electron to a lower spin state In the solid, the simplest method is to set U x to be proportional to the spin with the coefficient of proportionality being an adjustable parameter, E0 U x ẳ E0 jZ" Z# j ẵ17 Defining the spin, S ¼ jZ" À Z# j and assuming charge neutrality (T ẳ Z" ỵ Z# ị), the two-d-band binding energy on a site i is then as follows: Ui ẳ Ui" ỵ Ui# ỵ Uix ẳ ỵ Wi T ỵ Si2 ị TWi =2 E0 Si 4N ½18Š Differentiating this equation about Si gives us the optimal value for the magnetization of a given atom of Si ¼ 2NE0 =Wi , and the many-body energy of an atom with T ¼ 6, N ¼ (suppressing the i label) as U ¼ À6W =5 À 5E02 =W E0 =Wi ¼ À2W =5 À 4E0 E0 =Wi ! 0:4 0:4 ½19Š where neither band is allowed to have occupancy more than or less than For a material with T d-electrons (where T > 5), transfer of electrons between the spin bands becomes advantageous for W > 10E0 =ð10 À T Þ For smaller W, the spin " band is full and the energy is simply proportional to the bandwidth of the # band as in the FS model Similar cases apply to the T < case when the minority band may be empty There has been some controversy about the expression for U x In the two-band model, this is a promotion energy from the minority spin band to the majority In the atomic case, it is the energy to violate Hund’s rule, and the implicit reference state is the high-spin atom Electron transfer is bound by the number of electron, so the function has discontinuous slope at Si ¼ 0:4 By contrast, the approach of Dudarev and coworkers27 uses a Stoner model for the spin energy, which introduces quadratic and quartic terms in U x ðSi Þ In that case, the implicit reference state is the nonmagnetic solid, and any value of Si is acceptable Within the second-moment model, the bandwidth W is given by the square root of r, the sum of the squares of the hopping integral Applying this, and the usual pairwise repulsion V ðr Þ, gives an expression for the two-band energy 282 Interatomic Potential Development Ui ¼ X j À B= V ðrij Þ À pffiffiffiffiffi rj pffiffiffiffiffi rj H ð2W À 5E0 Þ À 4E0 H 5E0 2W ị ẵ20 where H is the Heaviside step function, B is a constant, and the zero of energy corresponds to the nonmagnetic atom Note that this form does not explicitly include S, and that it has the pEAM form with an embedding function FI xị ẳ x ð1 À B=xÞ Although this model incorporates magnetism and provides a way to calculate the magnetic moment at each site, it is possible to use it without actually calculating S The additional many-body repulsive term is similar to, for example, the many-body potential method of Mendelev et al It is also interesting that in the original FS paper, it was not possible to fit the properties of the magnetic elements Fe and Cr; an extra term was added ad hoc Later parameterizations of FS potentials for iron with a pure square root for F have not exactly reproduced the elastic constants.28 The implication of this work is that for secondmoment type models, there should be a one-toone relation between the local density r and the magnetic moment Figure 12 shows this relation for two parameterizations, r from Dudarev–Derlet and Mendelev et al and the magnetic moment calculated with spin-dependent DFT projected onto atoms It shows that there are two cases For atoms associated with local defects, the density varies quite sharply with r, while for the crystal under pressure the variation is slower It is noteworthy that the same broad features are present in both potentials, even though the Mendelev et al potential was fitted without consideration of magnetic properties, albeit with a FS-type embedding function This suggests that the magnetic effects were unwittingly captured in the fitting process 1.10.8.3 Nonlocal Magnetism The two-band model projects the magnetism onto each atom It does not properly describe magnetic interactions, so it cannot distinguish between ferro, para, and antiferromagnetism In order to so, we need to include an interionic exchange term Pauli repulsion arises from electron eigenstates being orthogonal While its nature on a single atom is complex, its interatomic effects can be modeled as a pairwise effect of repulsion between electrons of similar spins The secondary effect of magnetization is that there are more electrons in one band than in another, and more same-spin electron pairs to repel one another, and so the repulsion between those bands is enhanced Conceptually, this can be captured in two pairwise effects, the standard nonmagnetic screened-Coulomb repulsion of the ions plus the core–core repulsion, and an additional Si dependent term arising from Pauli repulsion between like-spin electrons V rij ị ẳ V0 rij ị ỵ Si" Sj" ỵ Si# Sj# ịVm rij ị ẵ21 Note that an antiferromagnetic state with Si ỵ Sj ẳ would have a lower repulsive energy In the tightbinding picture, this would be compensated by a much reduced hopping integral, and hence, lower W If we insist on Si > 0, then we suppress these solutions and can model ferromagnetic or diamagnetic iron Also, as with DFT-GGA/LDA, the spin is Ising-like At the time of writing, no good parameterization of this type of potential exists The difficulty is that determining the spins Si is a nonlocal process: the optimal value of the spin on site i depends on the spin at site j The only practical way to proceed appears to be to treat the spins as dynamical variables, in which case it is probably better to treat them as noncollinear Heisenberg moments 1.10.8.4 Three-Body Interactions It is worth noting that the ‘glue’ type potentials cannot be expanded in a sum of two-body, threebody, four-body, etc terms Three-body terms enter into the free electron picture through nonlinear response of the electron gas, and into the tightbinding picture in the fourth moment description and beyond At constant second moment, increasing the fourth and higher even moments of the DOS tends to lead to a bimodal distribution Bimodal distributions will be favored by materials with half-filled bands Thus, three-body terms are likely to be important in structures with few small-membered rings of atoms, and hence, small low moments The bcc structure is a borderline example of this, but the classic is the diamond structure Diamond has no rings of less than six atoms, resulting in a strongly bimodal DOS This bimodal structure in the tight-binding representation is also interpreted as bonding and antibonding states in a covalent picture Interatomic Potential Development 283 Magnetic moments in iron versus r Dudarev/Derlet potential case2 bcc Fe perfect crystal Vacancy 100 sia 110 sia 111 sia Octa Tetra m /mB 0 0.5 1.5 r (a) 2.5 Magnetic moments in iron versus r Ackland/Mendelev potential bcc Fe perfect crystal Vacancy 100 sia 110 sia 111 sia Octa Tetra m /mB 20 30 40 (b) 50 60 70 r Figure 12 Relationship between ab initio calculated magnetic moment per atom, and r from (a) Dudarev–Derlet magnetic potential and (b) Mendelev et al Interatomic potentials for carbon and silicon fall into this category However, once a band gap is opened, the Fermi energy and perfect screening are lost, and the rigid band approximation is less appropriate 1.10.9 Modified Embedded Atom Method The modified embedded atom method (MEAM) is an empirical extension of EAM by Baskes, which 284 Interatomic Potential Development includes angular forces As in the EAM, there are pairwise repulsions and an embedding function In the EAM, the ri is interpreted as a linear supposition of species-dependent spherically averaged atomic electron densities (here designated by fðr Þ); in MEAM ri is augmented by angular terms The spherically symmetric partial electron density rð0Þ is the same as the electron density in the EAM: 0ị ri ẳ X fð0Þ ðrij Þ j where the sum is over all atoms j, not including the atom at the specific site of interest i The angular contributions to the density are similar to spherical harmonics: they are given by similar formulas weighted by the x; y; and z components of the distances between atoms (labeled by a; b; g): 32 X X a ij r1ị ị2 ẳ f1ị rij Þ rij a j 32 32 X X X a b ij ij ðrð2Þ Þ2 ¼ fð2Þ ðrij Þ À fð2Þ ðrij Þ5 r ij j j a;b 32 32 X X X aij bij gij ð3Þ 3ị 2ị r ị ẳ f ðrij Þ f ðrij Þ5 j rij3 j a;b;g The fðlÞ are so-called ‘atomic electron densities,’ which decrease with distance from the site of interest, and the a; b; and g summations are each over the three coordinate directions (x, y, z) The functional forms for the partial electron densities were chosen to be translationally and rotationally invariant and are equal to zero for crystals with inversion symmetry about all atomic sites Although the terms are related to powers of the cosine of the angle between groups of three atoms, there is no explicit evaluation of angles, and all the information required to evaluate the MEAM is available in standard MD codes Typically, atomic electron densities are assumed to decrease exponentially, that is, flị Rị ẳ exp Àb ðlÞ ðR=re À 1Þ where the decay lengths (re and b ðlÞ ) are constants While there is no derivation of the MEAM from electronic structure, it also introduced the physically reasonable idea of many-body screening, which is missing in pair-functional forms such as EAM Thus, fðRij Þ is reduced by a screening factor determined by the other atoms k forming three-body triplets with i and j : primarily those lying between i and j This eliminates the need for an explicit cut-off in the ranges of V ðr Þ and fðlÞ ðr Þ For close-packed materials, the improvement of MEAM over standard EAM is marginal; the angular terms come out to be small For sp-bonded materials, a large three-body term can stabilize tetrahedrally coordinated structures, but since the physics arises from preferred 109 angles rather than preferred fourfold coordination, it suffers problems similar to Stillinger– Weber type potentials (discussed below) Very high angular components enable one to fit the complex phases of lanthanides and actinides It is tempting to attribute this to the correct capture of the f-electron physics, although the additional functional freedom may play a role in enabling fits to low symmetry structures 1.10.10 Potentials for Nonmetals While much of the work on structural materials has concentrated on metals, there are important issues involving nonmetallics for coatings, corrosion, and fuel In this section, we review other types of potentials 1.10.10.1 Covalent Potentials Empirical potentials for covalent materials have been much less successful than for metals As with the NFE pair potentials, the bulk of the energy is contained in the covalent bond, and potentials which well describe distortions from fourfold coordination tend to fail when applied to other bond situations, such as surfaces, high pressure phases, or liquids A commonly used example, the Stillinger–Weber potential,29 is written as  2 X XX V ðrij Þ þ F ðrij ÞF ðrik Þ À cos yijk Ui ¼ j j k where V ðrij Þ and F ðrij Þ are short-ranged pair potentials The form of the three-body term, with its minimum at 109 , ensures the stability of a tetrahedrally bonded network The stacking fault energy (equivalently, the difference between cubic and hexagonal diamond) is zero, so the ground state crystal structure is not unique; however, this is not far from correct, and hexagonal diamond can be found in carbon and silicon Moreover, it has little effect in many simulations since the large kinetic barrier against cubic–hexagonal phase transitions prevents them occurring in simulations Solidification and recovery from cascade damage are counterexamples Stillinger–Weber works well for fourfoldcoordinated amorphous networks, and vibrational properties of the diamond structures It gives too low Interatomic Potential Development density (and coordination) for the liquid and highpressure phases, because it fails to reproduce the rebonding of atoms at the surface to remove dangling bonds An alternate approach to stabilizing diamond via the 109 angle is to so through its tetravalent nature The simplest type is the restricted bond pair potential30: X X Ui ẳ Arij ị Brij ị j ¼1;4 j where the attractive part of the potential is summed over at most four neighbors (one per electron) This formalism describes well the collapse of the network under pressure or melting, but lacks shear rigidity (only the repulsion of second neighbors provides shear rigidity) There is also some ambiguity over which four neighbors should be chosen, which makes implementation difficult An embellishment on this is the bond-charge model, in which the electrons in the bonds repel one another This adds a three-center term of the form X Cðrjk Þ i where j and k are bonded neighbors of i This approach avoids explicitly introducing the tetrahedral angle into the potential Note that although this term is associated with atom i (and is often interpreted as a bondbending term at i ), in the simplest form forces derived from this term are independent of i The problem of defining ‘bonded neighbors’ can be circumvented, in the spirit of the embedded atom method, by having an embedding function that effectively cuts off after the bonding reaches four neighbors worth31 as in the Tersoff approach: X X Aðrij Þ À Bðrij Þ Ui ẳ j where Brij ị ẳ frij ị j X Gðrik ; rjk ; rij Þ k The bond ij is weakened by the presence of other bonds ik and jk involving atoms i and j The protetrahedral angular dependence is still necessary to stabilize the structure, and further embellishment by Brenner32 corrects for overbinding of radicals These potentials give a good description of the liquid and amorphous state, and have become widely used in many applications, in addition to elements such as Si and C, as well as covalently bonded compounds such as silicon carbide33 and tungsten carbide.34 1.10.10.2 285 Molecular Force Fields Potentials based on bond stretching, bond bending, and long-ranged Coulomb interactions are widely used in molecular and organic systems Chemists call these potentials ‘force fields.’ They cannot describe making and breaking chemical bonds, but by capturing molecular shapes, they describe the structural and dynamical properties of molecules well There are many commercial packages based on these force fields, for example, CHARMM35 and AMBER.36 They are primarily useful for simulating molecular liquids and solvation, but have seen little application in nuclear materials, on account of the long-range Coulomb forces, which are costly to evaluate in large simulations 1.10.10.3 Ionic Potentials With no delocalized electrons, ionic materials should be suitable for modeling with pair potentials The difficulty is that the Coulomb potential is long ranged This can be tackled by Ewald or fast multipole methods, but still scales badly with the number of atoms The simplest model is the rigid ion potential, where charged (q) ions interact via long-range Coulomb forces and short-ranged pairwise repulsions V ðr ị X qi qj V rij ị ỵ Uẳ 4pe0 rij ij For example, a common form of the pair potential in oxides consists of the combination of a (6-exp) Buckingham form and the Coulomb potential: U¼ X ij A exparij ị b=rij6 ỵ qi qj 4pe0 rij where a and b are parameters and rij is the distance between atoms i and j As with other potential, various adjustments are needed in order to obtain reasonable forces at very short distance; see for example, recent reviews of UO2.37 For nuclear applications, the most commonly studied material in the open literature is UO2, which is widely used as a reactor fuel It adopts a simple fluorite structure with a large bandgap, which makes potentialfitting to get the correct crystal structure reasonably straightforward Early work fitted the potentials to lattice parameter and compressibility, and later to elastic constants and the dispersion relation The elastic constants are c11 ¼ 395 GPa, c12 ¼ 121 GPa, and c44 ¼ 64 GPa.38 As previously described, the Cauchy relation generally applies to a pairwise potential, 286 Interatomic Potential Development C12 ¼ C44 , which is seldom true experimentally for oxides However, the Cauchy relation is on the basis of the assumption that all atoms are strained equally, which is not the case for a crystal such as UO2 where some atoms not lie at centers of symmetry Thus, the violation of the Cauchy relation in UO2 can be fitted by attributing it to internal motions of the atoms away from their crystallographic positions (The violation of the Cauchy relation is similar in oxides with and without this effect, so it is debatable whether this is the correct physical effect.) The earlier potentials were based on the Coulomb charge plus Buckingham described above; more recent parameterizations include a Morse potential While this gives more degrees of freedom for fitting, having two exponential short-range repulsions with different exponents appears to be capturing the same physics twice Comparison of the parameters39 shows that the prefactor for the U–U Buckingham repulsion varies by ten orders of magnitude when fitted Moreover, the original Catlow parameterization sets this term to zero This difference tells us that the small U atoms seldom approach one another close enough for this force to be significant Even the ionic charges vary between potentials by almost a factor of two, with more recent potentials taking lower values Despite the huge disparity in parameters, the size of cascades is similar and the recombination rate is high Polarizability is not incorporated in rigid ion potentials; they will always predict a high-frequency dielectric constant of 1, which is much smaller than typical experimental values The main consequence of this for MD appears in the longitudinal optic phonon modes The solution is that ions themselves change in response to environment A standard model for this is the shell model in which the valence electrons are represented by a negatively charged shell, connected to a positively charged nucleus by a spring (Typically this represents both atomic nucleus and tightly bound electrons.) In a noncentrosymmetric environment (e.g., finite temperature), the shell center lies away from the nuclear center, and the ion has a net dipole moment – it is polarized X X qi qj V rij ị ỵ ỵ ki ri rishell ị2 Uẳ 4pe r ij i ij In this case, rij may refer to the separation between nuclei i and j or the centers of the shells associated with i and j In MD, the shells have extremely low mass, and are assumed to always relax to their equilibrium position: this is a manifestation of the Born–Oppenheimer approximation used in DFT calculation Shell model potentials,40 which capture the dipole polarizability of the oxygen molecules, were developed by Grimes and coworkers, and have been through many extensions and reparameterizations since then Again, there have been many successful parameterizations with wildly differing values for the parameters; even the sign of the charge on the U core and shell changes.41 A particular issue with ionic potentials is that of charge conservation A defect involving a missing ion will lead to a finite charge If the simulation is carried out in a supercell with periodic boundary conditions, this will introduce a formally infinite contribution to the energy The simple way to deal with this is to ignore the long wavelength (k ¼ 0) term in the Ewald sum This, under the guise of a ‘neutralizing homogeneous background charge’ is routinely done in first principles calculation Alternately, a variable charge approach can be used42 in which the extra charge is added to adjacent atoms The original approach then involved minimizing the total energy with respect to these additional charges, which is computationally demanding A promising new development is to limit the range of the charge redistribution.43 While this screening approximation is difficult to justify fully in an insulator, it is very computationally efficient or a system involving dilute charged impurities, and appears to reproduce most known features of AlO 1.10.11 Short-Range Interactions In radiation damage simulations, atoms can come much closer together than in any other application Interatomic force models are often parameterized on data that ignore very short-ranged interactions, and the physics of core wavefunction overlap is seldom well described by extarpolation In radiation damage, we are normally dealing with a case where two atoms come very close together, but the density of the material does not change By analogy with a free electron gas, we see that the energy cost of compressing the valence wavefunctions is important for high-pressure isotropic compression but absent in the collision scenario In the first case, the ‘short range’ repulsion is a many-body effect, while in the second case, it is primarily pairwise Thus, we should not expect that fitting to the high-pressure equation of state will give a good representation of the forces in the initial stages of a cascade, or even for interstitial defects Indeed, when there was no accurate measurement of interstitial Interatomic Potential Development formation energies, older potentials gave a huge range of values based on extrapolation of near-equilibrium data and uncertain partition of energy between pairwise and many-body terms Accurate values for these formation energies are now available from first principles calculations and are incorporated in the fitting; therefore, a symptom of the problem is resolved At very short range, the ionic repulsive interaction can be regarded as a screened-Coulombic interaction, and described by multiplying the Coulombic repulsion between nuclei with a screening function wðr =aÞ: V ðr Þ ¼ Z1 Z2 e wðr =aÞ 4pe0 r where wðr Þ ! when r ! and Z1 and Z2 are the nuclear charges, and a is the screening length The most popular parameterization of w is the Biersack– Ziegler potential, which was constructed by fitting a universal screening function to repulsions calculated for many different atom pairs (Ziegler 1985) The BiersackZiegler potential has the form wxị ẳ 0:1818e3:2x ỵ 0:5099e0:9423x ỵ 0:2802e0:4029x ỵ 0:02817e0:2016x where 0:8854a0 Z10:23 þ Z20:23 and x ¼ r =a and a0 ¼ 0:529 A˚ is the Bohr radius This potential must then be joined to the longer ranged fitted potential There are many ways to this, with no guiding physical principle except that the potential should be as smooth as possible Typical implementations ensure that the potential and its first few derivatives are continuous The short-range interactions arising from high pressure come mainly from isotropic compression, and should be fitted to the many-body part To achieve this division in glue models, the function F ðrÞ should become repulsive at large r, but fðr Þ should not become very large at small Rij a¼ 1.10.12 Parameterization Having deduced the functional form of the potential from first principles, it remains to choose the fitting functions and fit their parameters to empirical data Most papers simply state that ‘the potential was parameterized by fitting to ’ The reality is different 287 Firstly, one must decide what functions to use for the various terms Here, one may be guided by the physics (atomic charge density tails in EAM, square root embedding in FS, Friedel oscillations), by the anticipated usage (short-range potentials will speed up MD, and discontinuities in derivatives may cause spurious behavior), or simply by practicality (Can the potential energy be differentiated to give forces?) Secondly, one must decide what empirical data to fit Cohesive energies, elastic moduli, equilibrium lattice parameters, and defect energies are common choices Accurate ab initio calculations can provide further ‘empirical’ data, notably about relative structural stability, but now increasingly about point defect properties (It is, of course, possible to calculate all the fitting data from ab initio means Potentials fitted in this way are sometimes referred to as ab initio While this is pedantically true, the implication that these potentials are ‘better’ than those fitted to experimental data is irritating.) Ab initio MD can also give energy and forces for many different configurations at high temperature Force matching44 to ab initio data is one of best ways to produce huge amounts of fitting data There have been many attempts to fully automate this process, but to date, none have produced reliably good potentials This is in part because of the fact that although MD only uses forces (differential of the potential), many essential physical features (barrier heights, structural stability, etc.) depend on energy, which in MD ultimately comes from integrating the forces Small systematic errors in the forces, which lead to larger errors in the energies, can then cause major errors in MD predictions Furthermore, if the potential is being used for kinetic Monte Carlo, the forces are irrelevant By using least squares fitting, all the data may be incorporated in the fit, or some data may be fitted exactly and others approximately However, since the main aim of a potential is transferability to different cases, the stability of the fitting process should be checked The best way to proceed is to divide the empirical data arbitrarily into groups for fitting and control, to fit using only a part of the data, and then to check the model against the control data This process can be done many times with different divisions of fitting and control Any parameter whose value is highly sensitive to this division should be treated with suspicion Structural stability is the most difficult thing to check, since one simply has to check as many structures as possible In addition to testing the ‘usual suspects,’ fcc, bcc, hcp, A15, o-Ti, MD, or lattice 288 Interatomic Potential Development dynamics can help to check for mechanical instability of trial structures 1.10.12.1 Effective Pair Potentials and EAM Gauge Transformation Although the various glue-type potentials attribute different aspect of the physics to the N-body and pairwise terms in the potential, if one has complete freedom in choosing the functions for V ðr Þ, F ðrÞ, and f, then it is possible to move energy between the two terms Johnson and Oh noted that the EAM potential Ui ẳ X Vij rij ị ỵ Fi hX fj ðrij Þ i is invariant under a transformation hX i hX i X fj ðrij Þ ! Fi fj rij ị ỵ A fj rij ị F"i For an alloy,45 Vab r ị ẳ fb r ị f r ị Va r ị ỵ a Vb r ị fa ðr Þ fb ðr Þ ! Thus, it is possible to choose a ‘gauge’ for the potential, for example, by setting F r0 ị ẳ for some reference density r The advantage of the gauge transformation is that it simplifies fitting the potential It eliminates terms in F ðr0 Þ for pressure and elastic moduli at the equilibrium volume: these terms are nonlinear in the fitting parameters Thus, the fitting process can be done by linear algebra The downside of the gauge transformation is that it destroys the physical intuition behind the form of the many-body term Moreover, the gauge is determined by a particular reference configuration, a simple concept for elements, but one which does not transfer readily to alloys The FS potentials not have this freedom, because the function F is predefined as a square root However, they introduced the ‘effective pair potential p Veff rij ị ẳ V r ị fðr Þ= r0 where r0 is a reference configuration (typically the equilibrium crystal structure) Many of the equilibrium properties which they used for fitting depend only on this quantity In addition to gauge transformation, MD depends only on the derivative of the total energy Energy can be partitioned between atoms in any way one likes, without changing the physical results However, on-atom properties, such as the magnetic moment in magnetic potentials, typically depend on the partition of energy between atoms Such quantities not have the gauge-invariance property 1.10.12.2 Steel Example: Parameterization for 1.10.12.2.1 FeCr Steel is of particular importance to radiation damage Stainless steel is based on FeCr alloys, which have been observed by first principles calculation to exhibit unusual energy of solution For small Cr concentrations, the energy of solution is negative; however, once the concentration exceeds about 10%, it changes sign Thus, the FeCr system has a miscibility gap, but even at K, there is a finite Cr concentration in the Fe-rich region The underlying physics of this is that it is favorable for a Cr atom to dissolve in ferromagnetic Fe, provided the Cr spin is opposite to the Fe Two adjacent Cr cannot be antiparallel to each other and to the Fe matrix Thus, nearby Cr atoms suffer magnetic frustration, which leads to repulsion between Cr atoms in FeCr not seen in pure Fe or pure Cr Reproducing this effect in a potential is a challenging problem In early work, EAM was regarded as being inappropriate for bcc metals (this turned out to be due to the use of rapidly decaying functions) The original FS functional form stabilized bcc elements, but they were unable to obtain a good fit for the elastic constants in Fe and Cr without introducing further parameters The two-band model can be applied to the FeCr system46 by assuming that the material can be treated as ferromagnetic, and using s and d as the two bands They adopted the functional form of the interactions from the iron potential by Ackland and Mendelev, scaling the Cr electron density by the ratio of the atomic numbers 24/26 The CrCr potential was refitted to elastic and point defect properties As the previous Fe parameterization incorporated with effect of s-electrons in a single embedding function, the socalled s-band density of this model in fact depends only on the FeCr cross potential It described the excess energy of alloying by a many-body rather than pairwise additive effect By choosing values which favor Fe atoms with a single Cr neighbor, this potential gives the skew solubility This is an ingenious solution: magnetic frustration is essentially a ỵ N-body effect Cr atoms repel when in an Fe rich ferromagnetic environment; this is neatly captured by the longranged Slater orbital used for the s-electron It is Interatomic Potential Development debatable whether this term is really capturing physics associated with the s-band A related approach47 created a potential in which the embedding function depends directly on the local Cr concentration The skew embedding function readily reproduced the phase diagram, which was the intention of the work However, the short-range ordering and the Cr–Cr repulsion which appears to underlie the physics of radiation damage are less well reproduced 1.10.12.2.2 FeC Carbon dissolves readily in iron, producing a strengthening effect that underlies all steel The physics of this is rather complex: the solution energy is very high (6 eV), and carbon adopts an interstitial position in bcc Fe with a barrier of 0.9 eV to migration It is attracted to tensile regions of the crystal and to vacancies It is repelled from compressive regions, including interstitial atoms, although the asymmetry of the interstitial means there are some tensile sites at larger distances which are favorable First principles calculation also shows that the carbon forms covalently bonded pairs in a vacancy site, and the energy gained from the bond more than compensated for the reduced space available to the second carbon atom These criteria prove rather demanding for parameterizing FeC potentials, even though they only cover compositions with vey low carbon concentrations An early pair potential by Johnson48 proved extremely successful, and it was only once first principles calculation revealed the repulsion between C and interstitials that a major problem was revealed Although interstitial atoms are specific to radiation damage applications, there is a strong implication that the binding to other overcoordinated regions such as dislocation cores may be wrong It appears to be very difficult to obtain the correct bonding of carbon in all the cases above with smooth EAM-type functions Even in recent potentials,49 like those by Johnson, carbon binds chemically to the interstitial There is a qualitative explanation for this Electronic structure calculation50 shows that the electrons pile up between the two nearest neighbors in the octahedral configuration, essentially forming two FeC bonds However, all the simple potentials described above obtain similar bonding from all six neighbors, stabilizing the octahedral site because the tetrahedral site has only four neighbors This approach favors carbon bonding to highly coordinated defects, and underlies the bonding to interstitials An EAM 289 potential with a Tersoff–Brenner style saturation in the C cohesion has addressed this problem.51 This is tuned to saturate at two near neighbors, and so favors the octahedral site but not overcoordination As a consequence, it does not bind carbon to the interstitial 1.10.12.3 Austenitic Steel Few potentials exist for fcc iron Calculating hightemperature phase transitions is a subtle process involving careful calculation of free energy differences, which makes it difficult to incorporate in the fitting process Although the bcc–fcc transition has been reported for one EAM iron potential,52 it is probably fortuitous and has been disputed.53 In any case, it is at far higher temperature than experimentally observed Worse, it is likely that magnetic entropy plays a significant role,54 and the magnetic degrees of freedom are seldom included in potentials Some very recent progress has been made; an analytic bond order potential53 shows bcc–fcc–bcc transitions for iron and an MEAM parameterization by Baskes successfully reproduces the bcc–fcc–bcc phase transitions in iron on heating by using temperature-dependent parameters It seems certain that the challenge of austenitic steel will be receiving more attention in the next few years 1.10.13 Analyzing a Million Coordinates 1.10.13.1 Useful Concepts Without True Physical Meaning For very large simulations, imaging is a problem, since showing all atoms in a massive simulation is likely to obscure the important regions There are a number of heuristic quantities arising from the interatomic potential which can be used to pick out the atoms associated with atypical configurations of interest  Most empirical potentials define the energy per atom  The atomic level stress55 X a b sab f r À mi via vib i ¼ 2Oi j ij ij where f is the force on atom i due to atom j which determines the glass transition.56  The concept of ‘local crystal structure’ can be used to locate twin boundaries, phase transitions, etc This may be done by common neighbor analysis, or 290 Interatomic Potential Development bespoke investigation of pair and angular distribution functions to search for particular configurations.57  The balance between pair and many-body energies in glue-type models, or more significantly the various angular density functions of MEAM  The magnetization in ‘magnetic’ potentials 15 Although uniquely defined for a given potential, many of these are not well-defined concepts in electronic structure Yet, they can be extremely useful in identifying the atoms in far-from-equilibrium environments 19 1.10.14 Summary 16 17 18 20 21 22 23 Interatomic potential development is a continuing challenge for materials modeling They represent the only way to perform MD, which in turn is crucial for the nonequilibrium and off-lattice processes, which dominate radiation damage Despite best efforts, few potentials can be reliably employed to predict quantitative energies beyond where they are fitted Their most useful role is to reveal processes and topologies that might be of importance in real materials References 10 11 12 13 14 Robinson, M T Phys Rev B 1989, 40, 10717 Elliott, R S.; 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(b) Ducastelle, F J Phys 1970, 31, 1055; (c) Ducastelle, F.; Cyrot-Lackmann, F J Phys Chem Solids 1971, 32, 285; see e.g., ‘The Physics of Metals’ by J Friedel Finnis, M W.; Sinclair, J E Philos Mag A 1984, 50, 45 Ackland, G J.; Vitek, V.; Finnis, M W J Phys F 1988, 18, L153 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 (a) Coulson, C A Proc R Soc A 1939, 169, 413–428; (b) Pettifor, D G Phys Rev Lett 1989, 63, 2480–2483; (c) Finnis, M W Prog Mater Sci 2007, 52(2–3), 133–153 (a) Ercolessi, F.; Parrinello, M.; Tosati, E Philos Mag A 1988, 58, 213; (b) Robertson, I J.; Heine, V.; Payne, M C Phys Rev Lett 1993, 70, 1944–1947 Ackland, G J J Phys CM 2002, 14, 2975 Willaime, F.; Massobrio, C Phys Rev Lett 1989, 63, 2244; Phys Rev B 1991, 43, 11653–11665 Mendelev, M I.; Ackland, G J Philos Mag Lett 2007, 87, 349–359 Here, as in molecular dynamics codes, it is convenient to write the potential as a function of r2, which avoids the need for square roots, see Ackland et al Philos Mag A 1987, 56, 735 Ackland, G J.; Vitek, V Phys Rev B 1990, 41, 10324 Ackland, G J.; Finnis, M W.; Vitek, V J Phys F 1988, 18, L153 McMahon, M I.; Nelmes, R J.; Rekhi, S Phys Rev Lett 2001, 87, 255502 Ackland, G J.; Reed, S K Phys Rev B 2003, 67, 174108 (a) Hellmann, H Einfuhrung in die Quantenchemie; Franz Deuticke: Leipzig, 1937; p 285; (b) Feynman, R P Phys Rev 1939, 56(4), 340 Olsson, P.; Wallenius, J.; Domain, C.; Nordlund, K.; Malerba, L Phys Rev B 2005, 72, 214119 Dudarev, S L.; Derlet, P M J Phys Condens Matter 2005, 17, 7097 Ackland, G J.; Bacon, D J.; Calder, A F.; Harry, T Philos Mag A 1997, 75, 713–732 Stillinger, F H.; Weber, T A Phys Rev B 1985, 31, 5262 Ackland, G J Phys Rev B 1989, 40, 10351 Tersoff, J Phys Rev B 1988, 38, 9802 (a) Tersoff, J Phys Rev B 1988, 37, 6991; (b) Brenner, D W Phys Rev B 1990, 42(15), 9458 (a) Tersoff, J Phys Rev B 1994, 49, 16349 (b) Gao, F.; Weber, W J Nucl Instr Meth Phys Res B 2002, 191, 504–508 Juslin, N.; et al J Appl Phys 2005, 98, 123520 Brooks, B R.; et al J Comp Chem 2009, 30, 1545–1615 Cornell, W D.; et al J Am Chem Soc 1995, 117, 5179 Govers, K.; Lemehov, S.; Hou, M.; Verwerft, M J Nucl Mater 2007, 366, 161 Wachtman, J B.; Wheat, M L.; Anderson, H J.; Bates, J L J Nucl Mater 1965, 16(1), 39–41 Devanathan, R.; Yu, J.; Weber, W J J Chem Phys 2009, 130, 174502 (a) Lewis, G V.; Catlow, C R A J Phys C 1985, 18, 1149; (b) Jackson, R A.; et al Philos Mag A 1986, 53, 27; (c) Morelon, N D.; Ghaleb, D.; Delaye, J M.; Van Brutzel, L Philos Mag 2003, 83, 1533; (d) Basak, C B.; Sengupta, A K.; Kamath, H S J Alloys Compd 2003, 360, 210; (e) Arima, T.; Yamasaki, S.; Inagaki, Y.; Idemitsu, K J Alloys Compd 2005, 400, 43; (f) Yakub, E.; Ronchi, C.; Staicu, D J Chem Phys 2007, 127, 094508 (a) Catlow, C R A.; Grimes, R W J Nucl Mater 1989, 165, 313; (b) Grimes, R W.; et al J Am Ceram Soc 1989, 72; (c) Grimes, R W.; Miller, R H.; Catlow, C R A J Nucl Mater 1990, 172, 123; (d) Ball, R G J.; Grimes, R W J Chem Soc Faraday Trans 1990, 86, 1257; (e) Grimes, R W.; Catlow, C R A Philos Trans R Soc Lond A 1991, 335, 609; (f) Grimes, R W.; Ball, R G J.; Catlow, C R A J Phys Chem Solids 1992, 53, 475; (g) Grimes, R W Mat Res Soc Symp Proc 1992, 257, 361; (h) Busker, G.; Grimes, R W.; Bradford, M R J Nucl Mater 2000, 279, 46; (i) Busker, G.; Grimes, R W.; Bradford, M R J Nucl Interatomic Potential Development 42 43 44 45 46 47 48 49 Mater 2003, 312, 156; (j) Meis, C.; Chartier, A J Nucl Mater 2005, 341, 25 Streitz, F H.; Mintmire, J W Phys Rev B 1994, 50, 11996 Elsener, A.; et al Model Simul Mater Sci Eng 2008, 16, 025006 Ercolessi, F.; Adams, J B Europhys Lett 1994, 26, 583 Johnson, R A Phys Rev B 1989, 39, 12554; Phys Rev B 1988, 37, 3924; Phys Rev B 1990, 41, 9717 Olsson, P.; Wallenius, J.; Domain, C.; Nordlund, K.; Malerba, L Phys Rev B 2005, 72, 214119 Caro, A.; Crowson, D A.; Caro, M Phys Rev Lett 2005, 95, 075702 Johnson, R A Phys Rev 1964, 134, 1329 Lau, T T.; Forst, C J.; Lin, X.; Gale, J D.; Yip, S.; Van Vliet, K J Phys Rev Lett 2007, 98, 215501; (b) Becquart, C S.; et al Comp Mater Sci 2007, 40, 119 50 291 Domain, C.; Becquart, C S.; Foct, J Phys Rev 2004, B69, 144112 51 Hepburn, D J.; Ackland, G J Phys Rev B 2008, 78, 165115 52 Lopasso, E M.; Caro, M.; Caro, A.; Turchi, P E A Phys Rev B 2003, 68, 214205 53 Muller, M.; Erhart, P.; Albe, K J Phys CM 2007, 19, 326220 54 Muller et al JPCM 2007, 19, 326220 55 Maeda, K.; Egami, T.; Vitek, V Philos Mag A 1980, 41, 883 56 Kulp, D T.; et al Model Simul Mater Sci Eng 1992, 1, 315 57 (a) Honeycutt, J D.; Andersen, A C J Phys Chem 1987, 91, 4950; (b) Pinsook, U Phys Rev B 1998, 58, 11252; (c) Ackland, G J.; Jones, A P Phys Rev B 2006, 73, 054104  ...268 Interatomic Potential Development 1. 10 .13 1. 10 .13 .1 1 .10 .14 References Analyzing a Million Coordinates Useful Concepts Without... importance in real materials References 10 11 12 13 14 Robinson, M T Phys Rev B 19 89, 40, 10 717 Elliott, R S.; Shaw, J A.; Triantafyllidis, N Int J Solids Struct 2002, 39 (13 ? ?14 ), 3845–3856 (a)... picture Interatomic Potential Development 283 Magnetic moments in iron versus r Dudarev/Derlet potential case2 bcc Fe perfect crystal Vacancy 10 0 sia 11 0 sia 11 1 sia Octa Tetra m /mB 0 0.5 1. 5 r