PAIR POTENTIALS

Purely repulsive potentials Hard spheres at constant volume The simplest possible pair potential is the hard sphere potential defined as:

!(r) =" for r # ro0 for r > ro

$%&

(EP1)

This means no interaction for separations

r > ro and infinite repulsion for

r ! ro (see Figure 1a).

! (r)

r"#

r0 r1rr0

! (r)

(a) (b)

Fig. 1. (a) Hard sphere potential. (b) Hard sphere potential with an attractive well. This potential is often used, without proper mathematical definition, to represent crystal structures assuming a fixed volume per atom. However, only close-packed structures, such as face-centered cubic and close-packed hexagonal are stable within the hard sphere model. When used for other structures one needs some ‘sticks’ or ‘glue’ to hold the structure together, which means that purely repulsive interactions are insufficient even at fixed volume. An attractive well can be added as shown in Fig. 1b for the range of separations r0 < r < r1. A number of pair-potentials were constructed to describe only repulsive interaction for small separations of atoms. The purpose of these potentials is two-fold. First,

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they have been used in studies of atomic collisions in which the attractive part of the atomic interactions plays only a minor role. Secondly, they can represent the short-range atomic interactions in the schemes where the attractive part is described by other means such as another pair potential, tight-binding method, embedding function etc. Most common potentials of this type are the Born-Mayer potential and its modifications and screened Coulomb potentials. Born-Mayer potential This potential was originally introduced to represent the closed shell repulsion between ions in ionic crystals (M. Born and J, E, Mayer, Z. Phys. 75, 1, 1932) but it has been used in many other cases, simply when a strong repulsion at short separations is described by an uncomplicated analytical function. The form of this potential is

!(r) = Ae"Br (EP2.1)

where A and B are two adjustable parameters. This potential was modified by Huntington (H. B. Huntington, Phys. Rev. 91, 1092 1953) by introducing into it the nearest-neighbor separation

r0 :

!(r) = Ae"B(r"r0)/r0 (EP2.2) In the case A is the value of the potential at the first nearest neighbors separation. Screened Coulomb potentials These potentials are most appropriate for metals where the electrons screen very effectively the Coulomb charges of the ions and thus the corresponding Coulomb interactions. In general, screened Coulomb potentials have the form

!(r) =Z1Z2r

"(r as ) (EP3)

where

Z1 and

Z2 are the ionic charges of the two atoms and

! is a screening function;

as is the corresponding screening radius beyond which the interaction is very weak. The simplest form of the screening function was proposed by Bohr (N. Bohr, Kgl. Dansk. Vid. Selsk. Mat.-Fys. Medd. 18, No.8 1948) as

! = exp(" r as ) . The screening function can be determined on the basis of the Thomas-Fermi model in which the screening of a charge by the free electrons gas is considered. In this model

! can only be found numerically but a good approximation is

!TF =14"exp(#kTFr) where k TF

2 =8k F"

(EP4)

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and

kF is the Fermi vector; the units are defined such that with

e2 4!"0 =1,

! = m = 1; the lengths are then in atomic units (5.29x10-11m) and energy in Hartrees = 2Ry (4.36x10-18J).

Potentials with repulsive and attractive parts

Modified hard spheres potential A modification of the hard sphere potential, which allows for an attractive part of the potential, is the square well potential (see Figure 1b) defined as

!(r) =" for r # ro

$% for ro < r < r10 for r & r1

'

()

*) (EP5)

However, based on studies of the interaction of atoms forming diatomic molecules (dimmers), a more realistic assumption is that the potential varies smoothly from strongly repulsive at short separations of atoms to attractive at intermediate separations and converging to zero for large separations, as shown in Fig. 2. Morse potential This potential, originally proposed for dimmer molecules (P. M. Morse, Phys. Rev. 34, 57, 1929) has the functional form

!(r) = De"2#(r"r0) " 2De"#(r"r0) (EP6) and its shape is shown in Fig. 2. When r = 0

! = De" r0 (e"r0 # 2) which can be

very large provided

e!r0 >> 2. For

r! "

!" 0 and the potential has a minimum

equal to -D for

r = r0 .

! (r)

r

Fig. 2. Pair potential of Morse type with attractive and repulsive part

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This form of the pair potential has often been used not only for molecules but also to describe atomic interactions in solids. The parameters D, α and

ro can be determined by fitting, for example, the cohesive energy at equilibrium, using equation (G20 with

U = 0, i. e.

Ep =12

!(rij)i, ji" j

# ), equilibrium lattice parameter from the conditions of

equilibrium

!"# = 0 (

!"# given by equation G23 with U = 0 and also zero velocities of the particles), and the bulk modulus given by equation (AG6). Such fitting was performed, for example, by Girifalco and Weizer (Phys. Rev. 114, 1123, 1959) for a number of FCC and BCC metals. However, they found that the stability of the lattice is very sensitive to the cut-off of the potential.

It should be noted that this potential, originally developed for molecules, has no physical justification for solids other than that it reflects the fact that there must be repulsive and attractive parts of the interaction and these are joined smoothly. Lennard-Jones potential Lennard-Jones potential (J. E. Lennard-Jones, Proc. Roy. Soc. A106, 463, 1924) has the general form

!(r) =" n

rn#"mrm

(EP7.1)

It was originally derived for inert gasses, in particular argon, the cohesion of which is due to the Van der Waals forces arising from the dipole interaction. The attractive part corresponds in this case to m=6 and the most common form of this potential is the so-called (6-12) form

!(r) = 4"#r

$ % & '

( ) 12

*#r

$ % & '

( ) 6+

, - -

.

/ 0 0

(EP7.2)

where σ corresponds to this value of r for which Φ = 0 and the minimum value of the potential is

!min = "# at rmin = 26 ! . Buckingham potential The general form of this potential is

!(r) = Aexp "Br( ) " Cr6

(EP8)

where A, B and C are adjustable parameters. The first term is the (Pauli) repulsion of ions at short separations described by the Born-Mayer potential. The second term

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describes attraction that can have various physical origins, for example Van der Waals attraction. Short range purely empirical pair potentials Empirical pair potentials of the type similar to the Morse potential shown in Fig. 2 but cut at a cut-off radius,

rcut , were constructed for many elements. The usual requirement is that the potential is cut-off smoothly, i. e. at least

!(rcut ) = d!(r) dr

r=r cut

= 0 is imposed; zero second derivative is required if lattice

vibrations are to be studied. These potentials are usually short range: Most commonly for BCC metals

rcut lies between the second and third neighbors and for the FCC metals between third and fourth neighbors. One of the most successful potentials of this type has been the Johnson's potential (R. A. Johnson, Phys. Rev. 134, 1329, 1964) described analytically by three smoothly spliced polynomials of the form

A(r ! B)3 + Cr + D (EP9)

This potential was constructed for several bcc metals by fitting the lattice parameter at equilibrium and partly elastic moduli but the Cauchy relation C12 = C44 (cubic system) is imposed if the energy of the system is represented by the pair potential only.

Fig. 3. Johnson’s potential

For metals it is more appropriate to describe by the pair potential only the energy changes associated with the variation of atomic configurations at constant average density of the material but not the whole total energy of the system. Such potentials were derived from first principles for sp-valent metals in the framework of weak pseudopotentials (see below). The potential energy is then written as

! (r)

r

Short-range cut-off

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Ep = U(!) +12

"(rij)i, ji# j

$ (EP10)

where U is the density dependent part of the energy in which ρ is the average density of the material. This part of the energy is the major contribution to the cohesion. Furthermore, Φ may also be a function of the density ρ. If the density dependence of the pair potential is neglected this description of the potential energy is valid provided the density of the material studied does not deviate substantially from the reference density, usually the density of the ideal equilibrium lattice for which the potential was constructed.

The physical meaning of the pair potential in equation (EP10) is entirely different from that based on interaction of atoms in molecules. It describes an effective interaction of atoms which form a medium of a given density composed of a large number of these atoms. This potential does not describe interaction of isolated pairs of atoms but it describes changes of the potential energy associated with the changes of the configuration of the atoms at constant density.

Evaluation of stresses and elastic moduli for the case that a pair potential together with a volume (density) dependent term represent the potential energy is summarized in the section dealing with General aspects of atomistic computer modeling. Pair potentials for s and p bonded metals: Justification for the empirical short-range potentials In this section we discuss materials with pure metallic bonding in which the electrons are entirely delocalized, the Fermi surface is close to a sphere and the system is well described in a nearly free-electron model. The following elemental metals belong to this category: Li, K, Na, Rb, Cs; Be, Mg, Zn, Cd, Hg; Al, Ga, In, Tl; Sn, Pb; Ca, Sr, Ba, as well as alloys formed from these elements, such as Li-Mg alloys.

In the materials listed above the potential, describing the interaction of electrons with the nuclei or ions, is represented by a pseudopotential that can be regarded as a weak perturbation when solving the Schrödinger. Such pseudopotential is defined as follows: (i) Core electrons are bonded together with the nucleus forming an ion of effective

positive charge, Z, which is equal in magnitude to the valence of the metal i. e. to the number of conduction electrons that are not included into the core.

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(ii) The true potential of the ion core is replaced by the pseudopotential which is constructed such as to induce the same electron density outside the atomic core as does the true atomic potential.

(iii) The pseudopotential is regarded as a weak perturbation for the conduction electrons so that these can be treated as nearly free and standard perturbation theory can be used to solve the Schrödinger equation.

The potential energy of the system can then be written as

Ep =12

!(rij ,")i, ji# j

$ + U(") (EP11)

where the density dependent term U(ρ) is the zeroth and first order term in the perturbation expansion of the energy and the pair potential is the second order term. Examples of pair potentials for potassium, magnesium and aluminum, derived using corresponding pseudopotentials, are shown in Figs. 4a-c.

Fig. 4a Pair potential for BCC potassium – positions of 1st, 2nd and 3rd neighbors

are marked (a = 5.233Å). The minimum of the potential is about 0.03eV, much smaller than the cohesive energy.

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Fig. 4b Pair potential for HCP magnesium – positions of 1st, 2nd and 3rd neighbors

are marked; a is the lattice parameter of the FCC lattice with the same volume per atom (a = 4.518Å). The minimum of the potential is about 0.035eV, much smaller than the cohesive energy.

Fig. 4c Pair potential for FCC aluminum – positions of 1st, 2nd, 3rd, 4th and 5th

neighbors are marked (a = 4.033Å). The minimum is at the second nearest neighbors and the contribution is anti-cohesive (positive)

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Analytical expressions can be obtained in the limits of

r! 0 and

r! ". For small values of r we obtain the screened Coulomb potential of the type (EP3)

!(r,") =Z2# ak TF

2

4$rexp(%kTFr) (EP12)

where

kTF is the Thomas-Fermi wave vector and

! a the volume per atom. The density dependence of the potential is hidden not only in the direct dependence on

! a but also in the dependence of

kTF on the density. For large values of r the functional form of the pair potential is determined by the singularity of the effective pseudopotential at

q = 2kF which leads to the oscillatory behavior of the type

!(r,") #cos(2k Fr)(2k Fr)

3 (EP13)

This potential depends on the density via the density dependence of the Fermi vector

kF . The oscillations of the pair-potential are the typical Friedel oscillations the nature of which is the same as oscillations of the density of the electron gas induced by the presence of a positive charge that is screened by the electrons. Such potentials may be relatively long range. Since the number of atoms at a distance r from an atom is, for large values of r, proportional to

r2; contribution of these atoms to the energy

associated with the pair potential is

!cos(2kF r)(2kF r)

and thus the convergence is very

slow and only in the sense of the principal value (Cauchy sense). However, methods of damping these oscillatory potentials have been developed.

Physical meaning of the pair potential for s and p bonded metals The pair potential part of the energy, determined as the second order perturbation, describes changes of the energy of the system associated with changing the configuration of the atoms at constant density (i. e. constant total volume). The bulk of the total potential energy is contained in

U(!) that remains unchanged if the configuration of atoms varies at constant volume of the system. Hence, the physical meaning of the pair potential interaction described in this section is for metals entirely different than pair potentials describing interaction of atoms in molecules, pair potentials employed in ionic solids etc. Pair potentials entering the potential energy given by equation (EP 11) describe an effective interaction of atoms that form a medium of a given density composed of a large number of these atoms. This type of pair potential is inapplicable when analyzing interaction of isolated pairs of atoms.

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Deficiencies of pair potentials for metals The volume dependent term, U, which represents the bulk of the cohesion, is a function of the total volume, or average density, and cannot be easily defined locally. This is the reason why most calculations employing pair-potentials have been performed at constant volume when the contribution of this term is constant. This precludes the use of this approach in situations where significant local variations of the density exist. In lattice defects the local changes in the density are, of course, common and the most extreme case are surfaces where the density sharply varies away from the ideal crystal density a few atomic spacing below the surface to zero after crossing the surface.

An exact solution of this problem can only be achieved by carrying out full quantum mechanical calculations. However, a development which preserves the simplicity of the description of the total energy on the level similar to equation (EP10) is the embedded atom method and other many-body central force potentials schemes. In these methods

U(!) is replaced by an embedding function which depends on the positions of atoms but it is not a simple pair potential.