The Static Lattice Simulation

Static lattice simulations are used to detennine the minimum energy configuration o f a structure at zero Kelvin. W hen the cell dim ensions are known, the static lattice energy is obtained via a constant volum e minim ization procedure, in which the initial atom coordinates are allow ed to relax until the structure reaches its minimum energy configuration. The minimization technique used in our simulations is the N ew ton

-Raphson method (Norgett and Fletcher, 1970) in which the first step is to expand the lattice energy to the second-order form about a point r;

U(r' ) = U(r) + g ^ - S + ^ S ^ - W - S (2.8)

where ô = r' - r = 5r, the Cartesian displacement o f the atomic positions (coordinate strain). The term g is the first derivative o f the potential energy with respect to atomic displacement, and defined as:

g , = ^ (2.9)

where each g, is an element o f the matrix g. The IV term in equation (2.8) represents the

set o f second derivatives o f the lattice energy in matrix form defined by:

IV = (2.10)

where s is the bulk strain, and Wrr is given by:

S ^ U

The remaining elements o f the above matrix may be similarly defined. These calculations are made repeatedly, with the atomic coordinates adjusted slightly at each iteration until all the forces acting on the atoms, g„ become zero so that the coordinate strain, ôr, is removed, i.e:

0 (2.12)

a dr

g = (2.13)

The atomic coordinates are updated by Ôr at each stage, and the revised coordinates after the { k + \Ÿ interaction, are related to the initial coordinates, r^, by the expression:

h + \ = h - S k ' ^ k

(2-14)

where Hk = (Wrr)k'\ the initial inverse o f the second derivative matrix defined by equation (2.10). The minimized coordinates are obtained via the repeated application o f this expression until the forces g, acting on the atoms are removed. However, the calculation o f the second derivative matrix, W, and its inverse Hk, is complex and

computationally expensive. Hence, Norgett and Fletcher (1970) developed a method

which significantly reduces computer time by using an updating procedure for the inverse matrix:

S r S r ^ H , ■ S g ■ S g ’’ - H ,

S r ^ ■ S g J r . - S g ^ ■ HI, ■ S g ' (2.16)

where Hk+i is known as the Hessian matrix', ô r and ôg are the vectors describing the differences in atomic positions and forces acting on each atom, respectively, for subsequent iterations, so that Ôr = rk+i - rk, and ôg = gk+i - gk- In our calculations, the Hessian matrix is updated after 1000 iterations, and this procedure continues until equation (2.12) is satisfied and all forces acting on the atoms are zero, so that a minimum energy equilibrium structure at constant cell volume is obtained.

For a constant pressure calculation, an additional minimization o f the bulk strain energy is required. This involves the relaxation o f both the atoms in the unit cell and the cell dimensions themselves, corresponding to the removal o f all mechanical strain acting on the unit cell. This is followed by a constant volume minimization, as described previously, until all forces acting on the atoms and the unit cell are zero. The first step in the constant pressure minimization is again the expansion o f the lattice energy to the second-order form (equation 2.8) and calculation o f the first and second derivatives o f the lattice energy with respect to strain. In this case, ô = ôsi, which represents the 6 independent strain elements o f the symmetric strain matrix:

£ = OS, _{~ ^ 6 ^ ^ 5} — Ss^ Se 2 —Se^ 2 ^ ^ 2 ^ 5e. (2.17)

The first derivative o f the lattice energy with respect to strain gives the mechanical pressure or stress, defined by:

dÔE, (2.18)

The second derivative of the lattice energy with respect to strain gives the elastic

constants (assuming the equilibrium condition, g = 0), the matrix C of which is given by:

d^U

(2.19)

From Catlow and Mackrodt (1982), the strained coordinates are given by the relationship:

r ' - ( / + £■)• r (2.20)

where e is the strain on the original coordinates and lattice vectors r; I is the identity

matrix; and r' represents the resultant coordinates and lattice vectors after applying the strain. The strain matrix follows Voigt notation and using equation (2.17), equation (2.20) may be written as:

1 + 2^6 1 c V 2^5 2^6 1 + £ ’2 — ₂F_{6 4} 2^5 2^4 1 + £

y

(2.21)

where %, y; and z are components o f r. The strains are calculated from the elastic constants and stress acting on the crystal assuming Hooke's Law, in which stress is proportional to strain, and the elastic compliance tensor (the inverse o f the elastic constants) is the constant o f proportionality.

The first derivative o f the lattice energy with respect to strain is more easily

determined by considering the square o f the displacement r':

T

(

2

.

22

)

Differentiation of equation (2.22) with respect to strain yields the mechanical or static pressure:

SSf Ôr Ô€i ôr^

(2.23)

and evaluation o f the differential at zero strain yields:

where: 1

2

(2.24)

i

1

2

3

4

5

6

X

_y

z

_y

X X rP X

_y

z z z

_y

If the Ewald Method is employed to determine the Coulombic component o f the lattice energy, further strain derivatives must be considered. This is because the Ewald method sums much o f the Coulombic interaction in reciprocal space and hence, the effect o f strain on the reciprocal lattice vectors G, and cell volume F, must also be evaluated, following the relations:

1 dG

2 ds,

= -G^G

(2.24)

with /, a and p defined above, and:

where / = 1, 2, 3. In addition to the mechanical strain, the elastic constant matrix C (equation 2.19) is required to calculate the bulk strains. Using Hooke’s Law, the bulk strains are determined from:

Se = ^ ^ - C - ' (2.26)

e s e

This equation may be substituted into equation (2.20) to give the updated atomic coordinates and cell dimensions. As the above expressions are approximate, the calculations proceed iteratively until all residual strains are removed; all internal strains are removed at each iteration by performing successive constant volume minimizations described above. Hence, a minimum energy structure is obtained free o f internal and bulk strain, and therefore at constant pressure.

2.2.3 Dynamic Simulation

The dynamic contribution to the energy o f the crystalline system by the vibrational motions o f its constituent atoms at temperatures above zero Kelvin may be

calculated via molecular dynamics, for medium- to high-temperature simulations {e.g.

Dove, 1988), or via lattice dynamics, for low- to medium-temperature simulations {e.g.

Price et a l, 1987; Dove, 1993). Lattice dynamics is a semi-classical method which uses the quasi-harmonic approximation, in which it is assumed that each atom moves in simple harmonic motion about its equilibrium position which remains fixed. The classical approach is derived from the Newtonian analysis o f the dynamics o f the constituent atoms or ions. The lattice dynamical method allows for the description o f the unit cell in terms o f independent quantised harmonic oscillators whose frequencies vary as a function o f unit cell, thus allowing for thermal expansion {e.g. Bom and Huang, 1954). The motions o f the individual atoms are treated collectively as lattice vibrations or phonons, and the equations o f motions o f these vibrating species are solved to obtain their vibrational frequencies within the periodic stmcture. The thermodynamic properties and stabilities o f crystalline solids are determined from the calculated vibrational frequencies via statistical mechanics, thus providing the quantitative link between the microscopic and macroscopic properties o f a phase.

In order to calculate the vibrational frequencies o f a system, the relative motions of each atomic species must first be considered. Such motions are controlled by the potential energy function, assuming an already equilibrated unit cell. The potential function describing the interactions o f 2 atoms i and j at original positions r, and ij, displaced by u, and Uy to new positions R, and Ry, respectively, may take the following form;

The displacement o f atom i by U/ from its equilibrium position, will experience a restoring force F„ where:

F, = £ (2.28)

The equations o f motion that must be solved for atoms i and y, with mass and my

respectively, have the form:

From the theory o f small displacements {e.g, Ziman, 1964), it may be shown that:

which may be interpreted as meaning that the sum on the right-hand side o f equation (2.30) represents the force acting on atom i due to the displacement Uy o f atom j. Similarly,

However, in accordance with Block's theorem {e.g. Ziman, 1964), equations (2.29-2.31)

u,{Rj) = e,{q)Qxp{i{q.R, - œ{q)t)) (2.32)

and,

Uj {Rj) = 6j (q) Qxp{i(q. Rj - (o{q)t)) (2.33)

where q is the reciprocal lattice vector, o(q ) is the frequency o f the vibrational mode, and e(q) is the polarization vector which describes the atomic displacements during the vibration and thus, the direction in which the atoms move. Manipulation o f equations (2.27-2.31) yields the expression;

- m a > \ { R ,) = - ' £ d{ R , - (i?,) (2.34)

and substitution of equations (2.32) and (2.33) into equation (2.34) gives:

mco^e Xq) exp(/(^. R, - cot)) = ^ D{R, - Rj )Cj {q) exp(/(^. Rj - cot)) (2.35) R,R j

where ZXq) is the dynamical matrix:

D{q) = Y , D ( R , - R j )exp(-i(q. R - q . R j)) (2.36)

R,Rj

Substitution o f equation (2.36) into equation (2.35) gives the final result:

mco^e,(q) = D(q)ej(q) (2.37)

Solution o f this equation for a given value o f wavevector q yields 3n eigenvalues which

are the squared frequencies (co^q)) o f each o f the normal modes o f the crystal, obtained from diagonalization o f the dynamical matrix, and 3n sets o f eigenvectors ( ^ q ) , Cy(q) and Cz(q) which describe the pattern o f atomic displacements for each normal mode.

Theoretical lattice dynamics (e.g. Kittel, 1971) demonstrates that the vibrational frequencies o f a system vary as a function o f wavevector q. Thus, in order to determine the thermodynamic properties o f a solid accurately, it is necessary to calculate the vibrational frequencies over all possible wavevectors. However, this is impossible due to

the size and complexity o f the calculation required. This problem is commonly overcome by using an approximation in which the phonon frequencies are calculated at selected points on an imaginary three-dimensional grid within the Brillouin zone, a

procedure known as zone sampling. Weighting factors are applied to the selected points

according to the number o f times the points are generated by the symmetry o f the Brillouin zone. At low-temperatures, the acoustic phonons with wavevectors close to the zone centre are the only thermally excited modes. Hence, a fine grid is required in this region in order to calculate the phonon frequencies and therefore the thermodynamic properties correctly.

Having determined the phonon frequencies for points within the irreducible Brillouin zone, it is possible to calculate the thermodynamic functions: vibrational energy (£), entropy (S), free energy (G), and heat capacity (Cv), using the following expressions based on statistical thermodynamics:

M

( 1 4 . )

where each function is summed over the total number o f phonon frequencies, M, and

where,

hct).

X —

/

kT

(2.42)

and the zero point energy.

(2.43)

i ^

is included in the vibrational energy and the free energy.

According to the harmonic approximation on which equations (2.38-2.43) are based, the structure has a thermal expansion coefficient o f zero. However, under the assumption o f the quasi-harmonic approximation, the volume dependence o f the thermodynamic and structural properties may be calculated. To minimize a structure at constant pressure above zero Kelvin, the kinetic pressure generated by the atomic vibrations must be included in the total pressure o f the system. The kinetic pressure is the first derivative o f the free energy, G, with respect to volume. Thus, for an isotropic material.

- (2.44)

By calculating G at a given volume, and then recalculating after making a small adjustment to the cell volume, the kinetic pressure may be determined. The thermally equilibrated structure is then obtained from a constant pressure energy minimization, in which the total pressure is now defined as the sum o f the mechanical, hydrostatic and kinetic pressures. The strain on the crystal system is then determined from the kinetic pressure and elastic constants (equation 2.26). A constant volume minimization is performed each time the cell volume is modified to ensure that the atoms remain at their potential energy minima and thus within the constraints o f the harmonic approximation. As the final static pressure is likely to be non-zero, correction terms must be applied to the elastic constants. Barron and Klein (1965) have shown that the elastic constants under an isotropic initial stress should be calculated as:

^ - ô „ ô ^ ) (2.45)

'apar y

where Fst is the static pressure, ô is the Kronecker delta, and a , p, a and x refer to the tensor notation o f the strain directions:

P 1 2 3 a 1 _{£i £6 £5} 2 _{€6 £2 £4} 3 _{£5 £4 £3}

From equation (2.45), the proposed corrections for the 21 independent elastic constants are shown in Table 2.1, and these corrections have recently been demonstrated by Wall et al. (1993) to be applicable to the elastic constants calculated from the second derivative o f the lattice energy with respect to strain at non-zero pressures and temperatures.

Elastic Constants Static Pressure Corrections

C i i , C 2 2 , C 3 3 ,

In document The lattice dynamics of garnets (Page 38-48)