# Lattice Constants and Bulk Moduli

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MDexp3_latticeJin Zhaohui

The ground state (T = 0 K)

Total-energy curves of the seven phases of Si as a function of the atomic volume normalized to expt. Dashed line is the common tangent of the energy curves for the diamond phase and the β-tin phase.

Reference volume is diamond-structure Si at P = 0.

The system moves from 1 → 2 → 3 → 4 under pressure.

from M. T. Yin and M. L. Cohen Phys. Rev. B 26, 5668 (1982)

Theoretical ground state energy versus volume curves, used to predict high pressure allotropic phase transitions.

Bulk moduli of the elements C. Kittel, Introduction to Solid State Physics, 8th ed., (2005).

Note that the modulus scale is logarithmic. Negative values are plotted along the base line.

Data from Handbook of elastic properties of solids, liquids, and gases, vol. II. New York: Academic Press; 2001. p. 97 [Chapter 7].

Bulk moduli of the elements

Procedure to compute a0 and B

We can calculate the lattice energy E (in eV) as a function of the volume V (in Å3). The equilibrium lattice constant a0 corresponds to the volume that gives the minimum of the E(V) curve.

Step 1: Given a structure, compute the energy for several values of the volume

(1) Generate a 3x3x3 FCC copper (Cu) supercell with 108 atoms.

(2) LAMMPS writes “data” file containing lattice, volume per atom and energy per atom:

# lattice (Å) volume (Å3) energy (eV) 3.591 11.57673851775 -3.48875019885056 3.592 11.586412672 -3.48888175993704 3.593 11.59609221425 -3.48900596416684 … …

Step 2: Fit to an analytic form, e. g. , the Murnaghan equation of state

These data are then least-squares-fitted to the Murnaghan equation of state. We also use a second order polynomial fit.

Step 3: calculate bulk modulus

The curvature of the E(V) curve near the minimum also tells us the bulk modulus of the crystal.

In order to obtain reliable minimum values for these discrete sets of points, one usually interpolates the obtained curves with equation of state functions, which are analytical functions derived from general thermodynamic considerations about the internal energy in the vicinity of the minimum. A popular form is the equation due to Murnaghan (Birch, Intermetallic compounds: Principles and Practice, Vol I: Principles. pages 195-210),

where V is the volume, B0 and B0’ are the bulk modulus and its pressure derivative at the equilibrium volume V0.

Fitting the Birch-Murnaghan (BM) equation of state

E = E0 + 9 8

The bulk modulus is defined by

(1)

where V is the volume of a cubic unit cell and P the pressure.

The cohesive energy E per atom can be calculated via VASP or LAMMPS as a function of the lattice parameter, a. For a cubic cell, the total energy is given by ε = ME, where M is the number of atoms in the unit cell the volume of which is given by V = a3.

Pressure is given by

(3)

Where a0 is the equilibrium lattice parameter corresponding to the minimum of E.

B ≡ − dP

Example:

From polynomial fitting of E vs a, one can easily get

which is 7.266 eV/2 for Cu (a0 = 3.635 , fcc, M = 4)

and 3.394 eV/2 for Si (a0 = 5.465 , diamond, M = 8).

Using Eq (3), the calculated B is 142.3 GPa for Cu and 88.44 GPa for Si.

Note that 1 eV/3 ≈ 160.22 GPa.

d2E da2

VASP 3.635 5.465

VASP 142.3 88.44

* Data taken from C. Kittel, Introduction to Solid State Physics, 8th ed., (2005).

Cu Al Fe Si Mg

lmp_serial -i in.lattice

a_0: lattice constant B_0: bulk modulus

equilibrium volume

Energy minimum

lattice energy vs atomic volume of FCC Cu

plot.bm.gnu and plot.2nd.gnu are the gnuplot scripts. (to quit gnuplot, pressing “q”)

LAMMPS calculations

Set potential

Set the iterations

Lattice spacing delta

Jump to the next iteration

LAMMPS input script

Write dump files with a lattice equal the guess value

Structure Expt. a0 (Å) Expt. B (GPa) a0 (Å) B (GPa)

Mg HCP 3.209 (c/a=1.623) 35 3.184 (c/a=1.628) ?

Al FCC 4.049 72 4.045 ?

Si Diamond 5.431 98 5.43 ?

Fe BCC 2.866 168 2.855 ?

Cu FCC 3.614 142 3.61 ?

Calculations of Bulk Moduli (T = 0 K)

The ground state (T = 0 K)

Total-energy curves of the seven phases of Si as a function of the atomic volume normalized to expt. Dashed line is the common tangent of the energy curves for the diamond phase and the β-tin phase.

Reference volume is diamond-structure Si at P = 0.

The system moves from 1 → 2 → 3 → 4 under pressure.

from M. T. Yin and M. L. Cohen Phys. Rev. B 26, 5668 (1982)

Theoretical ground state energy versus volume curves, used to predict high pressure allotropic phase transitions.

Bulk moduli of the elements C. Kittel, Introduction to Solid State Physics, 8th ed., (2005).

Note that the modulus scale is logarithmic. Negative values are plotted along the base line.

Data from Handbook of elastic properties of solids, liquids, and gases, vol. II. New York: Academic Press; 2001. p. 97 [Chapter 7].

Bulk moduli of the elements

Procedure to compute a0 and B

We can calculate the lattice energy E (in eV) as a function of the volume V (in Å3). The equilibrium lattice constant a0 corresponds to the volume that gives the minimum of the E(V) curve.

Step 1: Given a structure, compute the energy for several values of the volume

(1) Generate a 3x3x3 FCC copper (Cu) supercell with 108 atoms.

(2) LAMMPS writes “data” file containing lattice, volume per atom and energy per atom:

# lattice (Å) volume (Å3) energy (eV) 3.591 11.57673851775 -3.48875019885056 3.592 11.586412672 -3.48888175993704 3.593 11.59609221425 -3.48900596416684 … …

Step 2: Fit to an analytic form, e. g. , the Murnaghan equation of state

These data are then least-squares-fitted to the Murnaghan equation of state. We also use a second order polynomial fit.

Step 3: calculate bulk modulus

The curvature of the E(V) curve near the minimum also tells us the bulk modulus of the crystal.

In order to obtain reliable minimum values for these discrete sets of points, one usually interpolates the obtained curves with equation of state functions, which are analytical functions derived from general thermodynamic considerations about the internal energy in the vicinity of the minimum. A popular form is the equation due to Murnaghan (Birch, Intermetallic compounds: Principles and Practice, Vol I: Principles. pages 195-210),

where V is the volume, B0 and B0’ are the bulk modulus and its pressure derivative at the equilibrium volume V0.

Fitting the Birch-Murnaghan (BM) equation of state

E = E0 + 9 8

The bulk modulus is defined by

(1)

where V is the volume of a cubic unit cell and P the pressure.

The cohesive energy E per atom can be calculated via VASP or LAMMPS as a function of the lattice parameter, a. For a cubic cell, the total energy is given by ε = ME, where M is the number of atoms in the unit cell the volume of which is given by V = a3.

Pressure is given by

(3)

Where a0 is the equilibrium lattice parameter corresponding to the minimum of E.

B ≡ − dP

Example:

From polynomial fitting of E vs a, one can easily get

which is 7.266 eV/2 for Cu (a0 = 3.635 , fcc, M = 4)

and 3.394 eV/2 for Si (a0 = 5.465 , diamond, M = 8).

Using Eq (3), the calculated B is 142.3 GPa for Cu and 88.44 GPa for Si.

Note that 1 eV/3 ≈ 160.22 GPa.

d2E da2

VASP 3.635 5.465

VASP 142.3 88.44

* Data taken from C. Kittel, Introduction to Solid State Physics, 8th ed., (2005).

Cu Al Fe Si Mg

lmp_serial -i in.lattice

a_0: lattice constant B_0: bulk modulus

equilibrium volume

Energy minimum

lattice energy vs atomic volume of FCC Cu

plot.bm.gnu and plot.2nd.gnu are the gnuplot scripts. (to quit gnuplot, pressing “q”)

LAMMPS calculations

Set potential

Set the iterations

Lattice spacing delta

Jump to the next iteration

LAMMPS input script

Write dump files with a lattice equal the guess value

Structure Expt. a0 (Å) Expt. B (GPa) a0 (Å) B (GPa)

Mg HCP 3.209 (c/a=1.623) 35 3.184 (c/a=1.628) ?

Al FCC 4.049 72 4.045 ?

Si Diamond 5.431 98 5.43 ?

Fe BCC 2.866 168 2.855 ?

Cu FCC 3.614 142 3.61 ?

Calculations of Bulk Moduli (T = 0 K)