Elasticity of hcp nonmagnetic Fe under pressure

Transcription

1 PHYSICAL REVIEW B 68, Elasticity of hcp nonmagnetic Fe under pressure S. L. Qiu Department of Physics, Alloy Research Center, Florida Atlantic University, Boca Raton, Florida , USA P. M. Marcus IBM Research Division, T. J. Watson Research Center, Yorktown Heights, New York 10598, USA Received 8 November 00; revised manuscript received 4 March 003; published 4 August 003 First-principles total-energy calculations of structural and elastic properties of hcp nonmagnetic Fe have been made using the augmented-plane-wave plus local orbital method with the generalized gradient approximation. The uilibrium state, including both the volume and aspect ratio c/a, at a given pressure p is found from the minimum of the Gibbs free energy G with respect to the lattice constants a and c. Abrupt structure is found in c/a below 1000 kbar as a function of p, which agrees with experiment. The elastic constants c ij are found from second strain derivatives of G. Particular attention is paid to two corrections to the c ij, which are evaluated in new ways a pressure correction ruired when the c ij are evaluated from the energy rather than G and a correction for internal relaxation of the second basis atom in the unit cell. Results are compared with previous first-principles calculations of the c ij as functions of pressure. DOI: /PhysRevB PACS numbers: 6.0.Dc, Lp, Bb, Ks I. INTRODUCTION The hcp or phase of Fe dominates the high-pressure phase diagram. Its elastic properties are geophysically important, since the Earth s core appears to be hcp Fe. 1 Recent calculations of the structural and elastic properties as functions of pressure from first principles have been made up to 4000 kbar at zero temperature. 1, This paper recalculates these properties as functions of pressure from first principles, but pays special attention to two features of the calculation: a pressure correction and an internal relaxation correction. Comparison of our results for the elastic constants with Refs. 1 and shows significant discrepancies from Ref. 1, which we attribute to the treatment of these two features, but substantial agreement with Ref., which treats the two corrections differently from this paper. The first feature is the necessity of making a correction to the elastic constants at finite pressure if the elastic constants are computed from second strain derivatives of the total energy. In our formulation the elastic constants are second strain derivatives of a free energy and the correction is automatically included. Our elastic constants are the stress-strain coefficients which describe the elastic response to strains at a given pressure and enter the uations of motion. The second feature is the necessity of calculating the internal relaxation of the second basis atom in the strained hcp unit cell see Fig. 1. The internal relaxation is shown to have a significant effect on certain shear elastic constants i.e., a reduction which grows with increasing pressure. The calculation of the internal relaxation is simplified by the choice of a particular symmetrical orientation of the unit cell with respect to Cartesian axes and by use of symmetrical strains for calculating elastic constants. Our procedures give the structural and elastic properties directly as functions of pressure, rather than as functions of volume. The procedures then do not ruire the uation of state to determine the pressure at each volume. The uilibrium state is found at each pressure including the optimum shape factor c/a, which minimizes the free energy. A steep rise in c/a to a maximum at 400 kbar and gradual decrease beyond 400 kbar is found which corresponds well to experiment. Section II describes the calculation program and the procedures based on use of the epitaxial Bain path and the free energy for determining uilibrium states and elastic constants at a given pressure. Section III gives the results of the calculations of free energy along the epitaxial Bain paths, the uilibrium structures, and elastic constants as functions of pressure. Section IV discusses the comparison with previous results from theory and experiment. II. PROCEDURES First-principles total-energy calculations of hcp nonmagnetic NM Fe under hydrostatic pressure were carried out using the WIENk implementation of the full-potential augmented-plane-wave plus local orbital (APWlo) method with the Perdew-Burke-Ernzerhof exchange-correlation potential in a generalized gradient approximation GGA with relativistic corrections. 3 A plane-wave cutoff R MT K max 7, G max 14, and mixer0.05 were used in all the calculations; 40 k points in the irreducible wedge of the Brillouin zone IBZ were used in the epitaxial Bain path EBP calculations, while 960 k points in the IBZ were used in the elastic constant calculations. The k-space integration was done by the modified tetrahedron method. 3 Tests with larger basis sets and different Brillouin-zone samplings yielded only very small changes in the results. The convergence criterion on the energies is set at mry per atom. Our procedure uses the thermodynamic result that minimizing the Gibbs free energy G of the system at zero temperature with respect to the structural parameters while holding the pressure p constant gives the uilibrium state at p and zero temperature, 4,5 where for hcp structures Ga,c;pEa,cpVa,c /003/685/ /$ The American Physical Society

2 S. L. QIU AND P. M. MARCUS PHYSICAL REVIEW B 68, The use of the Gibbs free energy at T0 K is uivalent to using the enthalpy EpV to determine uilibrium. This uivalence shows the purely mechanical nature of uilibrium at T0 K. At finite T thermal effects enter and uilibrium ruires minimizing G. In Eq. 1, E is the total energy per atom and V the volume per atom. The first derivative of G with respect to a strain i around uilibrium then gives a deviation of the stress from p whose gradient with respect to strains drives the system back toward uilibrium. Since G is a quadratic function of strains around uilibrium, the first derivative is linear in the strains; then the gradients of the stress deviation with respect to strains are the coefficients of the strains; these coefficients are the stress-strain coefficients that enter the uations of motion. We call these coefficients, which are second strain derivatives of G at uilibrium, the elastic constants, since they give the response of the system to strains. If second derivatives of E are called elastic constants, then a pressure correction is ruired to obtain the stress-strain coefficients at p The EBP provides one simple way to find the minimum of G at p with respect to the structural parameters. The EBP is defined by the condition that the stress in the c direction 3 is p, i.e., 3 1 V Ea,c 3 4 a )a Ea,c c p. a Then (G/c) a 0 on the EBP Refs. 4 and 5 the contribution of the pv term in Eq. 1 cancels p]. At a minimum of G on the EBP the first derivative of G vanishes in a second direction along the EBP: hence the derivative must vanish in all directions in the a,c plane at an ordinary quadratic minimum, i.e., along the EBP: G EBP a;pe EBP a;ppv EBP a;p. 3 The notation indicates that p is a parameter held constant for the calculation of E EBP (a;p) and G EBP (a;p), and G EBP (a;p) is followed to a minimum, which is then the uilibrium state at p with lattice constants (a 0,c 0 ) and volume per atom V 0. The procedure for the elastic constants of the hcp lattice is as follows. A. Unit cell In a conventional two-atom hcp unit cell 90, 10, and ab and the two atoms are located at 0,0,0 and /3, 1/3, 1/, respectively, with vector components in units of the lattice constants (a,b,c). To study the effect of the internal relaxation of the second atom it is more convenient to use an unconventional two-atom hcp unit cell with 60 and to rotate the unit cell by 15 about the c (x 3 ) axis, so as to have the basal rhombus symmetrically oriented with respect to the orthogonal axes x 1 and x, as shown in Fig. 1; then the position of the second atom in the uilibrium state is at 1/3, 1/3, 1/. This arrangement makes the internal relaxation under strain one dimensional in the bisecting plane of the 60 angle provided one uses strains that preserve the reflection symmetry of the bisecting plane. The FIG. 1. hcp unit cell showing a basis of two atoms with 60 rotated 15 about the c or x 3 axis so that the basal rhombus is oriented symmetrically with respect to the orthogonal axes x 1 and x. This orientation makes the internal relaxation one dimensional for strains that preserve reflection symmetry of the bisecting plane. unconventional unit cell was used in all the calculations reported here. The change of orientation of the unit cell does not affect the c ij since the hcp lattice symmetry makes the c ij invariant to rotations around the c axis. B. Elastic constants c ij p A hcp structure has five independent elastic constants, 6 i.e., c 11, c 1, c 13, c 33, and c 44 with c 11 c, c 13 c 3, c 44 c 55 and c 66 (c 11 c 1 )/. The elastic constants are defined by 6 c ij 1 V 0 Ga,c;p i j a 0,c 0, i, j1 6, 4 where i and j are strains in Voigt s notation, which is related to the tensor notation by 1 11,, 3 33, 4 3, 5 13, and 6 1. The variation of G around the uilibrium point (a 0,c 0 ), where G has a minimum for small strains, is a quadratic function of the strains given by GG 0 V 0 6 c ij i j, 5 i, j1 where G 0 is the unstrained free energy

3 ELASTICITY OF hcp NONMAGNETIC Fe UNDER PRESSURE The lattice vectors of a hcp unit cell are given by the matrix R= a 1 b 1 c 1 d 1 a b c d a 3 b 3 c 3 d 3, 6 where the first three columns give the components of the lattice vectors in orthogonal axes x 1, x, and x 3 and the fourth column gives the position of the second atom. The strained values of the lattice vectors and the position of the second atom are given by 6,7 (I== )R= R=, i.e., a1 b1 c1 d1 a b c d a 3 b 3 c 3 d 3 a 1 b 1 c 1 d 1 a b c d a 3 b 3 c 3 d 3, 7 where I= is the identity matrix and = is the strain tensor. Then the volume per atom, V, and the angles,, and of the strained unit cell are determined by the strained lattice vectors. For the unconventional hcp unit cell shown in Fig. 1 the matrix of the lattice vectors is cos 15 a sin 15 0 a6/6 R= a a sin 15 a cos 15 0 a6/ c c/ The (I== ) matrices that give the elastic constants separately and preserve the reflection symmetry of the plane bisecting the 60 angle are as follows, using V 0 ()/4) c 0 a 0 : with 11 gives 0 0 c 11 c 1 1 V 0 )c 0 a 0, gives c 33 1 V 0 4 )c 0 a 0, with gives c 13 1 )c 0 a 0 1 c 11 c 1 c 33, with c 44 c V 0 c V 0 PHYSICAL REVIEW B 68, gives G 1 )c 0 a 0, gives )c 0 a 0 G. 13 Since c 66 (c 11 c 1 )/, c 11 and c 1 can be calculated from Eqs. 9 and 13. In the unrelaxed case, i.e., where the second atom is always at 1/3, 1/3, 1/ as fractions of a,b,c, five values of each strain, i.e., ij , 0.0, and 0.05, are used to calculate the free energy. Fitting the five points with a third-order polynomial and taking the derivatives in Eqs gives the unrelaxed elastic constants c ij. Here we avoid the value ij 0 to keep the same crystallographic symmetry for all five strains. C. Internal relaxation Johnson 8 calculated the internal relaxation in the hcp lattice using a simple embedded-atom method EAM model for Ti at zero pressure assuming an ideal c/a ratio and found a 10% reduction in c 66 and no reduction in c 44. Johnson found relaxations in two directions, but by using the special cell orientation and symmetrical strains described in Sec. II, the relaxation is along one line. This greatly simplifies the location of the minimum of G We have investigated the relaxation effect on the elastic constants as functions of pressure using the following procedures. For each of the five strains used for calculating the unrelaxed elastic constants we calculate the free energies with the second atom at five or six different positions: 0.93, 0.93, 0.5, 0.313, 0.313, 0.5, 1/3, 1/3, 1/, 0.353, 0.353, 0.5, 0.373, 0.373, 0.5, and 0.393, 0.393, 0.5, where 1/3, 1/3, 1/ is the unrelaxed position, which was used in the calculations of the unrelaxed elastic constants. The projections of all these positions onto the basal plane are on the bisector line of the basal rhombus because of the choice of the unconventional unit cell and use of symmetrical strains. Figure shows an example of the calcula

4 S. L. QIU AND P. M. MARCUS PHYSICAL REVIEW B 68, c 1 given by Eq. 7, c 33 given by Eq. 10, and c 13 given by Eq. 11 are not affected by the internal relaxation because the strains do not break the hexagonal symmetry. However, combining c 66 and c 11 c 1 makes c 11 and c 1 be individually affected by the relaxation. This procedure for the calculation of the effects of internal relaxation on the c ij was also used in Ref. 9 for hcp Mg at p0. FIG.. Example of the calculation of relaxed and unrelaxed values of c 66 at p000 kbar. a Free-energy curves corresponding to or ) at p000 kbar. At each strain free energies with the second atom at five or six different positions are calculated. The vertical dashed line indicates the vector components of the unrelaxed position as fractions of the lattice lengths 1/3, 1/3, 1/. The minimum of each curve in a gives a point in b at the corresponding strain. b Relaxed solid circles connected by solid curve and unrelaxed open squares with dashed curve free-energy curves whose curvatures give the relaxed and unrelaxed values of c 66 at p000 kbar respectively. In both a and b the solid and dashed curves interpolate between the calculated points. tions of relaxed and unrelaxed c 66 at p000 kbar. Figure a shows two free-energy curves as functions of d/a, the position of the second basis atom, corresponding to two out of the five strains, i.e., or ) at p 000 kbar. Fitting of each free-energy curve with a thirdorder polynomial gives the minimum free energy due to relaxation of the second basis atom for the corresponding 6 strain. We have checked that the minimum along the bisector is a true minimum and not a saddle point, which would also be consistent with the reflection symmetry of the bisector. The minimum free energies corresponding to all the five strains are plotted in Fig. b where the solid circles connected by a solid line represent the relaxed free-energy curve and the open squares on a dashed line correspond to the unrelaxed free-energy curve. The curvatures of these two curves give the relaxed and unrelaxed c 66 at p000 kbar, respectively, from Eq. 13. Since relaxation always reduces G clearly the relaxed c 66 is less than the unrelaxed one. Repetition of such calculations at different pressures yields the pressure dependence of the internal relaxation effect on c 66. A similar internal relaxation effect on c 44 has also been found that is much smaller than that on c 66. Note that c 11 D. Bulk modulus B The bulk modulus B, unlike the c ij, is given by a second derivative of E, but with respect to volume rather than strain. Now B measures the change in uilibrium V produced by a change in p, i.e., BV dp dv, 14 which uses an ordinary derivative since at each p there is a unique uilibrium V and unique uilibrium E. Then the de produced by dv is given by the negative of the external work done by the system, depdv, hence p de/dv, and BV d E dv V d G V d G dv, 15 V d pv dv dv where GEpV is used and p is kept constant. If we let and the other ij 0 so that c/a is held constant while V changes, then GG 0 (V/) (c 11 c 1 4c 13 c 33 ). Replacing dv in Eq. 15 by 3Vd, which is valid because G is a minimum at uilibrium, defines a constrained bulk modulus B (c) : B (c) 1 9V d G d 9 c 11c 1 c 13 c 33 /. 16 To allow for a change of c/a when V changes so as to minimize G, express GG 0 in terms of V/V and 1 by putting 1 and 3 V/V 1. Then minimizing G G 0 with respect to 1 at constant V/V to eliminate 1 gives BV d G dv c 33c 11 c 1 c c 11 c 1 c 33 4c 13 Equation 17 is also given in Ref.. Later B (c) (p) and B(p) will be compared with each other and with measurements. Since c 11 c 1, c 13, and c 33 are not affected by the internal relaxation, there is no internal relaxation effect on either B (c) or B. III. RESULTS Figure 3a shows the G EBP (c/a;p) curves of hcp NM Fe in the vicinity of the uilibrium state at pressures from 0 to 5000 kbar, where the reference energy E 0 is the energy per atom in the ground state of hcp NM Fe at zero pressure. For clarity, the G EBP (c/a;p) curves at p kbar are shifted toward E 0 by 37, 73, 108, 141, 173, 364, 709, and 1604 mry/atom, respectively. Fitting of each G EBP (c/a;p) curve with a third-order polynomial in c/a and finding the minimum of G yields the uilibrium parameters G min, c/a,

5 ELASTICITY OF hcp NONMAGNETIC Fe UNDER PRESSURE PHYSICAL REVIEW B 68, FIG. 3. a G EBP (c/a;p) curves of hcp NM Fe in the vicinity of the uilibrium state at pressures from 0 to 5000 kbar. E Ry/atom is the energy per atom in the ground state of hcp NM Fe at zero pressure. For compact presentation, the G EBP (c/a;p) curves at pressures from 100 to 5000 kbar are shifted toward E 0 by 37, 73, 108, 141, 173, 364, 709, and 1604 mry/atom, respectively. b (G min E 0 ) at the minima of the G EBP (c/a;p) curves shown in a. In both a and b the solid lines interpolate between the calculated points. FIG. 4. a Equilibrium volume V and b c/a as functions of pressure obtained from the G EBP (c/a;p) curves shown in Fig. 3a compared with the experimental data from Refs. 10 open circles and 11 open triangles. The solid lines interpolate between the calculated points solid circles. and V at each pressure. The data points in Figure 3b correspond to (G min E 0 ) at the minima of the G EBP (c/a;p) curves shown in Fig. 3a. The uilibrium values of V and c/a of the G EBP (c/a;p) curves shown in Fig. 3a are plotted in Figs. 4a and 4b, respectively, along with the experimental results from Refs. 10 and 11. The calculated uilibrium V(p) curve is in good agreement with the experimental data at p1000 kbar but deviates below 1000 kbar, as shown in Fig. 4a. The quantities a, c, c/a, V, and c ij and the energies (G min E 0 ), pv, and EE 0 at the minima of the G EBP (c/a;p) curves as functions of pressure are tabulated in Table I. In Fig. 4b the sharp rise to a maximum of the calculated uilibrium values of c/a solid circles between 100 and 400 kbar is in good agreement with the measurements open circles reported in Ref. 10. Above 500 kbar the experimental values open triangles of c/a from Ref. 11 oscillate with pressure, but the average values are close to the calculated c/a curve and there is a maximum at about 400 kbar. Figure 5 shows the elastic constants c ij and the bulk modulus B of hcp NM Fe at p0, 1000, 000, and 5000 kbar calculated with the Eqs. 9 13, 16, and 17; the internal relaxation is included in the c ij calculations. The solid lines interpolate between the calculated points, which are used in the following figures to compare with the experimental data and the previously calculated results. The values of B (c) (p) open squares calculated using Eq. 16 and the values of B(p) signs calculated using Eq. 17 are essentially identical see Table I. For clarity the values of c ij and B at p kbar are not plotted in Fig. 5 and following figures, but all the values are tabulated in Table I. Figures 6a and 6b show the relaxed and unrelaxed values of c 66 and c 44, respectively, as functions of pressure. The solid lines in a and b give the relaxed c 66 and c 44, which are the same as shown in Fig. 5. The dashed lines in a and b give the unrelaxed c 66 and c 44 showing that the relaxation effect increases smoothly with increasing pressure and causes a reduction of 13% in c 66 and 5% in c 44 at p 5000 kbar. The solid triangles and solid circles are the calculated results from Refs. 1 and, respectively. The open circles are the experimental data from Ref. 1. The open squares are the experimental data from Ref. 13, in which the data of c 66 are obtained from the measured values of c 11 and c 1. Figure 6a shows that the calculated values of c 66 from Ref. are close to our unrelaxed curve, but the values from Ref. 1 are well above it, while the experimental data are close to the relaxed curve. Figure 6b shows that the calculated values of c 44 from Ref. are again close to our unrelaxed curve and Ref. 1 values above it, while experiment is still higher. Figures 7a and 7b show the relaxed and unrelaxed values of c 11 and c 1 as functions of pressure. The solid lines in a and b give the relaxed c 11 and c 1, which are the same as shown in Fig. 5. The dashed lines in a and b give the

6 S. L. QIU AND P. M. MARCUS PHYSICAL REVIEW B 68, TABLE I. Lattice parameters a, c, c/a, and V, energies (G min E 0 ) and (EE 0 ), elastic constants c ij, and bulk modulus B at the minima of the G EBP (c/a;p) curves of hcp NM Fe. E Ry/atom is the energy per atom in the hcp NM ground state. Pressure kbar a Å c Å c/a V (Å 3 /atom) G min E mry/atom EE mry/atom c c c c c c B (c) B unrelaxed c 11 and c 1. The solid triangles and solid circles are the calculated results from Refs. 1 and, respectively. The open circles and open squares are the experimental data from Refs. 1 and 13, respectively. Figure 7a shows again that the calculated values of c 11 from Refs. 1 and are close to the unrelaxed curves and the experimental data are close to the relaxed curve from this work. Large discrepancies in the calculated values of c 1 from Ref. 1 and experiment are apparent, as shown in Fig. 7b but c 1 from Ref. is close to our values. As mentioned in Sec. II the internal relaxation affects c 44, c 66, c 11, and c 1 but not c 11 c 1 ) and not c 13 and c 33 because the strains used to calculate c 13 and c 33 do not break the hexagonal symmetry. The solid lines in Figs. 8a and 8b give c 13 and c 33 as functions of pressure, respectively, which are the same as shown in Fig. 5. The solid triangles and solid circles are the calculated results from Refs. 1 and, respectively. The open circles and open squares are the experimental data from Refs. 1 and 13, respectively. The calculated values of c 13 from this work and from Ref. are in good agreement with each other and the experimental results, but values from Ref. 1 are well above. In contrast all the calculated values of c 33 agree with each other, but deviate significantly from the measurements. Figure 9 shows the comparison of the calculated and the measured bulk modulus B. The B (c) (p) curve calculated with Eq. 16 and the B(p) curve calculated with Eq. 17 are surprisingly close and are shown in Fig. 9 as a single solid line, which is the same as shown in Fig. 5. The solid circles are the calculated values from Ref. 1. The open circles and open squares are the experimental data from Refs. 10 and 13, respectively. The crosses are obtained from the measured sound velocity and the density of Fe under pressure reported in Ref. 14. IV. DISCUSSION The procedure used here for finding uilibrium properties under pressure is simple and direct. The uilibrium structure is found from the minimum of a function G EBP (a;p) defined in Eq. 3, and gives directly as functions of p both the uation of state V(p) and the form factor of the hcp structure (c/a)(p). No internal relaxation occurs because hydrostatic pressure preserves the hexagonal symmetry. The elastic constants at p are found directly from second derivatives of G(a,c; p) at uilibrium. These elastic constants do not either ruire reference to other work for the pressure corrections needed when second strain derivatives of E(a,c) at uilibrium are used or ruire the addition of second-order strains, as is done in Ref.. Direct numerical calculation of the effects of internal relaxation on the c ij is greatly simplified by forcing the relaxations to be one dimensional. The plot of V(p) in Fig. 3a agrees with Refs. 1 and and shows good agreement with experiment above 1000 kbar, but shows that theory is lower below 1000 kbar. Ref

7 ELASTICITY OF hcp NONMAGNETIC Fe UNDER PRESSURE PHYSICAL REVIEW B 68, FIG. 5. Relaxed elastic constants c ij and bulk modulus B of hcp NM Fe at p0, 1000, 000, and 5000 kbar. The values of B (c) (p) open squares calculated with Eq. 16 and the values of B(p) signs calculated with Eq. 17 are essentially identical see Table I. The solid lines interpolate between the calculated points, which are used in the following figures to compare with the experimental data and the previously calculated results. FIG. 7. a Relaxed and unrelaxed values of c 11 and b c 1 as functions of pressure. The solid lines in a and b give the relaxed values of c 11 and c 1, which are the same as shown in Fig. 5. The dashed lines in a and b give the unrelaxed values of c 11 and c 1. The solid triangles and solid circles are the calculated results from Refs. 1 and, respectively. The open circles and open squares are the experimental data from Refs. 1 and 13, respectively. FIG. 6. a Relaxed and unrelaxed values of c 66 and b c 44 as functions of pressure. The solid lines in a and b give the relaxed values of c 66 and c 44, which are the same as shown in Fig. 5. The dashed lines in a and b give the unrelaxed values of c 66 and c 44. The solid triangles and solid circles are the calculated results from Refs. 1 and, respectively. The open circles and open squares are the experimental data from Refs. 1 and 13, respectively. erence finds that an antiferromagnetic phase of hcp Fe shows better agreement of V( p) with experiment below 1000 kbar. The plot of c/a vs p in Fig. 4b at closely spaced values of p shows an abrupt behavior below 1000 kbar that rises from at p0 to a maximum of at 400 kbar, a rise of 1.5%, and then falls more gradually a few tenths of a percent to fairly constant values above 1000 kbar. This behavior corresponds well with one set of measurements; 10 other measurements oscillate strongly, 11 but also show a maximum at 400 kbar. This behavior of c/a is consistent with Ref., which states that c/a changed from 1.58 at p 0 to at 3. Mbar, but does not plot or tabulate the values. However, in an earlier paper 15 using a tight-binding model fitted to first-principles calculations, (c/a)(p) is shown as a monotonically rising convex smooth curve Fig. 1 of Ref. 15 for hcp Fe from c/a1.565 at p0 to at p1000 kbar, with no indication of a maximum. After the uilibrium structure at p is found, the elastic constants are found from second strain derivatives at the uilibrium structure of G(a,c; p), which is the function defined in Eq. 1. Now internal relaxation will occur for all strains that break hexagonal symmetry, i.e., the strains in Eqs. 1 and 13. The pressure correction, namely the contribution of the pv term in G to the elastic constants, will be needed for the strains that change the volume to second order in, namely, the strains in Eqs. 9, 11, 1, and

8 S. L. QIU AND P. M. MARCUS PHYSICAL REVIEW B 68, FIG. 8. c 13 and c 33 as functions of pressure. The solid lines in a and b are the same as shown in Fig. 5. Both c 13 and c 33 are not affected by the internal relaxation. The solid triangles and solid circles are the calculated results from Refs. 1 and, respectively. The open circles and open squares are the experimental data from Refs. 1 and 13, respectively. 13. The strain in Eq. 10 produces only a volume change linear in and hence does not contribute. The pressure corrections are identical with those given by Barron and Klein. 16 Comparison of our elastic constants with the firstprinciples calculation in Ref. 1 shows that Ref. 1 has not taken account of either the pressure correction needed when elastic constants are calculated just from second derivatives of E or the effects of internal relaxation. Reference 1 refers to Fast et al. 7 for the procedures to calculate hcp elastic constants, which are developed in Ref. 7 at p0 and are found as second strain derivatives of E which at p0 do not need a pressure correction. At p4 Mbar, Ref. 1 finds c 11 c Mbar using a strain that changes V but not the hexagonal symmetry, to be compared with our value of 39.8 Mbar. The pressure correction omitted in Ref. 1 would add p4 Mbar to c 11 c 1, which would bring the value close to ours. Reference 1 finds c 11 c Mbar at p4 Mbar with a strain that both breaks hexagonal symmetry and changes V; this value of c 11 c 1 is to be compared to our 1.1 Mbar. However, Ref. 1 omits a pressure correction that would subtract p4 Mbar and omits a relaxation correction that for our calculation of c 66 ) subtracted 1.8 Mbar from c 11 c 1 ; these two corrections would bring c 11 c 1 from Ref. 1 close to our value. The first-principles calculation in Ref. like Ref. 1 also finds elastic constants as second strain derivatives of the energy E at finite p but avoids the need for a pressure correction by using only volume-conserving strains. The volume FIG. 9. Comparison of the calculated and the measured bulk modulus B The solid line represents both the B (c) (p) curve calculated with Eq. 16 and the B(p) curve calculated with Eq. 17, which is the same as shown in Fig. 5. The solid circles are the calculated data from Ref. 1. The open circles and open squares are the experimental data from Refs. 10 and 13, respectively. The crosses are obtained from the measured sound velocity and the density of Fe under pressure reported in Ref. 14 using the formula B. conservation is obtained by including in the main diagonal of the strain matrix some contributions quadratic in as well as the usual components linear in. The components in contribute to the elastic constants because E is not a minimum in the uilibrium state at finite pressure p. Hence the expansion of E in powers of the strain components has terms linear in the strain components, which are then proportional to and contribute the pressure correction to the c ij. In general the elastic constants of Ref. are fairly close to ours: e.g., at p3. Mbar, Ref. has c 11 c Mbar vs our 33.9 Mbar and has c 11 c Mbar vs our 10.7 Mbar. Reference does not take account of internal relaxation in the calculation of c 44, and their value of c 44 is close to our unrelaxed curve, however, we find a small relaxation effect which increases with pressure. As noted in Sec. III, c 66 values in Ref. are also close to our unrelaxed curve, although stated to be relaxed. In summary the use of the EBP generalized to finite p and of the Gibbs free energy has simplified and unified the calculation of uilibrium structure and elastic constants under pressure. The pressure correction is automatically included. The pressure can be chosen arbitrarily at which the properties are wanted. The use of a special orientation of the unit cell and of only strains that preserve the reflection symmetry of the plane bisecting the 60 angle has simplified the calculations of internal relaxation, which are done here as functions of pressure. ACKNOWLEDGMENTS The calculations were carried out using the computational resources BOCA3 Beowulf at Charles E. Schmidt College of Science, Florida Atlantic University. P.M.M. thanks IBM for providing facilities at the Thomas J. Watson Research Center

9 ELASTICITY OF hcp NONMAGNETIC Fe UNDER PRESSURE 1 See, for example, P. Söderlind, J. A. Moriarty, and J. M. Wills, Phys. Rev. B 53, Gerd Steinle-Neumann, L. Stixrude, and R. E. Cohen, Phys. Rev. B 60, P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Luitz, computer code WIENk, an augmented plane wavelocal orbitals program for calculating crystal properties, Karlheinz Schwarz, Technical Universität Wien, Austria, Hong Ma, S. L. Qiu, and P. M. Marcus, Phys. Rev. B 66, P. M. Marcus, Hong Ma, and S. L. Qiu, J. Phys.: Condens. Matter 14, L J. F. Nye, Physical Properties of Crystals Clarendon Press, Oxford, L. Fast, J. M. Wills, B. Johansson, and O. Eriksson, Phys. Rev. B 51, PHYSICAL REVIEW B 68, R. A. Johnson, Modell. Simul. Mater. Sci. Eng. 1, F. Jona and P. M. Marcus, Phys. Rev. B 66, A. P. Jephcoat, H. K. Mao, and Peter M. Bell, J. Geophys. Res. 91, H. K. Mao, Y. Wu, C. Chen, and J. F. Shu, J. Geophys. Res. 95, A. K. Singh, H. K. Mao, J. Shu, and R. J. Hemley, Phys. Rev. Lett. 80, H. K. Mao, J. Shu, G. Shen, R. J. Hemley, B. Li, and A. K. Singh, Nature London 396, J. M. Brown and R. G. McQueen, Geophys. Res. Lett. 7, ; J. Geophys. Res. 91, R. E. Cohen, L. Stixrude, and E. Wasserman, Phys. Rev. B 56, T. H. K. Barron and M. L. Klein, Proc. Phys. Soc. London 85,