The Grüneisen parameter and its higher order derivatives for the Earth lower mantle and core

Physics of the Earth and Planetary Interiors 262 (2017) 41–47

Contents lists available at ScienceDirect

Physics of the Earth and Planetary Interiors journal homepage: www.elsevier.com/locate/pepi

The Grüneisen parameter and its higher order derivatives for the Earth lower mantle and core J. Shanker, K. Sunil ⇑, B.S. Sharma Dr. .B.R. Ambedkar University, Khandari Campus, Agra 282002, India

a r t i c l e

i n f o

Article history: Received 6 May 2016 Received in revised form 13 October 2016 Accepted 11 November 2016 Available online 12 November 2016 Keywords: Grüneisen parameter Higher order derivatives Infinite pressure behaviour Earth lower mantle and core

a b s t r a c t Expressions have been obtained for the higher order derivatives of the Grüneisen parameter in the limit of infinite pressure. The usefulness of these expressions has been demonstrated by considering a relationship between reciprocal Grüneisen parameter and the pressure-bulk modulus ratio at finite pressures. It has been found that the relationship under study satisfies the boundary conditions at infinite pressure for the higher order Grüneisen parameters. In order to investigate the applicability of the present formulation at finite pressures, we have calculated the Grüneisen parameter and its higher order derivatives for the Earth lower mantle and core using the input data from Stacey and Davis (2004). Ó 2016 Elsevier B.V. All rights reserved.

1. Introduction The Grüneisen parameter c is related to thermoelastic properties of materials (Anderson, 1995; Barton and Stacey, 1985; Stacey, 1995). The pressure derivatives of c play a central role in predicting the thermal behaviour, equation of state, and melting at high pressures (Kushwah et al. 2015; Stacey, 2005; Vijay 2014). The free volume theory of c. (Vashchenko et al., 1963) has been generalized (Stacey, 2005; Stacey and Davis (2004)) to obtain c as a function of pressure P, bulk modulus K and pressure derivative of bulk modulus K 0 ¼ dK=dP. This generalized free-volume formula can be written as

c¼

K0 2


 P  16  3f 1  3K P 1  2f 3K

ð1Þ

where f is the free volume parameter. Different formulations (Barton and Stacey, 1985; Dugdale and MacDonald, 1953; Slater, 1939) were developed for c by taking f equal to 0,1,2 and 2.35. Value of f can also be determined by matching the zero pressure value of c for the given material. In fact, values of f are quite uncertain mainly because of some invalid assumptions implied in the derivation of various formulations of c as discussed at length by Stacey and Davis (2004). Eq. (1) has been used to obtain the third order Grüneisen parameter for materials at infinite pressure (Shanker et al., 2009). The expressions at infinite pressure for c1 as well as for ⇑ Corresponding author. E-mail address: [email protected] (K. Sunil). http://dx.doi.org/10.1016/j.pepi.2016.11.002 0031-9201/Ó 2016 Elsevier B.V. All rights reserved.

the third order Grüneisen parameter k1 are independent of the free volume parameter f. However, at finite pressures when we use Eq. (1) to determine the higher order pressure derivatives of c, we need pressure derivatives of f, such as df/dP and d2f/dP2. These derivatives are even more uncertain than f itself. In the present study we propose a simple formula for the reciprocal c versus pressure-bulk modulus ratio. This formula satisfies the boundary conditions at zero pressure and also at infinite pressure which is a basic requirement for any thermodynamic relationship or equation of state to be physically acceptable. It should be mentioned that the infinite pressure values of thermoelastic properties (Kumar et al., 2015; Stacey, 2005) are the extrapolated values in the limit of infinite pressure by considering the material to remain in the same structure or same phase. In fact, no material can exist at infinite pressure. This point has been elaborated very convincingly by Stacey and Davis (2004). 2. Infinite pressure behaviour At infinite pressure, Eq. (1) gives

1 2

c1 ¼ K 01 

1 6

ð2Þ

Eq. (2) has the status of an identity (Stacey and Davis, 2004). The subscript 1. presents values at infinite pressure. K 01 is the value of K 0 ¼ dK=dP at infinite pressure. Eq. (2) reveals that c becomes linear in K 0 in the limit of infinite pressure. The second Grüneisen parameter

42

J. Shanker et al. / Physics of the Earth and Planetary Interiors 262 (2017) 41–47

q ¼



dc dV

V

c





dc dP

K

¼ 

c

T



Eq. (14) then gives

ð3Þ T

K 3 K IV KK 00

where K ¼  VðdP=dVÞT . We can write (Shanker et al., 2012)

q ðK=cÞ ðdc=dPÞ ¼  0 KK 00 KðdK =dPÞ

ð4Þ 0

where K 00 ¼ d K=dP ¼ dK =dP. At infinite pressure, Eq. (4) can be written as follows 2



q KK 00

2





1

¼ 

dc 0 dK

c1

1



ð5Þ

q KK 00



¼ finite constant

ð6Þ

1

q and KK 00 both are functions of P, and in the limit of infinite pressure both of them tend to zero, but their ratio is finite. Using calculus, we have the following relationship



q KK 00





dq=dP K K 00 þ KK 000

¼



ð7Þ

0

1



V k¼ q 

dq dV



3

K ¼  q

T



dq dP



ð8Þ T

Using Eq (8) in Eq (7) we get

q KK 00





¼



kq 0

00

2

K KK þ K K

1

ð9Þ

000

1

We can rewrite Eq (9) as follows

K 01

k1 ¼ 

K 2 K 000 KK 00



!

ð10Þ 1

Eq. (10) has been derived also by using the generalized free volume formula (Shanker et al., 2009). The fourth order Grüneisen parameter is defined as follows

n ¼ 



V k

dk dV





¼  T

K dk k dP



ð11Þ



K 0 KK 00 þ K 2 K 000

1

  qdk=dP þ kdq=dP ¼  2 IV K K þ 3KK 0 K 000 þ K 02 K 00 þ KK 002 1

4

n1 ¼

1 4 K 3 K IV k1 KK 00

! þ K 01

K 2 K 000 KK 00

! 1

!2 3 K 2 K 000 5  KK 00

ð13Þ

1

It should be mentioned that KK 00 , K 2 K 000 and K 3 K IV all become zero in the limit of infinite pressure, but the ratios appearing on right of Eq. (13) remain finite constants. Using the principle of calculus we can write

K 2 K 000 KK 00

!

¼ 1

! ð15Þ 1

3. Relationship between reciprocal c and P/K We propose a simple relationship for the reciprocal gamma as follows

1

¼ A þ B

c

2KK 0 K 000 þ K 2 K IV K 0 K 00 þ KK 000

!

ð14Þ 1

P K

ð16Þ

where A and B are material dependent constants

A¼

1

ð17Þ

c0

B ¼ K 01



1

c1



1

 ð18Þ

c0

The subscripts 0 and 1. present values at zero pressure and infinite pressure, respectively. We have used the following identity at infinite pressure (Knopoff, 1963)

  P 1 ¼ 0 K 1 K1

ð19Þ

The most important reason for considering Eq. (16) at finite pressures is that it leads to the boundary conditions for higher order Grüneisen parameters at infinite pressure which are identical to those derived in the preceding section using the principle of calculus. Eq. (16) when differentiated with respect to pressure gives

q

c

P ¼ Bð1  K 0 Þ K

ð20Þ

where q is the second order Grüneisen parameter (Eq. (3)). Now, differentiating Eq. (20) with respect to P, we find

qk

c



   P K0P ¼ B ðKK 00 Þ þ K 0 1  K K c

q2

ð21Þ

k¼

 KK 00 P þ K0 þ q 1  K 0 P=K K

ð22Þ

Eqs. (11) and (22) yield the following expression for the fourth order Grüneisen parameter

4

1

1

K 2 K 000 KK 00



ð12Þ

where K IV ¼ d K=dP . We can rewrite Eq. (12) as follows

2

 K 01

where k is the third order Grüneisen parameter (Eq. (8)). Eqs. (20) and (21) then yield

T

Using the principle of calculus, the RHS of Eq, (9) gives

kq

1

!2

Eqs. (13) and (15) taken together yield n1 ¼ 0. Thus we find that the first and third order Grüneisen parameters, c1 and k1 remain positive finite whereas the second and the fourth order Grüneisen parameters, q1 and n1 tend to zero. These are the important boundary conditions which must be satisfied by all physically acceptable equations of state and thermodynamic relationships.

1

where K 000 ¼ d K=dP . Eq. (7) is based on the principle of calculus that if two functions of a common variable both vanish or both become infinite at the same value of the variable then at that point the ratio of the functions is equal to the ratio of their derivatives with respect to the variable. The third order Grüneisen parameter k is defined as follows 3

K 2 K 000 KK 00

¼

1

Since c becomes linear in K 0 in the limit of infinite pressure, we can write



!

2

n¼

2KK 00 ðKK 00 P=KÞ K 2 K 000 ðP=KÞ  þq  0 2 0 kð1  K P=KÞ kð1  K P=KÞ kð1  K 0 P=KÞ

ð23Þ

We have thus used Eq. (16) for determining higher order parameters q, k and n given by Eqs (20), (22) and (23) respectively. At infinite pressure, Eqs. (19) and (20) give q1 ¼ 0, and Eq. (22) becomes

k1 ¼ K 01 þ

1 K 01



KK 00 1  K 0 P=K



ð24Þ 1

43

J. Shanker et al. / Physics of the Earth and Planetary Interiors 262 (2017) 41–47

We have the following identity (Shanker et al., 2009)

K 2 K 000 KK 00

!

¼  2K 01  1

1 K 01



KK 00 1  K 0 P=K

 ð25Þ 1

Using Eq. (25) in Eq. (24) we get Eq. (10) for k1 which also has the status of an identity. In order to determine n1 from Eq. (23), we write k1 in place of k and q1 equal to zero. At infinite pressure, Eq. (23) becomes

"  ! #   2 1 KK 00 1 KK 00 1 K 2 K 000 n1 ¼  2 þ þ k1 1  K 0 P=K 1 K 02 1  K 0 P=K 1 K 01 1  K 0 P=K 1 1

ð26Þ Eq. (24) can be rewritten as follows



KK 00 1  K 0 P=K

 ¼ 1

K 01

ðk1 

K 01 Þ

ð27Þ

Eq. (10) can be rewritten as follows

K 2 K 000 KK 00

!

¼  ðk1 þ K 01 Þ

ð28Þ

1

Eqs. (27) and (28) taken together yield

K 2 K 000 1  K 0 P=K

!

¼ K 01 ðk21  K 02 1Þ

ð29Þ

1

Substituting Eqs. (27) and (29) in Eq. (26), we find n1 ¼ 0, i.e. the fourth order Grüneisen parameter vanishes at infinite pressure. 4. Results for the Earth lower mantle and core The input data used in calculations are taken from Stacey and Davis (2004). Thus we have taken c0 ¼ 1:4545, c1 ¼ 1:0387, K 00 ¼ 4:2, and K 01 ¼ 2:41 for the lower mantle. For the core, c0 ¼ 1:8345, c1 ¼ 1:33, K 00 ¼ 4:96, and K 01 ¼ 3:0 are used as input. Values of P,K, K 0 and KK 00 are also taken from Tables 1 and 6 of Stacey and Davis (2004). Values of K 2 K 000 are determined from Eq. (63) of Stacey and Davis (2004). Values of c; q; k and n at different values of pressure are calculated from Eqs. (16), (20), (22) and (23) respectively. The results have been given here in Tables 1 and 2. Values of c and q for the Earth lower mantle and core are compared with the corresponding values reported by Stacey and Davis (2004) in Figs. 1–4. There have been continuous efforts for investigating the volume dependence of gamma. Thus earlier workers (Brennan and Stacey, 1979; Brown and McQueen, 1986; Anderson 1997) supported the

assumption that q remains constant equal to 1. This assumption is incompatible with our recent knowledge. The fact that q ! 0 at V ! 0 suggested to Stacey and Isaak (2001) that the next derivative represented by k (Eq. (8)) might be constant. However, there is no fundamental reason for believing that k is constant. To ensure that q ! 0 at V ! 0; k must remain finite. Calculation of q requires KK 00 , and k is determined from K 2 K 000 . Since these derivatives of bulk modulus change with pressure, q and k also change. Stacey and Davis (2004) have calculated k for the Earth lower mantle by differentiating the reciprocal K-primed equation and the equation for the acoustic c. They found that k decreases moderately with the increase in pressure, giving a negative value for dk=dP. The significance of the fourth order Grüneisen parameter n becomes clear as it is directly related to dk=dP (Eq. (11)), and determined from K 3 K IV. Stacey and Davis (2004) obtained two sets of c and q each using different values of input parameters for the lower mantle. There is a difference of about 5% between the two sets of values of c, and a difference of about 30% between the two sets of values of q. Thus c is accurate to within ±5% and q within ±30%. In view of these uncertainties the comparison presented in Figs. 1–4 is satisfactory. The results obtained in the present study and those reported by Stacey and Davis (2004) both reveal that values of c for the lower mantle decreases from 1.35 to 1.12, and q from 0.82 to 0.27. For the core, c decreases from 1.48 to 1.38 and q from 0.25 to 0.10. As pointed out by Stacey (2005), the acoustic model is not satisfactory for the core because of the conduction electrons in metals. Stacey and Davis (2004) therefore developed an alternative method by considering a linear relationship between k and q. This relationship was then integrated successively to obtain expressions for q and c. Values of k0 and k1 were scaled to c0 =c1 . Values of k remain between k0 ¼ 4:44 and k1 ¼ 3:06 for the core. Our values of k0 and k1 are, respectively, higher and smaller than the Stacey-Davis values. At finite pressures our values of k are between 2 and 4 for the lower mantle, and between 2 and 3 for the core. These values of k are not much different from the Stacey-Davis values which are between 3 and 4 at finite pressures. Most of the calculations for the Earth interior were performed by Stacey and Davis (2004) using the reciprocal K-primed equation. k0 and k1 were the exceptions. Subsequently, the method was developed (Shanker and Singh, 2005; Shanker et al., 2009) for determining k0 and k1 using the Stacey reciprocal K-primed equation. The last term in the identity represented by Eq. (24) remains negative, and therefore k1 must be less than K 01 . It should be mentioned that the expressions for KK 00 and K 2 K 000 obtained from the Stacey reciprocal K-primed equation when substituted in Eq. (10) yield (Shanker et al., 2009)

Table 1 Values of c, and higher order derivatives for the Earth lower mantle. r (km)

P (GPa)

c Eq. (16)

q Eq. (20)

k Eq. (22)

n Eq. (23)

3480 3600 3630 3800 4000 4200 4400 4600 4800 5000 5200 5400 5600 5701

135.75 128.71 126.97 117.35 106.39 95.76 85.43 75.36 65.52 55.90 46.46 37.29 28.29 23.83 P=0

1.2159 1.2199 1.2209 1.2269 1.2345 1.2427 1.2518 1.2619 1.2731 1.2860 1.3008 1.3180 1.3386 1.3507 1.4545

0.3012 0.3096 0.3118 0.3247 0.3412 0.3595 0.3799 0.4033 0.4300 0.4613 0.4987 0.5434 0.5994 0.6333 0.9646

2.5582 2.5911 2.6009 2.6496 2.7113 2.7841 2.8634 2.9539 3.0576 3.1858 3.3244 3.4986 3.7172 3.8502 5.1646

1.1601 1.1940 1.2004 1.2519 1.3175 1.3891 1.4694 1.5610 1.6644 1.7720 1.9386 2.1160 2.3412 2.4811 3.8774

44

J. Shanker et al. / Physics of the Earth and Planetary Interiors 262 (2017) 41–47

Table 2 Values of c, and higher order derivatives for the Earth core. r (km)

P (GPa)

c Eq. (16)

q Eq. (20)

k Eq. (22)

n Eq. (23)

0 200 400 600 800 1000 1200 1221.5 1221.5 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 3480

363.85 362.90 360.03 355.28 348.67 340.24 330.05 328.85 328.85 318.75 306.15 292.22 277.04 260.68 243.25 224.85 205.60 185.64 165.12 144.19 135.75 P=0

1.4286 1.4288 1.4290 1.4298 1.4306 1.4320 1.4335 1.4337 1.4278 1.4294 1.4314 1.4339 1.4368 1.4403 1.4443 1.4489 1.4545 1.4607 1.4689 1.4786 1.4830 1.8345

0.1434 0.1436 0.1442 0.1455 0.1469 0.1491 0.1516 0.1520 0.1420 0.1447 0.1483 0.1522 0.1570 0.1629 0.1699 0.1778 0.1875 0.1982 0.2129 0.2303 0.2385 1.1277

2.3653 2.3660 2.3683 2.3747 2.3793 2.3879 2.3964 2.3846 2.3605 2.3711 2.3850 2.3987 2.4166 2.4394 2.4671 2.4960 2.5333 2.5707 2.6290 2.6966 2.7260 6.0876

0.5453 0.5470 0.5492 0.5534 0.5589 0.5670 0.5765 0.5798 0.5406 0.5506 0.5647 0.5780 0.5970 0.6191 0.6458 0.6744 0.7110 0.7512 0.8060 0.8707 0.9012 4.3214

1.40

Earth lower mantle

1.35 1.30

Present study

γ

1.25 1.20

Stacey, Davis (2004)

1.15 1.10

20

40

60

80

100

120

140

P(GPa) Fig. 1. Values of c as a function of pressure determined in the present study using the reciprocal gamma equation (16) compared with the corresponding values reported by Stacey and Davis (2004) for the Earth lower mantle.

Earth lower mantle

0.9 0.8 0.7 0.6

Stacey, Davis (2004)

q

0.5 0.4

Present study

0.3 0.2 0.1

20

40

60

80

100

120

140

P(GPa) Fig. 2. Values of q as a function of pressure determined in the present study using Eq. (20) compared with the corresponding values reported by Stacey and Davis (2004) for the Earth lower mantle.

45

J. Shanker et al. / Physics of the Earth and Planetary Interiors 262 (2017) 41–47

Earth core

1.50 1.48 1.46

Present study

γ

1.44 1.42

Stacey, Davis (2004)

1.40 1.38 100

150

200

250

300

350

P(GPa) Fig. 3. Values of c as a function of pressure determined in the present study using the reciprocal gamma Eq. (16) compared with the corresponding values reported by Stacey and Davis (2004) for the Earth core.

Earth core 0.28 0.26 0.24

Stacey, Davis (2004)

0.22

q

0.20

Present study

0.18 0.16 0.14 0.12 0.10 100

150

200

250

300

350

P(GPa) Fig. 4. Values of q as a function of pressure determined in the present study using Eq. (20) compared with the corresponding values reported by Stacey and Davis (2004) for the Earth core.

k1 ¼

K 02 1 K 00

ð30Þ

Eq. (30) gives k1 ¼ 1:38 for the lower mantle, and 1.81 for the core. We note from Eq. (30) that k1 < K 01 since K 01 < K 00 . We have used only four parameters as input viz. c0 , c1 , K 00 and 0 K 1 . Values of q0 ; k0 and n0 depend on these parameters. Eqs. (20), (22) and (23) at zero pressure reduce to the following relations

q0 ¼ c0 K 01 k0 ¼ n0 ¼

K 00



1

c1



1

c0

þ q0

2K 0 K 000 k0



ð31Þ ð32Þ

þ q0

ð33Þ

c1 and K 01 are related to each other by Eq. (2), so effectively there are only three input parameters used in present calculations. It is worth mentioning here that Eq. (16) is similar to the reciprocal K-primed equation given below (Stacey, 2000)

1 P ¼ aþb K K0

ð34Þ

where a ¼ 1=K 00 ; and

b ¼ K 01



1 1  K 01 K 00

 ð35Þ

Eqs. (16) and (34) yield the following relationship between 1=c and 1=K 0

 1

c



1

c0



1 1  K 01 K 00



 ¼

1 1  K 0 K 00



1

c1



1

c0

 ð36Þ

Eq. (36) is satisfied by the values of c and K 0 reported by Stacey and Davis (2004) for the lower mantle and core. This is demonstrated in Figs. 5 and 6. The plots are approximately straight lines supporting the validity of Eq. (36). It should be pointed out that Eq. (36) yields expressions for k1 (Eq. (10)) and for n1 (Eq. (13)) when differentiated successively with respect to pressure up to third order and fourth order, respectively. Eq. (34) due to Stacey (2000) when differentiated successively with respect to pressure

46

J. Shanker et al. / Physics of the Earth and Planetary Interiors 262 (2017) 41–47

Earth lower mantle

0.20 0.18 0.16 0.14 0.12 0.10 0.08 0.03

0.04

0.05

⎛ 1 1 ⎜⎜ − ′ ⎝ K K ′0 Fig. 5. Values of



1

c

 c1



0

plotted versus



1 K0

 K10



0.06

0.07

0.08

0.09

⎞ ⎟⎟ ⎠

for the Earth lower mantle using the data reported by Stacey and Davis (2004). The linear plot verifies Eq. (36).

0

Earth core

0.180 0.175 0.170 0.165 0.160 0.155 0.150 0.145 0.080

0.085

⎛ 1 1 ⎜⎜ − ′ ′0 K K ⎝ Fig. 6. Values of



1

c

 c1

0



plotted versus



1 K0

 K10



0.090

0.095

0.100

⎞ ⎟⎟ ⎠

for the Earth core using the data reported by Stacey and Davis (2004). The linear plot verifies Eq. (36).

0

gives the RHS of Eq. (10) equal to the RHS of Eq. (24), and the RHS of Eq. (13) equal to the RHS of Eq. (26) in the limit of infinite pressure. It should be mentioned that Eqs (24) and (26) are based on Eq. (16), the reciprocal gamma formula. Thus Eq. (16) is compatible with Eq. (34).

5. Conclusions It has been possible to have a simple relationship between 1=c and P/K at finite pressures. A strong point in favour of this relationship is that it leads to expressions for higher order Grüneisen parameters at infinite pressure which are identical to those derived from the basic principles of calculus (Shanker et al. 2009; Shanker et al. 2012). The results for the lower mantle and core given in Tables 1 and 2 are similar to those reported by Stacey and Davis (2004). c; q; k and n all decrease with the increase in pressure. At infinite pressure, q1 and n1 both become zero, but c1 and k1 remain positive finite. As pointed out by Stacey and Davis (2004),

q1 equal to zero is a confirmatory test for a positive finite value of c1 , the result n1 equal to zero confirms that k1 is positive finite. It should be mentioned that Eq. (16) is more suitable for geophysical applications since P and K both are available directly from the seismological data. The reciprocal gamma equation has been shown to be compatible with the Stacey reciprocal K-primed equation. Both the equations yield similar expressions for the higher order derivatives at infinite pressure. Acknowledgements We are thankful to both the reviewers for their helpful comments which have been very useful in revising the manuscript. References Anderson, O.L., 1995. Equation of State of Solids for Geophysics and Ceramic Science. Oxford University Press, New York, p. 405. Anderson, O.L., 1997. The volume dependence of thermal pressure of solids. J. Phys. Chem. Solids 58, 335–343.

J. Shanker et al. / Physics of the Earth and Planetary Interiors 262 (2017) 41–47 Barton, M.A., Stacey, F.D., 1985. The Grüneisen parameter at high pressure: a molecular dynamic study. Phys. Earth Planet. Inter. 39, 167–177. Brennan, B.J., Stacey, F.D., 1979. A thermodynamically based equation of state for the lower mantle. J. Geophys. Res. 84, 5535–5539. Brown, J.M., McQueen, R.G., 1986. Phase transitions, Grüneisen parameter, and elasticity for shocked iron between 77 GPa and 400 GPa. J. Geophys. Res. 91, 7485–7494. Dugdale, J.S., MacDonald, D.K.C., 1953. Thermal expansion of solids. Phys. Rev. 89, 832–834. Knopoff, L., 1963. The theory of finite strain and compressibility of solids. J. Geophys. Res. 68, 2929–2932. Kumar, S., Sharma, S.K., Pandey, O.P., 2015. Analysis of thermodynamic properties in the limit of infinite pressure. High Temp.-High Press. 44, 339–350. Kushwah, S.S., Upadhyay, A.K., Sharma, M.P., 2015. Equation of state and melting of some bcc transition metals. High Temp.-High Press. 44, 59–68. Shanker, J., Dulari, P., Singh, P.K., 2009. Extreme compression behaviour of equations of state. Physica B 404, 4083–4085. Shanker, J., Singh, B.P., 2005. A comparative study of Keane’s and Stacey’s equations of state. Physica B 370, 78–83.

47

Shanker, J., Sunil, K., Sharma, B.S., 2012. Formulation of the third order Gruneisen parameter at extreme compression. Physica B 407, 2082–2083. Slater, J.C., 1939. Introduction to Chemical Physics. McGraw- Hill, New York, p. 521. Stacey, F.D., 1995. Theory of the thermal and elastic properties of the lower mantle and the core. Phys. Earth Planet. Inter. 89, 219–245. Stacey, F.D., 2000. The K-primed approach to high pressure equations of state. Geophys. J. Int. 143, 621–628. Stacey, F.D., 2005. High pressure equations of state and planetary interiors. Rep. Prog. Phys. 68, 341–383. Stacey, F.D., Davis, P.M., 2004. High pressure equations of state with applications to the lower mantle and core. Phys. Earth Planet. Inter. 142, 137–184. Stacey, F.D., Isaak, D.G., 2001. Compositional constraints on the equation of state and thermal properties of the lower mantle. Geophys. J. Int. 146, 143–154. Vashchenko, V.Ya., Zubarev, V.N., 1963. Concerning the Grüneisen constant. Sov. Phys. Solid State 5, 653–655. Vijay, A., 2014. Analysis of the volume dependence of Grüneisen parameter and melting of hcp-iron. High Temp.-High Press. 43, 47–53.