EOS of simple Solids for wide Ranges in p and T

EOS of simple Solids for wide Ranges in p and T

Washington 2007

Wilfried B. Holzapfel Department Physik Universität-Paderborn Germany <[email protected]>

Problem:The accuracy of “primary” K(V)-scales is still seriously limited by the present range and precision in measurements of K and V under high p and T.

Solution:Semi-empirical EOS with theoretical and experimental input provide p(V,T)-relations of many “simple” materials for the determination of p with higher accuracy from measurements of V and T.Comparison of different markers gives estimates of the accuracy!


Washington 2007


EOS of “simple” or “regular” solids are well understood theoretically!

All thermo-physical data including the EOS must be modeled by the same thermodynamic potential (the same Gibbs function).

“Cold” (0 K) isotherms can be determined from a priory theoryor from semi-empirical effective potential forms (APL).

Thermal contributions are accurately modeled by thermodynamicswith experimental input from ambient pressure and theoreticalsupport for the volume dependence of the intrinsic anharmonicity.


Washington 2007


The approach:

1) Set up a thermodynamic model for the solid with anharmonicity!2) Use Kr and Kr’ values from ultrasonic or Brillouin measurements for one (cold) reference isotherm!3) Use ambient pressure data of Vo(T), αo(T),Cpo(T) and Ko(T) to determine thermo-physical parameters (θo, γo, ...) of the model!4) Refine Kr’ as best fitting Ko(T)!



The thermo-physical model for the Gibbs function: Total Free-Energy: F(V,T) = Ec(V) + Fth (V,T) with Ground State Energy Ec(V) and thermal Excitations in Fth(V,T)

Ec(V) from one cold isotherm: pc(V) = pAP2(V,Z,Vo,Ko,K’o)

Fth(V,T) = Fcond.el.(T,TFG(V)) + Fquasi-harm.phonon(V,T) + Fintr.anharm.(V,T)

Quasi-harmonic Phonons are modeled with an optimized pseudo-Debye-Einstein approximation: opDE, Intrinsic Anharmonicity with a Modified Mean Field approach: MMF

Washington 2007

Effective potential forms for cold isotherm

Washington 2007

Mie EOS (Mi3)

Effective Rydberg EOS (ER2)

Adapted Polynomial EOS (AP2)

Thomas-Fermi-limit is modeled by co!


m nMi3 0p (3/ n) K x (1 x )

ER 2 0 ER 22

1 xp 3 K exp(c (1 x))

x

ER 2 0c (3/ 2)(K 1)

AP2 0 0 25

1 xp 3 K exp(c (1 x)) 1 c x (1 x)

x

5 / 3FG0 FG 0p a (Z / V ) 0 0 FG0c ln(3 K / p ) 5

FGa 0.02337GPa nm

2 0 0c (3 / 2) (K 3) c

1/ 30x (V / V )

An optimized pseudo-Debye-Einstein model


Cold Isotherm:

The Mie-Grüneisen approach for the internal energy gives :

with

An optimized pseudo-Debye-Einstein model for quasi-harmonic phonons

with g=0.068 a=0.0434

and

is conveniently used here!

4

3( ) (1 )

exp( / ) 1E

opDEE

t fu t g g

a g t f t

30

0

3( )

4 (1 )Dh

ED

f gg

( , ) 3 ( ) ( )MG DU V T R V u t / ( )Dt T V

Washington 2007

Intrinsic AnharmonicityClassical Free Volume Approach with Modified Mean Field Potential g(r,V)

J.G. Kirkwood, J. Chem. Phys. 18, 380 (1950) Y. Wang, Phys. Rev. B 61, R11863 (2000)

Prag 2006


( , ) ( ) ( ) 2 ( ) ( ) ( )2 C C C C C

b s f rg r V E R r E R r E R E R r E R r

R

w i t h c o r r e l a t i o n p a r a m e t e r :

2,( , ) 4 e x p

B

g r Vv f V T r d r

k T

3

0 0( ) /R V R V V f o r f c c : 6 3

0 02R V

2

3( , ) l n l n ( , )

2 2B

v i b B

m k TF V T N k T v f V T

l n ( , )( , )

l nB

v i b

N k T v f V Tp V T

V V

( , ) 3 l n ( , )( , )

2 l nv i b

v i b B

F V T v f V TS V T N k

T T

3 l n ( , )( , ) ( , ) ( , )

2 l nv i b v i b v i b B

v f V TU V T F V T T S V T N k T

T

2

2

3 l n ( , ) l n ( , )( , )

2 l n ( l n )v i b B

v f V T v f V TC V T N k

T T

Intrinsic AnharmonicityClassical Free Volume Approach with Modified Mean Field Potential g(r,V)

J.G. Kirkwood, J. Chem. Phys. 18, 380 (1950) Y. Wang, Phys. Rev. B 61, R11863 (2000)

Prag 2006


l n ( , ) / l n

( 3 / 2 ) l n ( , ) / l n( , ) v i b

b a r i cv i b

p V v f V T V

U v f V T TV T

( , ) v Tt h e r m a l

V

K VV T

C

2

2 2

l n ( , ) / l n l n ( , ) / ( l n l n )( , )

( 3 / 2 ) l n ( , ) / l n l n ( , ) / ( l n )t h e r m a l

v f V T V v f V T V TV T

v f V T T v f V T T

Intrinsic Anharmonicity Two Series Expansions in the Classical Free Volume Approach


( , ) ( ) ( ) 2 ( ) ( ) ( )2 C C C C C


R

1 ) 2 4 62 ( ) 4 ( )

( , ) ( )2 ! 4 !

k V k Vg r V r r O r

22

( )2 ( ) Cb s f E R

k V RR R R

34

4 3

( )4 ( ) Cb s f E R

k V RR R R

2,( , ) 4 e x p

B

g r Vv f V T r d r

k T

4 2 ( )( )

3M P FB

k VV

k b s f m

2 )

4 224 ( ) 2 ( )

( , ) 4 1 e x p4 ! 2 !B B

k V r k V rv f V T r d r

k T k T

3 / 2

2 22( , ) 1 3 ( ) 6 . 3 ( )

2 ( )B

v f v f

k Tv f V T a V T a V T

k V

2

5 4 ( )( )

2 4 2 ( )B

v f

k k Va V

k V

2 2

0 03

0 0 0 0

2

4D D B

M P F

k mb s f

K K V

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Vibrational Free-Energy in the Classical Free-Volume Approachwith linear contributions from and only.


vfa avf

3 / 2

2( , ) 1 3 ( )

2 ( )B

v f

k Tv f V T a V T

k V

1 2 ( )( , ) 3 l n l n ( )

2B

v i b B v f

k T k VF V T N k T a V T

m

( , ) 3 1 ( )v i b B v fU V T N k T a V T

3( , ) ( ) ( ) ( )B

v i b v i b v f a v f

N k Tp V T V a V V T

V ( , ) 3 1 2 ( )V B v fC V T N k a V T

ln ( )( )

lnvf

avf

a VV

V

ln ( ) 1 ln 2( )

( )ln 2 ln

V k VV

V VDvib

0 0( ) 2( ) / 2( )D DV k V k V

( , ) ( ) (1 )V T V Tbar vib A

( , ) ( ) (1 2 )V T V Tthm vib A

( ) (1 ( ) / ( ))A vf avf viba V V V

0A Mie-Grüneisen Approximation implies

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Vibrational Grüneisen Parameterin the classical free volume approach


A c o n s t a n t c o r r e l a t i o n p a r a m e t e r

/ 0d d V i n t h e M M F - p o t e n t i a l

( , ) ( ) ( ) 2 ( ) ( ) ( )2 C C C C C


R

r e s u l t s i n 22

( )2 ( ) Cb s f E R

k V RR R R

a n d

( ) ( 2 / 3 ) ( 1 ) 1( )

2 ( 1 ( 2 / 3 ) ( 1 ) ( ( ) / ( ) ) 6C

v i bC C

K VV

p V K V

V a s h c h e n k o - Z u b a r e v ( 1 9 6 3 ) , B a r t o n - S t a c e y ( 1 9 8 5 )

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( , ) ( ) ( ) 2 ( ) ( ) ( )2 C C C C C


R


( )2 ( ) Cb s f E R

k V RR R R

a n d

( ) ( 2 / 3 ) ( 1 ) 1( )

2 ( 1 ( 2 / 3 ) ( 1 ) ( ( ) / ( ) ) 6C

v i bC C

K VV

p V K V




( , ) ( ) ( ) 2 ( ) ( ) ( )2 C C C C C


R


( )2 ( ) Cb s f E R

k V RR R R

a n d

( ) ( 2 / 3 ) ( 1 ) 1( )

2 ( 1 ( 2 / 3 ) ( 1 ) ( ( ) / ( ) ) 6C

v i bC C

K VV

p V K V




( , ) ( ) ( ) 2 ( ) ( ) ( )2 C C C C C


R


( )2 ( ) Cb s f E R

k V RR R R

a n d

( ) ( 2 / 3 ) ( 1 ) 1( )

2 ( 1 ( 2 / 3 ) ( 1 ) ( ( ) / ( ) ) 6C

v i bC C

K VV

p V K V


Higher Order Corrections in the Free Volume Approach


2_ 1 . _ 3c l a s s i c o r d e r a n h a r m o n i c i t y v fF R a T

Q u a n t u m c o r r e c t i o n s + h i g h e r o r d e r :

2( ) ( )3 4 ( ) ( 1 6 )T T

a n h v f D q h v f D q hD D

F R f a u f a u

w i t h q u a s i - h a r m o n i c o p t i m i z e d p s e u d o - D e b y e - E i n s t e i n u q h

q u a r t i c c o r r e c t i o n f a c t o r f 4

a n d h e x i c c o r r e c t i o n f a c t o r f 6

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Heat capacity of Cu at ambient pressure

Prag 2006

Experimental data +, o, x and • from Ly59, Ma60, Gr72 and Ch98

Present fit of Cpo and Cvo: solid and dashed line


0 200 400 600 800 1000 12000

0.5

1cpo T( )

cvo T( )

cp1 nc1

cp2 nc2

cp3 nc3

cp4 nc4

T T Tc1 nc1 Tc2 nc2 Tc3 nc3 Tc4 nc4

Cpo(T) 3R

T (K)

Heat capacity of Ag at ambient pressure

Experimental data o, x and • from Mo36, MF41, and Gr72


0 200 400 600 800 1000 12000

0.5

1cpo T( )

cvo T( )

cp1 nc1

cp2 nc2

cp3 nc3

T T Tc1 nc1 Tc2 nc2 Tc3 nc3


Cpo(T) 3R

T (K)

Heat capacity of Au at ambient pressure

Experimental data x and • from GG52, and Gr72



0 200 400 600 800 1000 12000

0.5

1cpo T( )

cvo T( )

cp1 nc1

cp2 nc2

cp3 nc3

T T Tc1 nc1 Tc2 nc2 Tc3 nc3 T (K)

Cpo(T) 3R

Vo(T) Vr

T (K)

Vo(T) Vr

x 10000

0 200 400 600 800 1000 12001.5

1

0.5

0

0.5

1

1.5

10 rvnv

D T( )

Tv nv T

0 200 400 600 800 1000 1200

1

1.02

1.04

1.06

vxnv

v0T T( )

Tv nv T

_

Relative atomic volume of Cu at ambient pressure

Experimental data • from TK75 and present fit as solid line

0 200 400 600 800 1000 1200

0 200 400 600 800 1000 1200

1

1.02

1.04

1.06

vxnv

v0T T( )

Tv nv T

0 200 400 600 800 1000 12001.5

1

0.5

0

0.5

1

1.5

10 rvnv

D T( )

Tv nv T

Vo(T) Vr

T (K)

Relative atomic volume of Ag at ambient pressure


0 200 400 600 800 1000 1200

Vo(T) Vr

x 10000

0 200 400 600 800 1000 1200

1

1.02

1.04

vxnv

v0T T( )

Tv nv T

0 200 400 600 800 1000 12001.5

1

0.5

0

0.5

1

1.5

10 rvnv

D T( )

Tv nv TT (K)

Relative atomic volume of Au at ambient pressure


Vo(T) Vr

x 10000

Vo(T) Vr

0 200 400 600 800 1000 1200

Ko(T) (GPa)

T (K)

0 200 400 600 800 1000 1200

100

120

140

KoT1 nK1

KoT2 nK2

K0a T( )

Koo T( )

TK1 nK1 TK2 nK2 T

Isothermal bulk modulus of Cu at ambient pressure

Experimental data o from CH66 and • from VT79

Present fit with and without anharmonic contributions: solid and dashed line

Isothermal bulk modulus of Ag at ambient pressure

Experimental data • from CH66 and o from BV81


0 200 400 600 800 1000 1200

80

100KoT1 nK1

KoT2 nK2

K0a T( )

Koo T( )

TK1 nK1 TK2 nK2 T

Ko(T) (GPa)

T (K)

0 200 400 600 800 1000 1200

120

140

160

180

KoT1 nK1

KoT2 nK2

K0a T( )

Koo T( )

TK1 nK1 TK2 nK2 TT (K)

Ko(T) (GPa)

Isothermal bulk modulus of Au at ambient pressure

Experimental data • BV81 and o from CH66


0 200 400 600 800 1000 12005

5.5

6

6.5

Ksotot T( )

KosT1 nK1

KosT31

T TK1 nK1 KosT30

Pressure derivative of the isothermal bulk modulus

for Cu at ambient pressure

Temperature dependent data • from VT79 with additional data for 300 K.

Present fit with anharmonic contributions: solid line

T (K)

Kó(T)

0 200 400 600 800 1000 12005.5

6

6.5

7

7.5

8

Ksotot T( )

KosT1 nK2

KosT31

T TK2 nK2 KosT30


for Ag at ambient pressure

Temperature dependent data • from BV81 with additional data for 300 K.


T (K)

Kó(T)

T 5 105 1305

0 200 400 600 800 1000 12005.5

6

6.5

7

7.5

Ksotot T( )

KosT1 nK1

KosT31

T TK1 nK1 KosT30


for Au at ambient pressure

Temperature dependent data • from BV81 with additional data for 300 K.


T (K)

Kó(T)

Hierarchy of Parameters in the Refinements

1. Fit of Cpo(T) at low T by Refinement of TDo and TDh

2. Fit of Vo(T) at low T by Refinement of r

3. Fit of Cpo(T) and Vo(T) at higher T by Refinement of Anharmonicity Parameters f4 and f6

4. Minor Refinement of K’r to improve Fit of Ko(T)

5. Final Check of Step 2. and 3.


Vr(cnm)

Kr(GPa)

TDo(K)

K’r

TDh(K)

r

f6

1000xAph

f4

A

TFeff(K)

Z

M

f2

q

AgCu Au29 47 7964 108 197

57213 62220 5562611,814 17,056 16,964133,1 101,0 166,45,40 6,20 6,21330 214 160328 223 183

2,00 2,46 3,112,11 3,07 3,878,00 0,80 1,580,36 0,34 0,290,60 0,72 -1,310,30 0,48 0,671,03 1,48 2,27

-2,87 -3,94 -5,45

Thermo-physical Parameters for

Discussion

1. Perfect representation of all thermo-physical data at ambient P(for “regular” Solids only!)

2. Uncertainties in (V) and Aph(V) are reduced by MMF-calculation(No experimental data give better constraints!)

3. Reliable basis for the calculation of thermo-physical data at any P

4. Perfect agreement with Shock Wave Data


Comment on Parametric EOS

If the cold isotherm pc(V) is represented by pAP2(V,Vo,Ko,K’o)

(or by any other second order parametric EOS)

no other isotherm is perfectly represented by the same formeven with best fitted “effective values” for Vr, Kr, K’r

deviating from the thermodynamic values!

Accurate representations of the present thermodynamic EOS by parametric EOS

need higher order forms with “effective” parameters!


PPT2 5 C3

2 PPT

5 5 C35

PPT8 5 C3

8

R3 v tf( ) r3 v tf C3tf

0 100 200 300 400 500 600

0.01

0.005

0

0.005

0.01

R3 v 10( )

N v( )

R3 v 5( )

R3 v 0( )

pAP3 xv v( )( )

Difference between thermodynamic and parametric EOS for Cu a) pAP2 b) pAP3

0 100 200 300 400 500 600

0.01

0.005

0

0.005

0.01

Reff v 10( )

N v( )

Reff v 5( )

Reff v 0( )

r500 v( )

pAP3 xv v( )( )

0 100 200 300 400 500 600

0.01

0.005

0

0.005

0.01.011

.011

rT v 10( )

N v( )

rT v 5( )

rT v 0( )

60010 pAP3 xv v( )( )

p

p

p

p

p / GPa

c) pAP2 + pAP3with data point from HHS-01 for 500K

p / GPa

0 K

500 K HHS 01: 500 K1000 K

0 K

500 K 1000 K

0 K

500 K

1000 K

0 100 200 300 400 500 600

0.01

0.005

0

0.005

0.01

rT v 10( )

N v( )

rT v 5( )

rT v 0( )

pAP3 xv v( )( )

0 100 200 300 400 500 600

0.015

0.01

0.005

0

0.005

Reff v 10( )

N v( )

Reff v 5( )

Reff v 0( )

r500 v( )

pAP3 xv v( )( )

0 100 200 300 400 500 600

0.01

0.005

0

0.005

0.01

rT v 10( )

N v( )

rT v 5( )

rT v 0( )

pAP3 xv v( )( )

0 100 200 300 400 500 600

0.015

0.01

0.005

0

0.005

Reff v 10( )

N v( )

Reff v 5( )

Reff v 0( )

r500 v( )

pAP3 xv v( )( )

0 100 200 300 400 500 600

0.015

0.01

0.005

0

0.005

Reff v 10( )

N v( )

Reff v 5( )

Reff v 0( )

r500 v( )

pAP3 xv v( )( )

p

p

p / GPap / GPa

Differences between thermodynamic and parametric EOS using pAP2+pAP3 for Ag and Au

Data from HHS 01 for 500K are given for comparison

0 K

500 K

1000 K

HHS 01: 500 K

0 K

500 K

1000 K

HHS 01: 500 K

0 11,695 141,7 5,25 5,29 0,45100 11,711 140,2 5,29 5,32 0,30200 11,758 136,8 5,34 5,36 0,15300 11,814 133,1 5,40 5,40 -0,01400 11,875 129,4 5,46 5,44 -0,19500 11,938 125,6 5,52 5,49 -0,40600 12,005 121,7 5,60 5,54 -0,63700 12,074 117,8 5,67 5,59 -0,89800 12,146 113,8 5,76 5,64 -1,19900 12,222 109,7 5,86 5,71 -1,55

1000 12,301 105,5 5,98 5,77 -1,97

Reference data for the parametric EOS of Cu

T Vr Kr K’r K’reff c33 310 nmK GPa

Reference data for parametric EOS of Ag


0 16,841 110,68 5,92 6,01 0,88100 16,879 108,56 6,00 6,06 0,58200 16,963 104,89 6,10 6,12 0,27300 17,056 101,11 6,20 6,19 -0,06400 17,156 97,28 6,31 6,27 -0,40500 17,260 93,41 6,42 6,35 -0,76600 17,369 89,51 6,55 6,43 -1,11700 17,484 85,58 6,67 6,52 -1,48800 17,604 81,63 6,81 6,62 -1,84900 17,731 77,64 6,95 6,72 -2,20

1000 17,864 73,64 7,11 6,83 -2,55

Reference data for parametric EOS of Au


0 16,791 180,6 5,89 6,00 1,52100 16,825 177,2 5,98 6,06 0,97200 16,892 172,0 6,10 6,13 0,39300 16,964 166,7 6,21 6,20 -0,17400 17,040 161,5 6,33 6,27 -0,72500 17,118 156,3 6,44 6,34 -1,24600 17,198 151,1 6,56 6,42 -1,75700 17,281 145,8 6,68 6,49 -2,26800 17,367 140,7 6,81 6,57 -2,79900 17,456 135,5 6,95 6,66 -3,38

1000 17,548 130,2 7,11 6,75 -4,11

EOS for Solids with Mean-Field Anharmonicity

Prag 2006

Software and Supportavailable on Request!

Cooperation welcome!


[WR57] J.M. Walsh, M.H. Rice, R.G. McQueen, and F.L. Yarger, Phys. Rev. 108 196 (1957)

[KK72] R.N. Keeler, and G.C. Kennedy, American Institut of Physics Handbook, 4938, ed. D.E. Gray, New York: McGraw Hill (1972)

[MB78] H.K. Mao, P.M. Bell, J.W. Shaner, and D.J. Steinberg, J. Appl. Phys. 49 3276 (1978)

[AM85] R.C.Albers, A.K. McMahan, and J.E. Miller, Phys. Rev. B31 3435 (1985)

[AB87] L.V. Al'tshuler, S.E. Brusnikin, and E.A. Kuz'menkov, J. Appl. Mech. and Tech. Phys. 28 129 (1987)

[NM88] W.J. Nellis, J.A. Moriarty, A.C. Mitchell, M. Ross, R.G. Dandrea, N.W. Ashcroft, N.C. Holmes, and G.R. Gathers,Phys. Rev. Lett. 60, 1414 (1988)

[Mo95] J.A. Moriarty, High Pressure Res. 13 343 (1995)

[WC000] Yi Wang, Dongquan Chen, and Xinwei Zhang, Phys. Rev. Lett. 84 3220-3223 (2000)

SESAME (provided by D.Young with permission)

Comparison of EOS data for Cu

0 100 200 300 400 500 600-0.15

-0.10

-0.05

-0.00

0.05

0.10

0.15Cup/p

p(GPa)

W R 5 7K K 72M B 78

A M 85A B 87N M 8 8

S E S A M E

W C 0 00

M o 95

1 .1sc

0 .9 sc

0 .15K

0 .15K

Relative deviations of different EOS data for Cu at 300 K with respect to an AP2 form using Ko=132.2 GPa and K’o=5.40 used by HH2002

Comparison of EOS data for Ag

WR57KK72

MB78AB87

[WR57] J.M. Walsh, M.H. Rice, R.G. McQueen, and F.L. Yarger, Phys. Rev. 108 196 (1957)


[MB78] H.K. Mao, P.M. Bell, J.W. Shaner, and D.J. Steinberg, J. Appl. Phys. 49 3276 (1978)


0 100 200 300 400 500 600p(GPa)

-0.15

-0.10

-0.05

-0.00

0.05

0.10

0.15p/p Ag

sc0.9.15K '

0

.15K '0

sc1.1

Relative deviations of different EOS data for Ag at 300 K with respect to an AP2 form using Ko=101.1 GPa and K’o=6.15 used by HH2002

Comparison of EOS data for Au

KK7 2JF82

AB87GN92

AI8 9


[JF82] J.C. Jamieson, J.N. Fritz, and M.H. Manghnani in: High Pressure Research in Geophysics, ed. S. Akimoto, M.H. Manghnani Center for Acad. Public., Tokyo (1992)


[GN92] B.K. Godwal, A.Ng, R. Jeanloz, High Pressure Res. 10 7501 (1992)

[AI89] O.L. Anderson, D.G. Isaak, S. Yamamoto, J. Appl. Phys. 65 1534 (1989)

.15K '0

0 100 200 300 400 500 600p(GPa)

-0.15

-0.10

-0.05

-0.00

0.05

0.10

0.15p/p Au

sc 0.9

sc1.1

5.1K '0

Relative deviations of different EOS data for Au at 300 K with respect to an AP2 form using Ko=166.7 GPa and K’o=6.20 used by HH2002

-0,2

-0,1

0,0

0,1

0,2

0 100 200 300 400 500 600

p(GPa)

pp

AP2JF82AI89HJ84MoDE2

Au

Relative deviations of different EOS data for Au at 1000 K with respect to the MoDE2 modelused by HH2002. Deviations with respect to their effective AP2 form are shown by the thin line.

EOS parameters for diamond from different fitsof theoretical E(V)-data with a fixed best value for Vor

Washington 2007


The approach:


Ko (GPa) K'o STD

AP3 433,17 3,69 0,003BE3 439,35 3,56 0,017AP2 443,34 3,50 0,032H02 444,21 3,45 0,036BE2 432,81 3,64 0,050ER2 419,77 3,92 0,087


Washington 2007


The approach:


Fit of Ko(T) for Diamond

Washington 2007


The approach:


0 1000 2000 3000 4000

350

400

450K0R TK1 m Koo T( )

K0a T( )

K0qh T( )

K0S TK1 m K0L TK1 m K0R TK1 m K0H TK1 m

TK1 m T T T TK1 m TK1 m

SK1

0

nK1m 1

m

K0a TK1m K0R TK1

m 2

Fit of Ko(T) for Diamond

Washington 2007


The approach:


0 1000 2000 3000 4000

350

400

450K0R TK1 m Koo T( )

K0a T( )

K0qh T( )

K0S TK1 m K0L TK1 m K0R TK1 m K0H TK1 m

TK1 m T T T TK1 m TK1 m

SK1

0

nK1m 1

m

K0a TK1m K0R TK1

m 2

Documents

EOS of simple Solids for wide Ranges in p and T