Chapter 10 　 Physics of Highly Compressed Matter

9.1 Equation of State of Matter in High Pressure

1. The pressure is equal to zero at the solid density and the experimental bulk modulus is reproduced.

2. The cold pressure at the density less than the solid density should be negative (tensile force).

3. The Fermi pressure of electron is reproduced to be a dominant term at high density in the limit of eF >> Te, when eF is the Fermi energy of electron.

4. The ideal gas EOS should be reproduced at high temperature Te >> eF .

5. The effective charge Z* is determined not only by the thermal ionization, but also by the pressure ionization.

More’s QEOS

F Fi , Ti Fe ,Te Fb

e Fe TeFe

Te

Pe 2 Fe

Se

Fe

Te

Pi

i

Ti

Pi

Ti

Formula of Equation of State Applicable to Wide Range of T and n

Total Free Energy

Thermodynamic Consistency

1. 0 < Ti < D (low-temperature solid phase)2. D < Ti < Tm (high-temperature solid phase)3. Tm < Ti (fluid phase)

Ion Equation of State (Cowan Model by More)

w Tm / Ti

i 3

2

Ti

Amp1w1/ 3

Pi Ti

Amp1Fw1/ 3

Pi 3sTi

Amp

Tm 0.322 b

10

3 / 1 4 (eV)

b 0.6Z1/ 9

/refref A / 9Z0.3 g / cm3

s b 2 / 1 F 3s 1

Melting Temperature

Electron Equation of State based on Thomas-Fermi Model

H

AZ, Te

H Te

Z4 / 3

Z* Z Z*,H

Pe Z10 / 3PeH e

Z7 / 3

Ae

H

Z*,H Z0*, H Zth

*,H

PeH Pe,0

H Pe,thH

eH e ,0

H e,thH

Z0*,H

1

Pe,0H 0.41 Z0

*,H 5/ 3Mbar

e,0H 4.2 1012 2 6 9

1 2 /3 U0 erg / g

H /0 1/ 2

0 0.148 g / cm 3

e,0H

1

2 Pe,0H d

Thomas Fermi ModelTakabe-Takami model,

Z th*,H 1 Z0

*,H Y

Pe,thH Z*,H 2

F

H H

mp

e,thH Z*,H

3

22

F

H

mp

Y 1/ 1 T0H / Te

H 1/ 2

T0H 0.0327exp 6.98 H 0.075

eV

F

H 3

5

2

2m

32

mp

2 / 3

H 2 / 315.58 H 2 / 3eV

rH is in the unit of g/cm3

Te

H

FH

1

2.25H 0.1545

1.23

Pe,thH Z*,H H

mpTe

H

e,thH

3

2

Z*,H

mpTe

H for 1

Pe,thH

Z0*,H

H

mp

TeH 2F

H

e,thH

3

2

Z0*,H 1

mp

TeH 2F

H for 1

Pe,thH e

He ,thH

varies from Ge = 2/3 for >> 1x to Ge = 2/3 g (= 0.821) for x << 1.

Bonding Correction

Pb Pb0s

2 / 3

exp b 1 R

Rs

b b0 1 exp b 1 R

Rs

B P

at s , Te 0

where Pb0 = eb0brs/3, rs the solid density, R/Rs = (rs / r)1/3. The parameters eb0 and b are determined so that the total pressure is equal to zero at = r rs and Te = 0 and the bulk modulus defined by

P

BBulk Modulus

Equation of State of DD

16

Image of Atoms in Hot-Dense Plasmas(Pressure Ionization)

10.2 Atomic Physics of Hot Dense Plasam

En IHZn

n

2

En0

Z n Z n,mPm 1

2mn n,nPn

En0

1

2

e2

rnn,n Pn

e2Pm

rmmn m,n

rn =a0n2 / Zn

Eion

PnEn

Eion IHn Zn / n 2

Average Atom ModelScreened Hydrogen Model

Pn gn 1 expEn

Te

1

Z*

1

2 2ni

2meTe

2

3/ 2

I1/ 2Te

I1/ 2 x y1/ 2

1 exp x y 0 dy

Z Z* Pnn

0 n i

m,m'm,m' m,m'

n im,m' n ixm 1 xm'

photo excitation cross-section sm,m'

∫fm,m'ndn = 1

xn = Pn / gn

23

10.3 Equation of State Experiments and Planetary Physics

28

Equation of StateGiant Planet

31

32

36

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