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Book Boris V. Vasiliev Astrophysics

Chapter 2 The

Chapter 2 The energy-favorable state of hot dense plasma 2.1 The properties of a hot dense plasma 2.1.1 A hot plasma and Boltzman’s distribution Free electrons being fermions obey the Fermi-Dirac statistic at low temperatures. At high temperatures, quantum distinctions in behavior of electron gas disappear and it is possible to consider electron gas as the ideal gas which obeys the Boltzmann’s statistics. At high temperatures and high densities, all substances transform into electron-nuclear plasma. There are two tendencies in this case. At temperature much higher than the Fermi temperature T F = E F k (where E F is Fermi energy), the role of quantum effects is small. But their role grows with increasing of the pressure and density of an electron gas. When quantum distinctions are small, it is possible to describe the plasma electron gas as a the ideal one. The criterium of Boltzman’s statistics applicability T ≫ EF k . (2.1) hold true for a non-relativistic electron gas with density 10 25 particles in cm 3 at T ≫ 10 6 K. At this temperatures, a plasma has energy and its EOS is the ideal gas EOS: E = 3 kT N (2.2) 2 P = NkT V (2.3) 15

But even at so high temperatures, an electron-nuclear plasma can be considered as ideal gas in the first approximation only. For more accurate description its properties, the specificity of the plasma particle interaction must be taken into account and two main corrections to ideal gas law must be introduced. The first correction takes into account the quantum character of electrons, which obey the Pauli principle, and cannot occupy levels of energetic distribution which are already occupied by other electrons. This correction must be positive because it leads to an increased gas incompressibility. Other correction takes into account the correlation of the screening action of charged particles inside dense plasma. It is the so-called correlational correction. Inside a dense plasma, the charged particles screen the fields of other charged particles. It leads to a decreasing of the pressure of charged particles. Accordingly, the correction for the correlation of charged particles must be negative,because it increases the compressibility of electron gas. 2.1.2 The hot plasma energy with taking into account the correction for the Fermi-statistic The energy of the electron gas in the Boltzmann’s case (kT ≫ E F ) can be calculated using the expression of the full energy of a non-relativistic Fermi-particle system [12]: E = 21/2 V me 3/2 π 2 3 ∫ ∞ 0 ε 3/2 dε e (ε−µe)/kT + 1 , (2.4) expanding it in a series. (m e is electron mass, ε is the energy of electron and µ e is its chemical potential). In the Boltzmann’s case, µ e < 0 and |µ e/kT | ≫ 1 and the integrand at e µe/kT ≪ 1 can be expanded into a series according to powers e µe/kT −ε/kT . If we introduce the notation z = ε and conserve the two first terms of the series, we obtain kT or ∫ ∞ I ≡ (kT ) 5/2 z 3/2 dz 0 e z−µe/kT + 1 ≈ ∫ ∞ ( ) ≈ (kT ) 5/2 z 3/2 e µe kT −z µe 2( − e kT −z) + ... dz (2.5) ( ) I µe ≈ e kT 3 (kT ) 5/2 Γ 2 + 1 Thus, the full energy of the hot electron gas is E ≈ 3V 2 0 (kT ) 5/2 √ 2 − 1 ( ) 2µe e kT 3 25/2 Γ 2 + 1 ≈ ≈ 3√ π 4 eµe/kT ( 1 − 1 2 5/2 eµe/kT ) . (2.6) ( me π 2 ) 3/2 ( e µe/kT − 1 2 5/2 e2µe/kT ) 16 (2.7)

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