ast403_equation of state notes

ast403_equation of state notes - EQUATION OF STATE Consider...

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EQUATION OF STATE Consider elementary cell in a phase space with a volume Δ x Δ y Δ z Δ p x Δ p y Δ p z = h 3 , (st . 1) where h = 6 . 63 × 10 - 27 erg s is the Planck constant, Δ x Δ y Δ z is volume in ordinary space measured in cm 3 , and Δ p x Δ p y Δ p z is volume in momentum space measured in ( g cm s - 1 ) 3 . According to quantum mechanics there is enough room for approximately one particle of any kind within any elementary cell. More precisely, an average number of particles per cell is given as n av = g e ( E - μ ) /kT ± 1 , (st . 2) with a ”+” sign for fermions, and a ” ” sign for bosons. The corresponding distributions are called Fermi-Dirac and Bose-Einstein, respectively. Particles with a spin of 1 / 2 are called fermions, while those with a spin 0 , 1 , 2 ... are called bosons. Electrons and protons are fermions, photons are bosons, while larger nuclei or atoms may be either fermions or bosons, depending on the total spin of such a composite particle. In the equation (st.2) E is the particle energy, k = 1 . 38 × 10 - 16 erg K - 1 is the Boltzman constant, T is temperature, μ is chemical potential, and g is a number of different quantum states a particle may have within the cell. The meaning of temperature is obvious, while chemical potential will become more familiar later on. In most cases it will be close to the rest mass of a particle under consideration. If there are anti-particles present in equilibrium with particles, and particles have chemical potential μ then antiparticles have chemical potential μ 2 m , where m is their rest mass. For free particles their energy is a function of their momentum only, with the total momentum p given as p 2 = p 2 x + p 2 y + p 2 z . (st . 3) The number density of particle in a unit volume of 1 cm 3 , with momenta between p and p + dp is given as n ( p ) dp = g e ( E - μ ) /kT ± 1 4 πp 2 h 3 dp, (st . 4) because the number of elementary cells within 1 cm 3 and a momentum between p and p + dp , i.e. within a spherical shell with a surface 4 πp 2 and thickness dp is equal to 4 πp 2 dp h - 3 . The number density of particles with all momenta, contained within 1 cm 3 is n = integraldisplay 0 n ( p ) dp, ρ = nm (st . 5) where ρ is the mass density of gas. Note, that if we know density and temperature, then we can calculate chemical potential with the eqs. (st.4) and (st.5), provided we know how particle energy depends on its momentum, i.e. the function E ( p ) is known. In the equation (st.4) g, k, T, μ, π, h are all constant, and the energy E depends on the momentum p only. As our particles may be relativistic as well as non-relativistic, we have to use a general formula for the relation between E and p . We have E E total = E 0 + E k , (st . 6) where the rest mass E 0 = mc 2 , c = 3 × 10 10 cm s - 1 is the speed of light, and E k is the kinetic energy of a particle. For a particle moving with arbitrary velocity there is a special relativistic relation: st — 1
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E 2 total = ( mc 2 ) 2 + ( pc ) 2 . (st . 7) Combining the last two equations we obtain E = mc 2 bracketleftbigg 1 + parenleftBig p mc parenrightBig 2 bracketrightbigg 1 / 2 , E k = mc 2 bracketleftBigg
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