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Classical and Quantum Gases Fundamental Ideas – Density of States – Internal Energy – Fermi-Dirac and Bose-Einstein Statistics – Chemical potential – Quantum concentration Density of States Derived by considering the gas particles as wave-like and confined in a certain volume, V. – Density of states as a function of momentum, g(p), between p and p + dp: V 2 g p dp g s 3 4p dp h – gs = number of polarisations 2 for protons, neutrons, electrons and photons Internal Energy The energy of a particle with momentum p is given by: Ep p c m c 2 2 2 2 4 Hence the total energy is: E 0 E p f E p g p dp Average no. of particles in state with energy Ep No. of quantum states in p to p +dp Total Number of Particles N 0 f E p g p dp Average no. of particles in state with energy Ep No. of quantum states in p to p +dp Fermi-Dirac Statistics For fermions, no more than one particle can occupy a given quantum state – Pauli exclusion principle Hence: f Ep 1 1 exp E p kT Bose-Einstein Statistics For Bosons, any number of particles can occupy a given quantum state Hence: f Ep 1 1 exp E p kT F-D vs. B-E Statistics 100 Fermi-Dirac Bose-Einstein 1 0.01 0.1 1 E/kT 10 0.1 0.01 Occuapncy 10 0.001 0.0001 The Maxwellian Limit Note that Fermi-Dirac and Bose-Einstein statistics coincide for large E/kT and small occupancy – Maxwellian limit f E p exp E p kT Ideal Classical Gases Classical occupancy of any one quantum state is small – I.e., Maxwellian Equation of State: N P kT V Valid for both non- and ultra-relativistic gases Ideal Classical Gases Recall: – Non-relativistic: Pressure = 2/3 kinetic energy density Hence average KE = 2/3 kT – Ultra-relativistic Pressure = 1/3 kinetic energy density Hence average KE = 1/3 kT Ideal Classical Gases Total number of particles N in a volume V is given by: N 0 exp E p kT V g s 3 4 p 2dp h V N g s 3 2mkT exp h 3 2 mc 2 kT Ideal Classical Gases Rearranging, we obtain an expression for , the chemical potential g n s Q 2 mc kT ln n 3 2 2 mkT where nQ 2 h (the quantum concentration) Ideal Classical Gases Interpretation of – From statistical mechanics, the change of energy of a system brought about by a change in the number of particles is: dE dN Ideal Classical Gases Interpretation of nQ (non-relativistic) – Consider the de Broglie Wavelength h h nQ p mkT 1 1 3 2 – Hence, since the average separation of particles in a gas of density n is ~n-1/3 – If n << nQ , the average separation is greater than and the gas is classical rather than quantum Ideal Classical Gases A similar calculation is possible for a gas of ultra-relativistic particles: g s nQ kT ln n kT where nQ 8 hc 3 Quantum Gases Low concentration/high temperature electron gases behave classically Quantum effects large for high electron concentration/”low” temperature – Electrons obey Fermi-Dirac statistics – All states occupied up to an energy Ef , the Fermi Energy with a momentum pf – Described as a degenerate gas Quantum Gases Equations of State: – (See Physics of Stars secn 2.2) – Non-relativistic: h P 5m 2 2 3 3 53 8 n – Ultra-relativistic: P hc 3 2 3 n 4 8 4 3 Quantum Gases Note: – Pressure rises more slowly with density for an ultra-relativistic degenerate gas compared to non-relativistic – Consequences for the upper mass of degenerate stellar cores and white dwarfs Reminder Assignment 1 available today on unit website Next Lecture The Saha Equation – Derivation – Consequences for ionisation and absorption Next Week Private Study Week - Suggestions – Assessment Worksheet – Review Lectures 1-5 – Photons in Stars (Phillips ch. 2 secn 2.3) The Photon Gas Radiation Pressure – Reactions at High Temperatures (Phillips ch. 2 secn 2.6) Pair Production Photodisintegration of Nuclei