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Classical and Quantum Gases n Fundamental Ideas –Density of States –Internal Energy –Fermi-Dirac and Bose-Einstein Statistics –Chemical potential –Quantum concentration

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Density of States n 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: –g s = number of polarisations n 2 for protons, neutrons, electrons and photons

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Internal Energy n The energy of a particle with momentum p is given by: n Hence the total energy is: Average no. of particles in state with energy E p No. of quantum states in p to p +dp

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Total Number of Particles Average no. of particles in state with energy E p No. of quantum states in p to p +dp

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Fermi-Dirac Statistics n For fermions, no more than one particle can occupy a given quantum state –Pauli exclusion principle n Hence:

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Bose-Einstein Statistics n For Bosons, any number of particles can occupy a given quantum state n Hence:

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F-D vs. B-E Statistics

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The Maxwellian Limit n Note that Fermi-Dirac and Bose-Einstein statistics coincide for large E/kT and small occupancy –Maxwellian limit

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Ideal Classical Gases Classical occupancy of any one quantum state is small Classical occupancy of any one quantum state is small –I.e., Maxwellian n Equation of State: n Valid for both non- and ultra-relativistic gases

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Ideal Classical Gases n Recall: –Non-relativistic: n Pressure = 2/3 kinetic energy density n Hence average KE = 2/3 kT –Ultra-relativistic n Pressure = 1/3 kinetic energy density n Hence average KE = 1/3 kT

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Ideal Classical Gases n Total number of particles N in a volume V is given by:

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Ideal Classical Gases Rearranging, we obtain an expression for, the chemical potential Rearranging, we obtain an expression for, the chemical potential

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Ideal Classical Gases Interpretation of Interpretation of –From statistical mechanics, the change of energy of a system brought about by a change in the number of particles is:

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Ideal Classical Gases n Interpretation of n Q (non-relativistic) –Consider the de Broglie Wavelength –Hence, since the average separation of particles in a gas of density n is ~n -1/3 –If n << n Q, the average separation is greater than and the gas is classical rather than quantum

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Ideal Classical Gases n A similar calculation is possible for a gas of ultra-relativistic particles:

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Quantum Gases n Low concentration/high temperature electron gases behave classically n Quantum effects large for high electron concentration/low temperature –Electrons obey Fermi-Dirac statistics –All states occupied up to an energy E f, the Fermi Energy with a momentum p f –Described as a degenerate gas

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Quantum Gases n Equations of State: –(See Physics of Stars sec n 2.2) –Non-relativistic: –Ultra-relativistic:

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Quantum Gases n 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

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Reminder n Assignment 1 available today on unit website

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Next Lecture n The Saha Equation –Derivation –Consequences for ionisation and absorption

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Next Week n Private Study Week - Suggestions –Assessment Worksheet –Review Lectures 1-5 –Photons in Stars (Phillips ch. 2 sec n 2.3) n The Photon Gas n Radiation Pressure –Reactions at High Temperatures (Phillips ch. 2 sec n 2.6) n Pair Production n Photodisintegration of Nuclei

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