Classical and Quantum Classical and Quantum GasesGases

Fundamental IdeasFundamental Ideas– Density of StatesDensity of States– Internal EnergyInternal Energy– Fermi-Dirac and Bose-Einstein Fermi-Dirac and Bose-Einstein

StatisticsStatistics– Chemical potential Chemical potential – Quantum concentrationQuantum concentration

Density of StatesDensity of States

Derived by considering the gas particles as Derived by considering the gas particles as wave-like and confined in a certain volume, wave-like and confined in a certain volume, V.V.– Density of states as a function of momentum, Density of states as a function of momentum, gg((pp), ),

between between pp and and pp + + dpdp::

g p dp gVh

p dps 324

– ggss = number of polarisations= number of polarisations 2 for protons, neutrons, electrons and photons2 for protons, neutrons, electrons and photons

Internal EnergyInternal Energy

The energy of a particle with The energy of a particle with momentum momentum pp is given by: is given by:E p c m cp

2 2 2 2 4 Hence the total energy is:Hence the total energy is:

E E f E g p dpp p

0Average no. of particles in state with energy Ep

No. of quantum states in p to p +dp

Total Number of ParticlesTotal Number of Particles

N f E g p dpp

0Average no. of particles in state with energy Ep

No. of quantum states in p to p +dp

Fermi-Dirac StatisticsFermi-Dirac Statistics

For fermions, no more than one particle For fermions, no more than one particle can occupy a given quantum statecan occupy a given quantum state– Pauli exclusion principlePauli exclusion principle

Hence:Hence:

f Ep EkTp

1

1exp

Bose-Einstein StatisticsBose-Einstein Statistics

For Bosons, any number of For Bosons, any number of particles can occupy a given particles can occupy a given quantum statequantum state

Hence:Hence: f Ep E

kTp

1

1exp

F-D vs. B-E StatisticsF-D vs. B-E Statistics

0.0001

0.001

0.01

0.1

1

10

100

0.01 0.1 1 10

E/kT

Occ

uapn

cy

Fermi-DiracBose-Einstein

The Maxwellian LimitThe Maxwellian Limit

Note that Fermi-Dirac and Bose-Note that Fermi-Dirac and Bose-Einstein statistics coincide for large Einstein statistics coincide for large EE//kTkT and small occupancy and small occupancy– Maxwellian limitMaxwellian limit

f Ep

E

kTp

exp

Ideal Classical GasesIdeal Classical Gases

Classical Classical occupancy of any one occupancy of any one quantum state is smallquantum state is small– I.e., MaxwellianI.e., Maxwellian

Equation of State:Equation of State:

PNV

kT Valid for both non- and ultra-Valid for both non- and ultra-

relativistic gasesrelativistic gases

Ideal Classical GasesIdeal Classical Gases Recall:Recall:

– Non-relativistic:Non-relativistic: Pressure = 2/3 kinetic energy densityPressure = 2/3 kinetic energy density Hence average KE = 2/3 Hence average KE = 2/3 kTkT

– Ultra-relativisticUltra-relativistic Pressure = 1/3 kinetic energy densityPressure = 1/3 kinetic energy density Hence average KE = 1/3 Hence average KE = 1/3 kTkT

Ideal Classical GasesIdeal Classical Gases

Total number of particles Total number of particles N N in a in a volume volume VV is given by: is given by:

N gVh

p dp

N gVh

mkT

E

kT s

smckT

p

exp

exp

0 32

3

4

23

22

Ideal Classical GasesIdeal Classical Gases

Rearranging, we obtain an Rearranging, we obtain an expression for expression for , the chemical , the chemical potentialpotential

mc kTg n

n

nmkTh

s Q

Q

2

2

322

ln

where

(the quantum concentration)

Ideal Classical GasesIdeal Classical Gases

Interpretation of Interpretation of – From statistical mechanics, the change From statistical mechanics, the change

of energy of a system brought about by of energy of a system brought about by a change in the number of particles is:a change in the number of particles is:

dE dN

Ideal Classical GasesIdeal Classical Gases

Interpretation of Interpretation of nnQ Q (non-relativistic)(non-relativistic)– Consider the de Broglie WavelengthConsider the de Broglie Wavelength

h

ph

mkTnQ1

2

13

– Hence, since the average separation of particles in a gas of Hence, since the average separation of particles in a gas of density density nn is ~ is ~nn-1/3-1/3

– If If nn << << nnQ Q , the average separation is greater than , the average separation is greater than and the and the gas is classical rather than quantumgas is classical rather than quantum

Ideal Classical GasesIdeal Classical Gases

A similar calculation is possible for A similar calculation is possible for a gas of ultra-relativistic particles:a gas of ultra-relativistic particles:

kTg n

n

nkThc

s Q

Q

ln

where 83

Quantum GasesQuantum Gases

Low concentration/high temperature electron Low concentration/high temperature electron gases behave classicallygases behave classically

Quantum effects large for high electron Quantum effects large for high electron concentration/”low” temperatureconcentration/”low” temperature– Electrons obey Fermi-Dirac statisticsElectrons obey Fermi-Dirac statistics

– All states occupied up to an energy All states occupied up to an energy EEff , the Fermi , the Fermi Energy with a momentum Energy with a momentum ppff

– Described as a degenerate gasDescribed as a degenerate gas

Quantum GasesQuantum Gases

Equations of State: Equations of State: – (See Physics of Stars sec(See Physics of Stars secnn 2.2) 2.2)– Non-relativistic:Non-relativistic:

Phm

n

2 23 5

3

538

– Ultra-relativistic:Ultra-relativistic:

Phc

n

4

38

23 4

3

Quantum GasesQuantum Gases

Note:Note:– Pressure rises more slowly with Pressure rises more slowly with

density for an ultra-relativistic density for an ultra-relativistic degenerate gas compared to non-degenerate gas compared to non-relativisticrelativistic

– Consequences for the upper mass of Consequences for the upper mass of degenerate stellar cores and white degenerate stellar cores and white dwarfsdwarfs

ReminderReminder

Assignment 1 available today on Assignment 1 available today on unit websiteunit website

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The Saha EquationThe Saha Equation– DerivationDerivation– Consequences for ionisation and Consequences for ionisation and

absorptionabsorption

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Private Study Week - SuggestionsPrivate Study Week - Suggestions– Assessment WorksheetAssessment Worksheet– Review Lectures 1-5Review Lectures 1-5– Photons in Stars (Phillips ch. 2 secPhotons in Stars (Phillips ch. 2 secnn 2.3) 2.3)

The Photon GasThe Photon Gas Radiation PressureRadiation Pressure

– Reactions at High Temperatures (Phillips ch. Reactions at High Temperatures (Phillips ch. 2 sec2 secnn 2.6) 2.6)

Pair ProductionPair Production Photodisintegration of NucleiPhotodisintegration of Nuclei