Classical and Quantum Classical and Quantum Gases Gases Fundamental Ideas Fundamental Ideas – Density of States Density of States – Internal Energy Internal Energy – Fermi-Dirac and Bose-Einstein Fermi-Dirac and Bose-Einstein Statistics Statistics – Chemical potential Chemical potential – Quantum concentration Quantum concentration
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. - PowerPoint PPT Presentation
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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-
– 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
Next LectureNext Lecture
The Saha EquationThe Saha Equation– DerivationDerivation– Consequences for ionisation and Consequences for ionisation and
absorptionabsorption
Next WeekNext Week
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