boson and fermion “gases”
DESCRIPTION
Boson and Fermion “Gases”. If free/quasifree gases mass > 0 non-relativistic P(E) = D(E) x n(E) --- do Bosons first let N(E) = total number of particles. A fixed number (E&R use script N for this) - PowerPoint PPT PresentationTRANSCRIPT
P461 - Quan. Stats. II 1
Boson and Fermion “Gases” • If free/quasifree gases mass > 0 non-relativistic
P(E) = D(E) x n(E) --- do Bosons first
• let N(E) = total number of particles. A fixed number (E&R use script N for this)
• D(E)=density (~same as in Plank except no 2 for spin states) (E&R call N)
• If know density N/V can integrate to get normalization. Expand the denominator….
dEee
EDdEEDEnN
kTE
0 0 / 1
)()()(
1
)2(4/
2/12/13
0 3
kTEee
dEEm
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11(
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2/3 ee
h
VmkTN
P461 - Quan. Stats. II 2
Boson Gas • Solve for e by going to the classical region (very
good approximation as m and T both large)
• this is “small”. For helium liquid (guess) T=1 K, kT=.0001 eV, N/V=.1 g/cm3
• work out average energy
• average energy of Boson gas at given T smaller than classical gas (from BE distribution ftn). See liquid He discussion
2/3
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Ne
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......)1(
)()(/)()(
2/3
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23
00
mkTh
VNkTE
dEEDEndEEDEEnE
P461 - Quan. Stats. II 3
Fermi Gas • Repeat for a Fermi gas. Add factor of 2 for S=1/2.
Define Fermi Energy EF = -kT change “-” to “+” in distribution function
• again work out average energy
• average energy of Fermion gas at given T larger than classical gas (from FD distribution ftn). Pauli exclusion forces to higher energy and often much larger
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)()(/)()(
2/3
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kTEE Feh
EmVEDEn
P461 - Quan. Stats. II 4
Fermi Gas • Distinguishable <---> Indistinguishable
Classical <----> degenerate
• depend on density. If the wavelength similar to the separation than degenerate Fermi gas (“proved” in 460)
• larger temperatures have smaller wavelength --> need tighter packing for degeneracy to occur
• electron examples - conductors and semiconductors - pressure at Earth’s core (at least some of it) -aids in initiating transition from Main Sequence stars to Red Giants (allows T to increase as electron pressure independent of T) - white dwarves and Iron core of massive stars
• Neutron and proton examples - nuclei with Fermi momentum = 250 MeV/c - neutron stars
3/1 nseparationph
particle
P461 - Quan. Stats. II 5
Degenerate vs non-degenerate
P461 - Quan. Stats. II 6
Conduction electrons• Most electrons in a metal are attached to individual
atoms.
• But 1-2 are “free” to move through the lattice. Can treat them as a “gas” (in a 3D box)
• more like a finite well but energy levels (and density of states) similar (not bound states but “vibrational” states of electrons in box)
• depth of well V = W (energy needed for electron to be removed from metal’s surface - photoelectric effect) + Fermi Energy
• at T = 0 all states up to EF are filled
W
EF
V
Filled levels
P461 - Quan. Stats. II 7
Conduction electrons • Can then calculate the Fermi energy for T=0 (and it
doesn’t usually change much for higher T)
• Ex. Silver 1 free electron/atom
3/22
0
2/1
3
2/13
0
0
3
8
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)2(82)(
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/108
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3/2
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282
19
3/2
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28
31
234
322
3
23
P461 - Quan. Stats. II 8
Conduction electrons
• Can determine the average energy at T = 0
• for silver ---> 3.3 eV
• can compare to classical statistics
• Pauli exclusion forces electrons to much higher energy levels at “low” temperatures. (why e’s not involved in specific heat which is a lattice vibration/phonons)
FF
FE
E
E
E
EE
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k
eVT o000,40
/106.8
3.33.35
P461 - Quan. Stats. II 9
Conduction electrons
P461 - Quan. Stats. II 10
Conduction electrons• Similarly, from T-dependent
• the terms after the 1 are the degeneracy terms….large if degenerate. For silver atoms at T=300 K
• not until the degeneracy term is small will the electron act classically. Happens at high T
• The Fermi energy varies slowly with T and at T=300 K is almost the same as at T=0
• You obtain the Fermi energy by normalization. Quark-gluon plasma (covered later) is an example of a high T Fermi gas
....
)2(2
11
2
32/3
3
3/5 mkTV
NhkTE
82012
3 kTE
)0()300( FF EE
P461 - Quan. Stats. II 11
Fermi Gases in Stars • Equilibrium: balance between gravitational
pressure and “gas” (either normal or degenerate) pressure
• total gravitational Energy:
• density varies in normal stars (in Sun: average is 1 g/cm3 but at r=0 is 100 g/cm3). More of a constant in white dwarves or neutron stars
• will have either “normal” gas pressure of P=nkT (P=n<E>) or pressure due to degenerate particles. Normal depends on T, degenerate (mostly) doesn’t
• n = particle density in this case
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tan5
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Epressure
P461 - Quan. Stats. II 12
Degenerate Fermi Gas Pressure• Start with p = n<E>
• non-relativistic relativistic
• P depends ONLY on density
3/43/12
3/53/21
3/124
33/215
3
2
2
2/1
2/1
3/13/2
0
2
0
2/1
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)(
)(:
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casesbothppDstatesdensity
FF
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EE FF
P461 - Quan. Stats. II 13
Degenerate Fermi Gas Pressure non-relativistic relativistic
• P depends ONLY on density
• Pressure decreases if, for a given density, particles become relativistic
• if shrink star’s radius by 2 density increases by 8 gravitational E increases by 2
• if non-relativistic. <E> increases by (N/V)2/3 = 4
• if relativistic <E> increases by (N/V)1/3 = 2 non-relativistic stable but relativistic is not. can
collapse
3/43/12
3/53/21
3/124
33/215
3
3/13/2
nnnKnnnKP
nKEnKEE
nnE
FF
F
P461 - Quan. Stats. II 14
Older Sun-like Stars • Density of core increases as H-->He. He inert (no
fusion yet). Core contracts
• electrons become degenerate. 4 e per He nuclei. Electrons have longer wavelength than He
• electrons move to higher energy due to Pauli exclusion/degeneracy. No longer in thermal equilibrium with p, He nuclei
• pressure becomes dominated by electrons. No longer depends on T
• allows T of p,He to increase rapidly without “normal” increase in pressure and change in star’s equilibrium.
• Onset of 3He->C fusion and Red Giant phase (helium flash when T = 100,000,000 K)
Hemm
eHemm
e
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H
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e pp
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P461 - Quan. Stats. II 15
White Dwarves• Leftover cores of Red Giants made (usually) from
C + O nuclei and degenerate electrons
• cores of very massive stars are Fe nuclei plus degenerate electrons and have similar properties
• gravitational pressure balanced by electrons’ pressure which increases as radius decreases radius depends on Mass of star
• Determine approximate Fermi Energy. Assume electron density = 0.5(p+n) density
• electrons are in this range and often not completely relativistic or non-relativistic need to use the correct E2 = p2 + m2 relationship
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MeVrelnonE
meEarthvolumem
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V
N
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mh
F
p
Sun
8.0)(
3.0)(
/1051
2
1
3/1
83
3/238
335
2
P461 - Quan. Stats. II 16
White Dwarves + Collapse• If the electron energy is > about 1.4 MeV can have:
• any electrons > ET “disappear”. The electron energy distribution depends on T (average E)
• the “lost” electrons cause the pressure from the degenerate electrons to decrease
• the energy of the neutrinos is also lost as they escape “cools” the star
• as the mass increases, radius decreases, and number of electrons above threshold increases
MeVEnpe Threshold 4.1
#e’s
EF ET
P461 - Quan. Stats. II 17
White Dwarves+Supernovas• another process - photodisentegration - also
absorbs energy “cooling” star. Similar energy loss as e+p combination
• At some point the not very stable equilibrium between gravity and (mostly) electron pressure doesn’t hold
• White Dwarf collapses and some fraction (20-50% ??) of the protons convert to neutrons during the collapse
• gives Supernovas
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nHeFe
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lightLneutrinosL
lightLlightL
SNSN
SunSN
P461 - Quan. Stats. II 18
White Dwarves+Supernovas
P461 - Quan. Stats. II 19
Neutron Stars-approx. numbers• Supernovas can produce neutron stars
- radius ~ 10 km - mass about that of Sun. always < 3 mass Sun - relative n:p:e ~ 99:1:1
• gravity supported by degenerate neutrons
• plug into non-relativistic formula for Fermi Energy 140 MeV (as mass =940 MeV, non-rel OK)
• look at wavelength
• can determine radius vs mass (like WD)
• can collapse into black hole
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Sun
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h
p
h2
1409402
1240
2
P461 - Quan. Stats. II 20
Neutron Stars• 3 separate Fermi gases: n:p:e p+n are in the same
potential well due to strong nuclear force
• assume independent and that p/n = 0.01 (depends on star’s mass)
• so need to use relativistic for electrons
• but not independent as p <---> n
• plus reactions with virtual particles
• free neutrons decay. But in a neutron star they can only do so if there is an available unfilled electron state. So suppresses decay
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enpepn
P461 - Quan. Stats. II 21
Neutron Stars• Will end up with an equilibrium between n-p-e
which can best be seen by matching up the Fermi energy of the neutrons with the e-p system
• neutrons with E > EF can then decay to p-e-nu (which raises electron density and its Fermi energy thus the balance)
• need to include rest mass energies. Also density of electrons is equal to that of protons
• can then solve for p/n ratio (we’ll skip algebra)
• gives for typical neutron star:
223/2
23/2
23/1
28
3
28
3
8
3
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m
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n
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/102344
317
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mkg