Transcript ppt
Lecture 22. Ideal Bose and Fermi gas (Ch. 7)
Gibbs factor
the grand partition function
of ideal quantum gas:
Z Z i Zi exp ni i
fermions: ni = 0 or 1
1.
2.
3.
4.
N E
Z exp
k BT
N E
exp
k
T
B
i
ni
kBT
bosons: ni = 0, 1, 2, .....
Outline
Fermi-Dirac statistics (of fermions)
Bose-Einstein statistics (of bosons)
Maxwell-Boltzmann statistics
Comparison of FD, BE and MB.
The Partition Function of an Ideal Fermi Gas
The grand partition function for all particles in the ith
single-particle state (the sum is taken over all possible
values of ni) :
ni i
Z i exp
ni
k BT
i
1 exp
k BT
If the particles are fermions,
n can only be 0 or 1:
Z
Putting all the levels
together, the full partition
function is given by:
i
1 exp
i
k BT
Z FD
FD
i
Fermi-Dirac Distribution
The probability of a state to be occupied by a fermion:
ni i
1
P i , ni exp
ni 0, 1
Zi
k BT
The mean number of fermions in a particular state:
1
1
ni
Zi
1 exp i
Zi
1 exp i
exp i
1 exp i
1
exp i 1
Fermi-Dirac distribution
( is determined by T
and the particle density)
1
nFD
1
exp
k BT
Fermi-Dirac Distribution
At T = 0, all the states with <
have the occupancy = 1, all the
states with > have the
occupancy = 0 (i.e., they are
unoccupied). With increasing T, the
step-like function is “smeared” over
the energy range ~ kBT.
1
~ kBT
0
T=0
(with respect to )
=
The macrostate of such system is completely defined if we
know the mean occupancy for all energy levels, which is
often called the distribution function:
f E n E
While f(E) is often less than unity, it is not a probability:
f E n
i
n=N/V – the average
density of particles
The Partition Function of an Ideal Bose Gas
The grand partition function for all particles in the ith
single-particle state (the sum is taken over all possible
values of ni) :
ni i
Zi exp
k
T
ni
B
If the particles are Bosons, n can be any #, i.e. 0, 1, 2, …
ni i
i
2 i
Zi exp
1 exp
exp
ni 0
k BT
k BT
k BT
2
If
x
1,
1
x
x
1
1 x
Zi
BE
i
1 exp
k BT
Putting all the levels
i
together, the full partition Z BE 1 exp
i
k BT
function is given by:
1
1
min i
Bose-Einstein Distribution
The probability of a state to be occupied by a Boson:
ni i
1
P i , ni exp
ni 0,1,2,
Zi
k BT
The mean number of Bosons in a particular state:
1
i
1
ni
Zi 1 exp
1 exp i
Zi
k BT
i
1 exp
kBT
exp i
1 exp i
Bose-Einstein distribution
2
1
exp i 1
min
The mean number of particles in a
1
nBE
given state for the BEG can exceed
exp
1 unity, it diverges as min().
k BT
Comparison of FD and BE Distributions
2
1
nFD
1
exp
k BT
<n>
BE
1
FD
nBE
n n
0
-6
when
-4
k BT
-2
0
2
()/kBT
1, exp
k
T
B
Maxwell-Boltzmann
distribution:
4
1
6
1
exp
1
k BT
nFD nBE
nMB
1
exp
k
T
B
1
exp
k
T
B
Maxwell-Boltzmann Distribution (ideal gas model)
Recall the Boltzmann distribution (ch.6) derived from canonical ensemble:
2 mk B T
Z1 V
2
h
3/ 2
V
VQ
V
1 N
Z
Z1 F k BT ln Z Nk BT ln
N!
NVQ
V
F
k BT ln
N T ,V
NVQ
1 N
N
ln
exp
Z1
Z1
1
The mean number of particles in a particular state of N particles in
volume V:
nMB N P
N
exp exp exp exp
Z1
nMB exp
MB is the low density limit where the
k
T
B
difference between FD and BE disappears.
Maxwell-Boltzmann distribution
nVQ
1 i.e. N Z1
1 and 0
nVQ 1
Comparison of FD, BE and MB Distribution
2
<n>
MB
1
nFD
1
exp
k BT
BE
1
1
nBE
exp
1
k BT
FD
n n
0
-6
-4
-2
0
2
()/kBT
4
6
nMB exp
k
T
B
what are the possible values of MB , FD , and BE ? assume 0
MB 0
FD F 0
BE min 0
Comparison of FD, BE and MB Distribution
(at low density limit)
<n>
1.0
= - kBT
The difference between FD, BE
and MB gets smaller when gets
more negative.
MB
FD
BE
0.5
i.e., when
0.0
0
1
/kBT
2
3
MB is the low density limit where
the difference between FD and BE
disappears.
0.2
MB
FD
BE
<n>
= - 2kBT
0.1
nVQ
0.0
0
1
2
/kBT
0, nFD nBE nMB
3
1 i.e. N Z1
1
Comparison between Distributions
Boltzmann
nk
1
exp
k BT
Bose
Einstein
nk
1
1
exp
k BT
Fermi
Dirac
nk
1
1
exp
k BT
indistinguishable
Z=(Z1)N/N!
nK<<1
indistinguishable
integer spin 0,1,2 …
indistinguishable
half-integer spin 1/2,3/2,5/2 …
spin doesn’t matter
bosons
fermions
localized particles
don’t overlap
wavefunctions overlap
total symmetric
wavefunctions overlap
total anti-symmetric
photons
atoms
free electrons in metals
electrons in white dwarfs
unlimited number of
particles per state
never more than 1
particle per state
gas molecules
at low densities
“unlimited” number of
particles per state
nK<<1
4He
“The Course Summary”
Ensemble
Macrostate
microcanonical
U, V, N
(T fluctuates)
canonical
T, V, N
(U fluctuates)
grand
canonical
Probability
Pn
The grand potential
S U ,V , N kB ln
1
En
1 kB T
Pn e
Z
1
T, V,
Pn e
(N, U fluctuate)
Z
k BT ln Z
Thermodynamics
En N n
kB T
F T ,V , N kB T ln Z
T ,V , kB T ln Z
(the Landau free energy) is a generalization
of F=-kBT lnZ
- the appearance of μ as a variable, while
d SdT PdV Nd
computationally very convenient for the grand canonical
ensemble, is not natural. Thermodynamic properties of
systems are eventually measured with a given density of particles. However, in the
grand canonical ensemble, quantities like pressure or N are given as functions of the
“natural” variables T,V and μ. Thus, we need to use
in terms of T and n=N/V.
/ T ,V N
to eliminate μ