Lecture 22. Ideal Bose and Fermi gas (Ch. 7)

Z Z i

i

expi

i ii

n B

nZ

k T

the grand partition function

of ideal quantum gas:

Gibbs factor

Tk

EN

B

exp

Tk

ENZ

B

exp

fermions: ni = 0 or 1 bosons: ni = 0, 1, 2, .....

Outline1.Fermi-Dirac statistics (of fermions)2.Bose-Einstein statistics (of bosons)3.Maxwell-Boltzmann statistics4.Comparison of FD, BE and MB.

The Partition Function of an Ideal Fermi Gas

If the particles are fermions, n can only be 0 or 1:

TkZ

B

iFDi

exp1

The grand partition function for all particles in the ith single-particle state (the sum is taken over all possible values of ni) :

in B

iii Tk

nZ

exp

Putting all the levels together, the full partition function is given by:

i B

iFD TkZ

exp1

Fermi-Dirac Distribution

1exp

1

Tk

n

B

FD

Fermi-Dirac distribution

The mean number of fermions in a particular state:

1,0exp

1,

i

B

ii

iii n

Tk

n

ZnP

The probability of a state to be occupied by a fermion:

( is determined by T and the particle density)

1 11 exp

1 exp

exp 1

1 exp exp 1

i i ii i

i

i i

n ZZ

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.

T =0

~ kBT

= (with respect to )

1

0

nEfi

n=N/V – the average density of particles

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: EnEf

While f(E) is often less than unity, it is not a probability:

The Partition Function of an Ideal Bose Gas

If the particles are Bosons, n can be any #, i.e. 0, 1, 2, …

The grand partition function for all particles in the ith single-particle state (the sum is taken over all possible values of ni) :

Putting all the levels together, the full partition function is given by:

exp

i

i ii

n B

nZ

k T

0

2exp 1 exp exp

i

i i i ii

n B B B

nZ

k T k T k T

1

1 expBE ii

B

Zk T

2 1If 1, 1

1x x x

x

1

1 exp iBE

i B

Zk T

min i

Bose-Einstein Distribution

Bose-Einstein distribution

The mean number of Bosons in a particular state:

The probability of a state to be occupied by a Boson:

The mean number of particles in a given state for the BEG can exceed unity, it diverges as min().

1

2

11 exp 1 exp

exp 11 exp

exp 11 exp

ii i i

i B

ii

B ii

n ZZ k T

k T

1, exp 0,1,2,i ii i i

i B

nP n n

Z k T

1

exp 1BE

B

n

k T

min

Comparison of FD and BE Distributions

1

exp 1BE

B

n

k T

1exp

1

Tk

n

B

FD

when 1, exp 1B Bk T k T

-6 -4 -2 0 2 4 60

1

2

FD

BE

<

n >

()/kBT

1

expFD BE

B

n n

k T

1

expMB

B

n

k T

Maxwell-Boltzmann distribution:

n n

Maxwell-Boltzmann Distribution (ideal gas model)

1

exp exp exp expMB

Nn N P

Z

, 1 1

1ln ln expB

T V Q

F V N Nk T

N NV Z Z

expMBB

nk T

Maxwell-Boltzmann distribution

3/ 2

1 2

2 B

Q

mk T VZ V

h V

NZN

Z 1!

1 ln ln 1B B

Q

VF k T Z Nk T

NV

The mean number of particles in a particular state of N particles in

volume V:

MB is the low density limit where the difference between FD and BE disappears.

11 i.e. 1 and 0QnV N Z 1QnV

Recall the Boltzmann distribution (ch.6) derived from canonical ensemble:

Comparison of FD, BE and MB Distribution

1

exp 1BE

B

n

k T

1exp

1

Tk

n

B

FD

expMBB

nk T

MB FD BEwhat are the possible values of , , and ? assume 0

-6 -4 -2 0 2 4 60

1

2

FD

MB BE

<

n >

()/kBT

min 0BE 0MB 0FD F

n n

Comparison of FD, BE and MB Distribution (at low density limit)

0 1 2 30.0

0.5

1.0

MB FD BE

= - kBT

< n

>

/kBT

0 1 2 30.0

0.1

0.2

MB FD BE

= - 2kBT

< n

>

/kBT

MB is the low density limit where the difference between FD and BE disappears.

The difference between FD, BE and MB gets smaller when gets more negative.

i.e., when 0, FD BE MBn n n

11 i.e. 1QnV N Z

Comparison between Distributions

Boltzmann FermiDirac

Bose Einstein

indistinguishableZ=(Z1)N/N!

nK<<1

spin doesn’t matter

localized particles don’t overlap

gas moleculesat low densities

“unlimited” number ofparticles per state

nK<<1

indistinguishableinteger spin 0,1,2 …

bosons

wavefunctions overlaptotal symmetric

photons 4He atoms

unlimited number ofparticles per state

indistinguishablehalf-integer spin 1/2,3/2,5/2 …

fermions

wavefunctions overlaptotal anti-symmetric

free electrons in metalselectrons in white dwarfs

never more than 1particle per state

1exp

1

Tk

n

B

k 1exp

1

Tk

n

B

k

Tk

n

B

k exp

1

“The Course Summary”

Ensemble Macrostate Probability Thermodynamics

micro-canonical

U, V, N

(T fluctuates)

canonical T, V, N

(U fluctuates)

grand canonical

T, V, (N, U fluctuate)

ln,, BkNVUS

ZTkNVTF B ln,,

ZTkVT B ln,,

1nP

Tk

E

nB

n

eZ

P

1

Tk

NE

nB

nn

eZ

P

1

The grand potential ZTkB ln (the Landau free energy) is a generalization of F=-kBT lnZ

NdPdVSdTd

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 to eliminate μ in terms of T and n=N/V.

NVT ,/

- the appearance of μ as a variable, while computationally very convenient for the grand canonical ensemble, is not natural. Thermodynamic properties of