Thermal and Statistical Physics

110 Pages·2006·2.411 MB·English

by  Blumler P.

School of Physical Sciences
PH 605
Thermal and Statistical
Physics
Part II: Semi-Classical Physics
Quantum Statistics
course-webpage:
http://wwwnmr.ukc.ac.uk/nmr/staff/pb/teach/PH605/PH605.html
Dr. Peter Blümler ([email protected])
Room 123, Phone: 3228
Syllabus
3. Semi-Classical Physics.............................................................................................2
3.1 The Boltzmann Distribution (derived!)........................................................................2
3.1.1 A simple example ..............................................................................................2
3.1.2 Generalisation....................................................................................................3
3.2 The Semi-Classical Perfect Gas..................................................................................8
3.2.1 Definition of the semi-classical, mon-atomic, perfect gas......................................8
3.2.2 Distinguishable / indistinguishable particles?.........................................................9
3.2.3 Contributions of different types of motion to Z ..................................................13
1
3.2.4 The density of states........................................................................................14
3.2.5 Partition function for translational motion, Ztr....................................................17
1
3.2.6 Partition function of internal motion, Z .............................................................20
int
3.2.7 Partition function of (molecular) rotation, Z .....................................................21
rot
3.2.8 Partition function of (molecular) vibration, Z ...................................................24
vib
3.2.9 Partition functions and comparison to experimental data.....................................28
3.3 Entropy and Energy of the Semi-Classical Gas..........................................................29
3.3.1 Entropy of a mon-atomic gas, the Sackur-Tetrode equation...............................29
3.3.2 The entropy of mixing-the Gibbs paradox..........................................................31
3.3.3 The principle of the equipartition of energy........................................................32
3.4 Validity and Limit of the Semi-Classical Description..................................................35
3.4.1 The classical limit.............................................................................................35
3.4.2 Maxwell velocity distribution in a classical gas...................................................38
3.4.3 Rotational specific heat of diatomic molecules- ortho/para 1H ............................41
2
4. Quantum Statistics...................................................................................................45
4.1 Ideal Solids..............................................................................................................45
4.1.1 Einstein's theory of an ideal crystal...................................................................45
4.1.2 Debye's theory of an ideal crystal.....................................................................48
4.2 Quantum Statistics....................................................................................................59
4.2.1 Bose-Einstein statistics....................................................................................62
4.2.2 Fermi-Dirac statistics......................................................................................64
4.2.3 Comparison of Boltzmann, BE and FD statistics................................................66
a
4.2.4 Determination of ..........................................................................................68
4.2.5 Systems with variable particle number...............................................................70
4.2.6 The Grand partition function, Z..........................................................................71
4.3 Application to Fermion/Boson-Systems....................................................................76
4.3.1 Free electrons in metals....................................................................................76
4.3.2 Pauli-paramagnetism.......................................................................................83
4.3.3 The perfect photon gas - black-body radiation....................................................85
4.3.4 Bose-Einstein condensation.............................................................................88
4.3.5 Superconductivity and superfluidity, BEC...........................................................95
4.3.6 Thermodynamics of stars...............................................................................103
PH 605: Thermal & Statistical Physics 3. Semi-Classical Physics page:2
Recommended Books / Background Reading for this second part:
• F. Mandl, "Statistical Physics", Wiley, 1988 [QC175, 17 copies]
• R. Baierlein: "Thermal Physics", Cambridge University Press, 1999 [ISBN:0-521-65838-1]
3 Semi-Classical Physics
3.1 The Boltzmann distribution (derived!)
The Boltzmann distribution was introduced in the last section of part I (see Dr. Mallett’s script).
recap: micro-state: certain assignment of particles to certain (energy) state
macro-state: realised by many micro-states („sum of micro-states“)
Ludwig Boltzmann concluded from the 2nd law of Thermodynamics that the macro-state with the
most micro-states is the most stable in equilibrium (remember S= k lnW ).
B
3.1.1 A simple example
There are 3 (distinguishable, independent and identical) particles A, B and C.
They are allowed to occupy 4 different energy states: e = 0, e , e = 2e and e = 3e
0 1 2 1 3 1
(e.g. harmonic oscillator).
The total energy of the system amounts to 3e .
1
The occupation numbers are N , N , N and N .
0 1 2 3
Now we are going to try to find the number of macro-states by which the system can be realised.
3 3
(cid:229) (cid:229)
macro-state N N N N Ni E = Ni e i
0 1 2 3
i=0 i=0
I 2 0 0 1 3 3e
1
II 1 1 1 0 3 3e
1
III 0 3 0 0 3 3e
1
We see that there are only 3 possible macro-states for the system.
The next question is then: How many micro-states are possible to realise each macro-state?
Note: We recall/realise that exchange of particles in the same micro-state doesn’t generate a new
micro-state!
energy state macro-state I macro-state II macro-state III
e A B C - - - - - - -
3
e - - - A A B B C C -
2
e - - - B C A C A B ABC
1
e BC AC AB C B C A B A -
0
no. of micro-states 3 6 1
© Dr. Peter Blümler School of Physical Sciences University of Canterbury
PH 605: Thermal & Statistical Physics 3. Semi-Classical Physics page:3
Hence macro-state II has the highest statistical weight (W or thermodynamic probability or number
of arrangements; often written as W for German „Wahrscheinlichkeit“ = probability) , W = 6, in
II
this example.
3.1.2 Generalisation
Inspired from this example we want to generalise this for N particles.
For single occupation N = 1 it can be directly concluded that the highest statistical weight is given by
i
all permutations, or W = N!
However, if we consider cases in which the occupation number can eventually become larger than 1
(N > 1) we are overestimating by this method. This is because the permutation of particles in each
i
individual micro-state doesn’t generate a new micro-state.
Hence, we need the following correction:
N!
W = [3.1.1]
N !N !N !....
0 1 2
Note: The meaning of this equation can easily be checked on the previous example.
3! 6
macro-state I: W = = =3 3
I
2!0!0!1! 2
3! 6
macro-state II: W = = =6 3
II
1!1!1!0! 1
3! 6
macro-state III: W = = =1 3
III
0!0!3!0! 6
We also know that in equilibrium the Boltzmann (entropy) equation tells us that the most probable is
realised (for maximum entropy), or
S= k lnW [3.1.2]
B max
Starting from these facts we now want to derive the equation for the Boltzmann-distribution:
Given: N particles (distinguishable, independent and identical)
in r energy states: e 0,e 1,e 2,...,e r- 1
with occupation numbers N0,N1,N2,...,Nr- 1
additionally we can establish the following boundary conditions:
r- 1
(cid:229)
a) total number of particles N = N = constant [3.1.3]
i
i=0
r- 1
(cid:229)
b) total energy N e = E = constant [3.1.4]
i i
i=0
© Dr. Peter Blümler School of Physical Sciences University of Canterbury
PH 605: Thermal & Statistical Physics 3. Semi-Classical Physics page:4
our goal is summarised in eq. [3.1.2]: We have to find the maximum statistical weight, or
N!
W = fi maximum [3.1.5]
r- 1
(cid:213)
N !
i
i=0
To simplify this task, we realise that when W has a maximum, lnW also must have a maximum
(because the logarithm is a monotonic function).
This enables us to use
Stirling’s approximation:
lnN! = Nln N - N for large N: [3.1.6]
Hence, eq. [3.1.5] becomes:
r- 1 r- 1
N! (cid:213) (cid:229)
lnW =ln =lnN!- ln N !=lnN!- lnN !
r- 1 i i
(cid:213) i=0 i=0
N !
i
i=0
[3.1.6] r(cid:229) - 1( )
= NlnN - N - N lnN - N
i i i
i=0
Note: we will later see that for realistic conditions the last step (applying Stirling’s formula to N,
i
hence N = large) is satisfied.
i
r- 1 r- 1 r- 1
(cid:229) ( ) (cid:229) (cid:229)
lnW = NlnN - N - N ln N - N = Nln N - N - N ln N + N
i i i i i i
i=0 i=0 i=0
123
=N [3.1.3]
r- 1
(cid:229)
lnW = NlnN - N lnN [3.1.7]
i i
i=0
The maximum statistical weight (eq. [3.1.5]) is then given for:
(cid:230) ¶ lnW (cid:246)
(cid:231)(cid:231) (cid:247)(cid:247) = 0 [3.1.8]
Ł ¶ N ł
i N,E
Rather than differentiating with respect to the occupation numbers Ni ((cid:228)¥), it is instructive to
consider small changes (symbol d ) of the occupation number. Hence eq. [3.1.8] becomes
© Dr. Peter Blümler School of Physical Sciences University of Canterbury
PH 605: Thermal & Statistical Physics 3. Semi-Classical Physics page:5
productrule
(cid:229) (cid:229) (cid:229)
- d lnW = d N lnN = N d ln N + lnN d N =0
i i i i i i
123
d N
i
N
i
(cid:229) (cid:229)
- d lnW = d N + ln N d N =0 [3.1.9]
i i i
Equation [3.1.9] can be combined with the constant boundary conditions in eqs. [3.1.3] and [3.1.4].
(cid:229) (cid:229)
i) - d lnW = d N + ln N d N =0 (maximum)
i i i
(cid:229)
ii) - d N = d N =0 (const. must not change)
i
(cid:229)
iii) - d E = e d N =0 (const. must not change)
i i
The easiest way to solve such an equation system or to combine the conditions is the method of
„undetermined (Lagrange) multipliers“ (recap: M. Boas: „Mathematical Methods in the Physical
Sciences“, 2nd ed., Wiley 1983, page 174ff.).
This gives:
(cid:229) (cid:229) (cid:229) (cid:229)
d N + lnN d N + l d N + b e d N =0
i i i i i i
(cid:229) [ ]
d N 1+ lnN +l +be =0 [3.1.10]
i i i
where l and b are the (yet) undetermined multipliers. The first term (d N ) in [3.1.10] can be chosen
i
arbitrarily to be any number as long as the last two are chosen to fulfil ii) and iii) in the conditions
above. But generally the following condition must hold:
1+ln N + l +be =0
i i
lnN =- (l +1)- be
i i
[ ] [ ] [ ]
N =exp - (l +1) exp - be ” a exp - be [3.1.11]
i i i
[ ]
with a ” exp - (l +1) .
What is left to do? We have to find expressions for a and b .
Determination of a :
r- 1 r- 1 r- 1
(cid:229) (cid:229) [ ] (cid:229) [ ]
From eq. [3.1.3] N = a exp - be =a exp - be = N
i i i
i=0 i=0 i=0
N
a = [3.1.12]
r- 1
(cid:229) - be
e i
i=0
© Dr. Peter Blümler School of Physical Sciences University of Canterbury
PH 605: Thermal & Statistical Physics 3. Semi-Classical Physics page:6
Determination of b :
r- 1
(cid:229) - be
N e e i
From eq. [3.1.4] E = (cid:229)r- 1N e = a r(cid:229) - 1 e e- be i [3.1=.12] i=0 i [3.1.13]
i i i r- 1
i=0 i=0 (cid:229) e- be i
i=0
(cid:229)r- 1 - be ¶ (cid:229)r- 1 - be
N e e i e i
now a trick: E = i=0 i =- N ¶ b i=0 =- N ¶ ln(cid:231)(cid:230) (cid:229)r- 1 e- be i (cid:247)(cid:246)
r- 1 r- 1 ¶ b (cid:231) (cid:247)
(cid:229) e- be i (cid:229) e- be i Ł i=0 ł
14243
i=0 i=0 [3.1.12] = N
a
N doesn’t depend on b , but a does, hence,
( )
E = - N ¶ ln a - 1 = N(cid:231)(cid:231)(cid:230) ¶ lna (cid:247)(cid:247)(cid:246) [3.1.14]
¶ b Ł ¶ b ł
From eq. [3.1.2] we also know:
S =k lnW [3.=1.7]k ί NlnN - (cid:229)r- 1N ln N Ͽ [3.1=.11]k ί Nln N - a (cid:229)r- 1e- be i ln(a e- be i)Ͽ
B B i i B
Œ œ Œ œ
[ º i=0 ] ß [ º i=0 ß ]
(cid:229) - be ( ) (cid:229) - be (cid:229) - be
=k Nln N - a e i lna - be = k Nln N - a lna e i +ab e e i
B i B i
14243 14243
N E
[3.1.12]= [3.1.13]=
a a
[ ]
S = k NlnN - Nlna +b E [3.1.15]
B
from the previous part (Maxwell relations: we recall, we use E here for the total internal energy
-rather than U- because it typically used in QM notation)
(cid:230) ¶ U (cid:246) (cid:230) ¶ E(cid:246)
T =(cid:231) (cid:247) =(cid:231) (cid:247)
Ł ¶ S ł Ł ¶ Sł
V V
(cid:230) ¶ E(cid:246)
(cid:231) (cid:247)
(cid:231) (cid:247)
(cid:230) ¶ E(cid:246) Ł ¶ b ł
T =(cid:231) (cid:247) = V,N [3.1.16]
Ł ¶ S ł (cid:230) ¶ S(cid:246)
V,N (cid:231) (cid:247)
(cid:231) (cid:247)
Ł ¶ b ł
V,N
[ ]
(cid:230) ¶ S(cid:246) [3.1.15] ¶ (cid:230) ¶ E(cid:246) (cid:230) ¶ E (cid:246)
(cid:231)(cid:231) (cid:247)(cid:247) = k - N lna +E +b (cid:231)(cid:231) (cid:247)(cid:247) = k b (cid:231)(cid:231) (cid:247)(cid:247)
Ł ¶ b ł B ¶ b Ł ¶ b ł B Ł ¶ b ł
V,N V,N V,N
14243
[3.1.4]=- E
© Dr. Peter Blümler School of Physical Sciences University of Canterbury
PH 605: Thermal & Statistical Physics 3. Semi-Classical Physics page:7
and with eq. [3.1.16] we get:
(cid:230) ¶ E (cid:246) (cid:230) ¶ E(cid:246)
(cid:231) (cid:247) (cid:231) (cid:247)
(cid:231) (cid:247) (cid:231) (cid:247)
Ł ¶ b ł Ł ¶ b ł
1
T = V,N = V,N =
(cid:230) ¶ S(cid:246) (cid:230) ¶ E(cid:246) k b
(cid:231)(cid:231) (cid:247)(cid:247) k b (cid:231)(cid:231) (cid:247)(cid:247) B
Ł ¶ b ł B Ł ¶ b ł
V,N V,N
1
b = [3.1.17]
k T
B
if we now insert this equation and eq. [3.1.12] into eq. [3.1.11]:
(cid:230) e (cid:246)
Nexp(cid:231)(cid:231) - i (cid:247)(cid:247)
[ ] Ł k T ł
N =a exp - be = B
i i (cid:229)r- 1 (cid:230) e (cid:246)
exp(cid:231)(cid:231) - i (cid:247)(cid:247)
Ł k T ł
i=0 B
we get the
Boltzmann-distribution:
(cid:230) e (cid:246) (cid:230) e (cid:246)
exp(cid:231)(cid:231) - i (cid:247)(cid:247) exp(cid:231)(cid:231) - i (cid:247)(cid:247)
N Ł k T ł Ł k T ł
i = B ” B [3.1.17]
N r(cid:229) - 1exp(cid:231)(cid:231)(cid:230) - e i (cid:247)(cid:247)(cid:246) ZN
Ł k T ł
i=0 B
where Z is called ‘partition function’ (Z from German: „Zustandssumme“ = sum of states of
N
existence)
We see, how a statistical distribution can be derived from very simple assumptions. The only
assumption was eq. [3.1.2]:
S= k lnW
B max
which is the statistical interpretation of the second law of Thermodynamics.
We will apply the same formalism to obtain other (quantum) statistical distributions later.
© Dr. Peter Blümler School of Physical Sciences University of Canterbury
PH 605: Thermal & Statistical Physics 3. Semi-Classical Physics page:8
3.2 The Semi-Classical Perfect Gas
Now we want to apply Boltzmann statistics to simple systems. The most simple system is the
perfect (or ideal) gas.
3.2.1 Definition of the semi-classical, mon-atomic, perfect gas
semi - classical, mon - atomic , perfect gas
1442443 1442443 14243
QM todetermine no internal energy number of particles
energylevels or structure is much smaller than
available energy levels
(no degeneracy)
This is some hybrid between quantum-mechanical and classical behaviour, however we assume to be
able to explain more effects more precisely than in the pure classical picture.
Definition:
• identical gas particles (molecules or atoms)
• only very weak (no) interaction between the particles (particles are separated, low density),
(in energetic terms this means that the potential energy of interaction is negligible compared to
their kinetic energy V<<K)
• there are much more available energy levels than particles (classical continuum of QM-
describable discrete energy levels).
The gas shall have the following properties:
¥
(cid:229)
• consists of N particles N = N [3.2.1]
r
r=1
• the individual energies -each particle can exist in- are: e £ e £ e £ £ e £
1 2 3 K r K
• forming a complete set of discrete quantum states, with no. 1, 2, 3, ..... , r,....
• hence the state of the system is:
N particles in (quantum) state 1 with energy e
1 1
N particles in (quantum) state 2 with energy e
2 2
N particles in (quantum) state 3 with energy e
3 3

M M M M M
N particles in (quantum) state r with energy e
r r

M M M M M
¥
(cid:229)
• it has a (total) internal energy of E = E(N ,N , ,N , ) = N e [3.2.2]
1 2 K r K r r
r=1
© Dr. Peter Blümler School of Physical Sciences University of Canterbury
PH 605: Thermal & Statistical Physics 3. Semi-Classical Physics page:9
3.2.2 Distinguishable / Indistinguishable Particles?
Because the particles are chosen to be non-interacting, we can select a single particle (independent
of the others) to represent a typical particle.
Hence,
(cid:229) ¥ (cid:230) e (cid:246)
micro-state: Z (T,V)= exp(cid:231)(cid:231) - r (cid:247)(cid:247) [3.2.3]
1
Ł k T ł
r=0 B
(cid:230) (cid:246)
(cid:229) E
macro-state:* Z (T,V)= exp(cid:231)(cid:231) - r (cid:247)(cid:247) [3.2.4]
N
Ł k T ł
r B
*however, the latter we do not know how to calculate!
For the calculation of the partition function, Z, we have to distinguish two very different cases:
a) the system consists of distinguishable, identical, non-interacting particles (i.e. the exchange of
particles results in a new state, e.g. ideal crystal)
„
state a state b
1 2 3 4 5 6 1 2 3 4 5 6
7 8 9 10 11 12 7 8 9 10 11 12
13 14 15 16 17 18 13 14 16 15 17 18
19 20 21 22 23 24 19 20 21 22 23 24
25 26 27 28 29 30 25 26 27 28 29 30
b) the system consists of indistinguishable, identical, non-interacting particles (i.e. the exchange
of particles does NOT result in a new state, e.g. ideal gas)
state a = state b
© Dr. Peter Blümler School of Physical Sciences University of Canterbury