Physical Chemistry III: Statisticalthermodynamic

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25-11-2002
Physical Chemistry III Statistical-thermodynamic
Statistical thermodynamic of Solids Kinetic
energy Introduction of structured solids Law of
Dulong and Petit (Heat capacity)
1819 Einstein Model of Crystals
1907 Born and von Karman approach
1912 Debye Model of Crystals
1912 Electronic energy Fermi
level 1926 Fermi-Dirac distribution
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Structured Solids (The 14 Bravais lattices)
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Law of Dulong and Petit 1
The crystal stores energy as - Kinetic energy
of the atoms under the form of vibrations.
According to the equipartition of energy, the
kinetic internal energy is f . ½ . k. T
where f is the degree of freedom. Each
atom or ion has 3 degrees of freedom EK 3/2
N k T
- Elastic potential energy. Since the kinetic
energy convert to potential energy and vice
versa, the average values are equal
Epot 3 N (½ K x2) 3 N x (½ k T)
The stored molar energy is then E EK
Epot 3 NA k T 3 R T ? C dE/dT 3 R
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Law of Dulong and Petit 2
Within this law, the specific heat is independent
of - temperature - chemical
element - crystal structure
At low temperatures, all materials exhibit a
decrease of their specific heat Classical
harmonic oscillator ? Quantum Statistical
mechanics
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Einstein Model approximation
Each molecule in the crystal lattice is supposed
to vibrate isotropically about the equilibrium
point in a cell delimited by the first neighbors,
which are considered frozen. System of N
molecules the system can be
treated as 3N independent one-
dimensional harmonic
oscillator Motions in the x, y and z axis
are Independent and equivalent
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Einstein Model molecular processing
System of 1-Dim Harmonic Oscillator Quantized
expression of the energy ev hu (v1/2)
v 0, 1, 2, ... Partition function (without
attributing 0 to the ground state) q S
e-(hu(v1/2)/kT) e-(hu/2kT) S e-(hu/kT)
v Considering the vibrational temperature q
qE hu/k The molecular internal energy
Umolecular - dLn(q) / db N, V k T2
dLn(q) / dT N, V
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Einstein Model energy of the system
System of N 3-Dim Harmonic Oscillators Q q3N
? U 3N Umolecular
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Einstein Model the heat capacity of
the system
System of N 3-Dim Harmonic Oscillators The heat
capacity of the crystal is then C dU / dT
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Einstein Model comparison with
experiment
The value of qE 1325 K was given to produce an
agreement with the experiment at 331,1 K. qE or
wE kqE/h is the parameter that distinguishes
different substances wE ? A (a E/m)1/2 where E
is Youngs modulus m is atomic mass and a is the
lattice parameter Einstein model gives also a
qualitatively quite good agreement on term of qE
calculated from the elastic properties
Comparison of the observed molar heat capacity of
diamond () with Einsteins model. (After
Einsteins original paper-1907)
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Einstein Model results and limitation
The Einstein Model of crystals takes into account
the alteration of the heat capacity by -
temperature - chemical element -
crystal structure This model explained the
decrease of the heat capacity at low
temperature. However This decrease is too fast!
The experimental results evolve as T3 Reason is
that the Einstein model does not consider the
collective motion and only consider one
vibrational frequency.
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Born and von Kármán approach
System of N atoms possess 3N degrees of freedom,
all expressing vibrational motion. Thus, the
whole crystal has 3N normal modes of vibration
characterized by their frequencies ui
wi/2p THE LATTICE VIBRATIONS OF THE CRYSTAL ARE
EQUIVELENT TO 3N INDEPENDENT OSCILLATORS E S
hui (1/2 (e(hui /kt)-1)-1)
3N
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Propagation of sound wave in solids notion
This propagation could be solved using the
classical concepts since the atomic structure
(dimensions) can be ignored in comparison to the
wavelength of a sound wave. The 3-D wave
equation ?2 ?(r) k2 ?(r) 0
where k is the magnitude of the wave vector k
2p/l Wave phase velocity v
l u l w/2p w/k
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Propagation of sound wave in solids standing
waves in a box
The 3-D wave equation of motion solved in a cubic
box with the side L Fn1 n2 n3 (r) A sin(n1p x /
L) sin(n2p y / L) sin(n3p z / L) The wave vector
in the Cartesian coordinates is k(pn1/L, pn2/L,
pn3/L) In the k space, formed by the allowed
values of k(ni 1, 2, ...), is composed of
cubic point lattice with the separation of p/L
and the volume of Vu (p/L)3.
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Propagation of sound wave in solids Density of
states
Defining the density of states come to the
determination of the number of normal modes of
standing waves with the lying magnitude between k
and kdk. f(k) dk (1/8) (4pk2) dk /(p/L)3
Vk2 dk/(2p2) In term of circular frequency
f(k) dk f(w) dw (Vk2/2p2)
(dk/dw) dw V w2 dw /(2 v2
vg p2) Where vg dw/dk is the group velocity
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Propagation of sound wave in solids Density of
states
In a non dispersing medium vg v f(k) dk
f(w) dw V w2 dw /(2 v3 p2) The wave vector has
three independent modes 1 longitudinal and 2
transversal modes f(w) dw V w2 dw /(2 p2)
(1/vL3 2/vT3 ) In an isotropic Medium vL vT
vm f(w) dw 3V w2 dw /(2 vm3 p2)
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Debye Model
Lattice vibrations are regarded as standing waves
of the atomic planes displacement It is assumed
that all normal mode frequencies satisfy the
equation of the density of states An upper limit
for frequencies is, however, set such as wD
f(w) dw 3N ? f(w) dw 9Nw2 dw/wD3 Now the
sum can be replaced with an integral S3N.....
wD .....f(w) dw
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Debye Model The energy of the
crystal
3N
E S ei
1
hwi
S (1/2)hwi ???
3N
hwi
e -1
1
kT
hw
wD
(1/2)hw ??? f(w) dw
hw
e -1
0
kT
9 8
9 xD3
x3
xD

• ? NkqD ? NkT ?? dx
• Where qD hwD/k xD hwD/kT
  x hw/kT

e -1
x
0
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Debye Model The heat capacity of the
crystal
T?? X? 0
T?0 X??
xD3/3
p4/15

• E 9Nk?D/8 3NkT
• ? Cv 3Nk

• E 9Nh?D/8
• Vibrational zero-point energy
• ? Cv dE/dTv 0

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Debye Model The heat capacity at low temperature
Cv dE/dT) v T enters this expression
only in the exponential term (b)
xD
Cv 3Nk ? ??? dx
3 xD3
x4 ex
(ex -1)2
0
ref5
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Debye Model-Experiment
The Debye Model gives good fits to the
experiment however, it is only an interpolation
formula between two correct limits (T 0 and
infinite)
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Einstein-Deby Models
Lattice structure of Al Cubic Closest Packing

• QE / Qelst 0,79
• QD / Qelst 0,95

• The lattice parameter a 0,25 nm
• The density r2,7 g/cm3
• The wave velocity v 3,4 km/s
• Qelst

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Periodic Table
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Lattice parameter
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Deby Temperature
9 8
9 xD3
x3
xD
E ? NkqD ? NkT ?? dx Where qD
hwD/k xD hwD/kT x
hw/kT
e -1
x
0
3
12 5
T qD
CV ? p4 N k ?
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The limit of the Debye Model
? The electronic contribution to the heat
capacity was not considered
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Electronic contribution Fermi level
At absolute zero temperature, electrons pack into
the lowest available energy, respecting the Pauli
exclusion principle each quantum state can have
one but only one particle Electrons build up a
Fermi sea, and the surface of this sea is the
Fermi Level. Surface fluctuations (ripples) of
this sea are induced by the electric and the
thermal effects. So, the Fermi level, is the
highest energetic occupied level at zero absolute
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Electronic contribution Fermi function
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Electronic contribution to the internal energy
Orbitals are filled starting from the lowest
levels, and the last filled or orbital will be
characterized by the Fermi wave vector KF The
total number of electron in this outer orbital
is
Because electrons can Adopt 2 spin orientations
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Electronic contribution to the internal energy
The wavefunction of free electron is Its
substitution in the Schrödinger equation
?
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Electronic contribution to the internal energy
Fermi Energy
Fermi Temperature
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Temperature effect on electrons
Only electrons near from Fermi level are affected
by the temperature.
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Electronic contribution in the heat capacity of a
metal
Ref 3
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Summary
The nearest model describing the thermodynamic
properties of crystals at low temperatures is the
one where the energy is calculated considering
the contribution of the lattice vibrations in the
Debye approach and the contribution of the
electronic motion (this is of importance when
metals are studied).
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Summary
Terms that replaced the partition function
are Density of state (collective motion) Fermi
function (electronic contribution)
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References
1 http//hyperphysics.phy-astr.gsu.edu/hbase/therm
o/ 2 http//www.plmsc.psu.edu/www/matsc597c-1997/
systems/ 3 http//web.mit.edu/5.62/www/notes/25.pd
f 4 http//www.cartage.org.lb/en/themes/Sciences/P
hysics/ SolidStatePhysics/Electrons/ElectronicHeat
/ElectronicHeat.htm 5 Statistical Physics, F.
Mandl 6 Thermodynamique statistique à partir de
problèmes et de résumé de cours C. Chahine, P.
Devaux 7 An introduction to statistical
thermodynamics, T. L. Hill 8 Statistical
mechanics, J. E. Mayer, M. G. Mayer