Download - Dulong Einstein Debye
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Physical Chemistry III: Statistical-thermodynamic
Statistical thermodynamic of Solids:
Kinetic energyIntroduction of structured solidsLaw of Dulong and Petit (Heat capacity) 1819Einstein Model of Crystals 1907Born and von Karman approach 1912Debye Model of Crystals 1912
Electronic energyFermi level 1926Fermi-Dirac distribution
25-11-2002
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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 freedomEK = 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 areIndependent and equivalent
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Einstein Model molecular processing
System of 1-Dim Harmonic Oscillator
Quantized expression of the energy:v = h (v+1/2) v = 0, 1, 2, ...
Partition function (without attributing 0 to the ground state)
q = e-(h(v+1/2)/kT) = e-(h/2kT) e-(h/kT) v
Considering the vibrational temperature = h/k
The molecular internal energy Umolecular = - d[Ln(q)] / d ]N, V = k T2 d[Ln(q)] / dT ]N, V
q = e-/2T
1- e-/T
Umolecular = k (1/2 + )1
e/T - 1
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System of N 3-Dim Harmonic Oscillators
Q = q3N U = 3N Umolecular
Einstein Model energy of the system
0.0 0.5 1.0 1.5 2.0 2.52000
4000
6000
8000
10000
12000
14000
16000
18000
20000
3/2 Nh + 3 NkT
3/2 Nh
Inte
rnal
ene
rgy
kT/h
U = 3 N k (1/2 + )1e/T - 1
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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
U = 3 N k (1/2 + )1e/T - 1
C = 3 Nk (/T)2 ––––––e/T
(e – 1)2/T0.0 0.5 1.0 1.5 2.0 2.5
0.0
0.2
0.4
0.6
0.8
1.03Nk
Hea
t cap
acity
T/
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Einstein Model comparison with experiment
Comparison of the observed molar heat capacity of diamond (+) with Einstein’s model. (After Einstein’s original paper-1907)
The value of = 1325 K was given to produce an agreement with the experiment at 331,1 K.ork ħ is the parameter that distinguishes different substances:
am
whereE is Young’s modulusm is atomic massand a is the lattice parameterEinstein model gives also a qualitatively quite good agreement on term of calculated from the elastic properties
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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.
Einstein Model results and limitation
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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 i =i/2
THE LATTICE VIBRATIONS OF THE CRYSTAL ARE EQUIVELENT TO 3N INDEPENDENT OSCILLATORS
E = hi (1/2 + (e(hi /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 = 2/
Wave phase velocity v = =/2k
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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 Ln1 n2 n3 (r) = A sin(n1x / L) sin(n2y / L) sin(n3z / L)
The wave vector in the Cartesian coordinates is k(n1/L, n2/L, n3/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 /L and the volume of Vu= (/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 k+dk.
f(k) dk = (1/8) (4k2) dk /(/L)3 = Vk2 dk/(22)
In term of circular frequency: f(k) dk = f() d = (Vk2/22) (dk/d) d = V 2 d /(2 v2 vg 2)
Where vg= d/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() d = V 2 d /(2 v3 2)
The wave vector has three independent modes: 1 longitudinal and 2 transversal modes
f() d = V 2 d /(2 2) (1/vL3 + 2/vT
3 ) In an isotropic Medium vL = vT =vm
f() d = 3V 2 d /(2 vm3 2)
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Debye ModelLattice 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
ƒD f() d = 3N f() d = 9N2 d/D
3
Now the sum can be replaced with an integral
3N..... = ƒD .....f() d
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Debye Model The energy of the crystal
E i
3N
1
[(1/2)ħi + ]3N
1
ħiħi
kTe -1 ƒ [(1/2)ħ + ] f() dħ
e -1ħkT0
D
NkD + NkT ƒ dx
Where D =ħD/k xD = ħD/kT x = ħ/kT
98
9xD
3 e -1x3
x0
xD
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Debye Model The heat capacity of the crystal
T0X
4/15E = 9NħD/8 Vibrational zero-point energy
Cv = dE/dT]v = 0
xD3/3
E = 9NkD/8 + 3NkT
Cv = 3Nk
TX 0
dxe
xNkT
xNkE
Dx
xD
D
0
3
3 1
9
8
9
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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 ()
Cv= 3Nk { ƒ dx } 3xD
3x4 ex
(ex -1)20
xD
When T<<D ƒ dx = 4/15x4 ex
(ex -1)20
xD
CV = 4 N k 125
TD
3ref5
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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 AlCubic Closest Packing
The lattice parameter a = 0,25 nmThe density =2,7 g/cm3
The wave velocity v =3,4 km/selst
E / elst = 0,79
D / elst = 0,95
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Cubic close packed, (a)
Hexagonal close packed (a, c)
Body centered cubic (a)
Cu (3.6147) Be (2.2856, 3.5832) Fe (2.8664)
Ag (4.0857) Mg (3.2094, 5.2105) Cr (2.8846)
Au (4.0783) Zn (2.6649, 4.9468) Mo (3.1469)
Al (4.0495) Cd (2.9788, 5.6167) W (3.1650)
Ni (3.5240) Ti (2.506, 4.6788) Ta (3.3026)
Pd (3.8907) Zr (3.312, 5.1477) Ba (5.019)
Pt (3.9239) Ru (2.7058, 4.2816)
Pb (4.9502) Os (2.7353, 4.3191)
Re (2.760, 4.458)
Lattice parameter
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Deby Temperature
CV = 4 N k 12 5
TD
3
NkD + NkT ƒ dx
Where D =ħD/k xD = ħD/kT x = ħ/kT
98
9xD
3 e -1x3
x0
xD
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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
The Fermi function f(E), drown from the Fermi-Dirac statistics, express the probability that a given electronic state will be occupied at a given temperature.
0 200 400 600 800 10000.0
0.5
1.0
E-EF < 0
E-EF > 0
f(E
)
Temperature
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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
FK
T dkkL
dkkfN0
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2
32)(2
k FT
VN
3
23
3 23 3VNk T
F
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Electronic contribution to the internal energy
The wavefunction of free electron is:
Its substitution in the Schrödinger equation:
).(),( tkxiAetx
2
222 ),(
2),(),(
x
tx
mtxtxE
22
2k
mE
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Electronic contribution to the internal energy
3 22
2
3
22
2
322
V
N
mmT
FF kE
Fermi Energy
Fermi Temperature
kEF
FT
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Temperature effect on electrons
Metal K Na Li Au Ag CuNT /V (10^22 cm^3) 1.34 2.5 4.6 5.9 5.8 8.5
KF (1/A°) 0.73 0.9 1.1 1.2 1.2 1.35EF (eV) 2.1 3.1 4.7 5.5 5.5 7TF (K) 24400 36400 54500 64000 64000 81600
Only electrons near from Fermi level are affected by the temperature.
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Electronic contribution in the heat capacity of a metal
TkEmV
C
dNEfET
C
EfET
C
ET
C
Fev
VN
iiev
VNiii
ev
VNee
v
22/12/3
22
,0
,1
,
2
2
)(2
)(2
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).
TTCv .. 3
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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/thermo/2 http://www.plmsc.psu.edu/~www/matsc597c-1997/systems/3 http://web.mit.edu/5.62/www/notes/25.pdf4 http://www.cartage.org.lb/en/themes/Sciences/Physics/SolidStatePhysics/Electrons/ElectronicHeat/ElectronicHeat.htm
5 Statistical Physics, F. Mandl6 Thermodynamique statistique à partir de problèmes et de résumé de
cours C. Chahine, P. Devaux7 An introduction to statistical thermodynamics, T. L. Hill8 Statistical mechanics, J. E. Mayer, M. G. Mayer