1

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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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 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

4

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

5

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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Periodic Table

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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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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

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

23

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