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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I.
Pascual Lattice vibrations
Thermal properties of solids: The heat capacity
HeatHeat capacitycapacity isis aa measurablemeasurable
physicalphysical quantityquantity
thatthatcharacterizescharacterizes thethe abilityability of aof a
bodybody toto storestore heatheat asas itit changeschanges inin
temperaturetemperature..
ItIt isis defineddefined asas thethe rate ofrate of changechange
ofof temperaturetemperature asas heatheat isis
addedadded to ato a bodybody atat thethe givengiven
conditionsconditions andand statestate ofof thethe bodybody..
InIn thethe International System ofInternational System of
UnitsUnits,, heatheat capacitycapacity isis expressedexpressed
inin unitsunits ofofjoulesjoules perper kelvinkelvin..
AtAt roomroom temptemp.. nearlynearly allall thethe solidssolids
havehave thethe samesame heatheat capacitycapacity
CCVV,, closeclose to 3Nto 3NAAkkBB oror 25 JK25
JK--11molmol--11
Heat capacity at
constant volume
Behavior at low temperatures ofBehavior at low temperatures
of
the heat capacity of diamondthe heat capacity of diamond
Molar heat capacity in JKMolar heat capacity in JK--11
Experiments find T
3
behavior
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Pascual Lattice vibrations
Thermal properties of solids: The Einstein model
ThenThen
Does not reproduceproperly the behavior at
very low temperatures
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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I.
Pascual Lattice vibrations
Thermal properties of solids: The Debye model
/a/a--/a/a
DD
k
The problem then lies in calculating the mean thermal energy of
a set of
oscillator with a frequency given by the dispersion relation
4.38.
TOTALTOTAL
ENERGYENERGY==
D
0
EnergyEnergy perper
phononphonon
NumberNumber ofof
phononsphonons perper
frequencyfrequency
dd XX XX
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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I.
Pascual Lattice vibrations
Thermal properties of solids: The Debye model
Number
Numberofofphonons
phononsper
perfrequency
frequency
Including the dispersion relation
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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I.
Pascual Lattice vibrations
Thermal properties of solids: The Debye model
TOTALTOTAL
ENERGYENERGY==
D
0
EnergyEnergy
perper
phononphonon
NumberNumber ofof
phononsphonons perper
frequencyfrequency
dd XX XX
High
HighTT
Low
Low
TT
Right behavior at
low temperatures
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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I.
Pascual Lattice vibrations
Das Debye-Modell der spezifischen Wrme
D
Phononen-Zustandsdichte eines realen Festkrpers und die
Debye-Nherung.
Experimentalphysik IV; SoSe2008; Petra Tegeder
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SS 09 - 20 140: Experimentalphysik IV K. Franke & J.I.
Pascual Lattice vibrations
Molare Wrmekapazitt von festem Argon geplottet
gegen T3.Das experimentelle Ergebnis stimmt sehr gut mit
demDebyeschen T3-Gesetz berein.
Debye-Modell
Beispiele:
Experimentalphysik IV; SoSe2008; Petra Tegeder
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Pascual Lattice vibrations
Summary of models describing the Heat Capacity of a crystal
Dulong-Petit: Classic approach
treating atoms in a lattice asindependent particles
Einstein: Quantum model treatingthe lattice as a set of
independentQuantum harmonic oscillators
Debye: Quantum model treatingthe lattice as a set of
coupledQuantum harmonic oscillators.Considers then the lattice of
Natoms as a set of N phonons with a
quadratic density of phonon states
by assuming a linear dispersionrelation, which is only a
goodapproximation at low temperatures.
