Experiment 3Bench 3MZU-PE / CAL-3Ready

Phonons and Lattice Heat Capacity

Lattice Dynamics Rig + Calorimeter

Drive a ball-and-spring chain to see phonon modes, then measure heat capacity in an adiabatic calorimeter.

Duration: 3 hours
Kittel: Chapters 4 & 5
Topics: Lattice vibrationsDispersionDebye model

Lattice dynamics rig

Monatomic chain

Running

AtomsAtoms move along the chain — watch the springs compress

Dispersion analyser

ω versus k across the first Brillouin zone

MONATOMIC CHAIN

ω at cursor

0.618√(C/M)

Group velocity dω/dk

2.988

Wavelength

10.00a

Adiabatic calorimeter

Copper · 25 g (0.393 mol)

Ready

Sample thermometer

Sample T

Pulse energy Q

Rise ΔT

KHI

C = Q/nΔT

J/mol·K

Heat pulse

Heater power500 mW

Pulse length10 s

Heater

Debye prediction

23.39J/mol·K

Measured / 3R

—

Θ_D

343K

Heat capacity plotter

Debye · Einstein · Dulong-Petit, with your measured points

COPPER · Θ_D = 343 K

DebyeEinstein (Θ_E = 0.75 Θ_D)Dulong-PetitYour calorimeter points

Chain drive

MZU-PE / CAL-3 · mechanical section

Wavevector k0.20 π/a

DIATMONO

Chain type

OPTACO

Branch

Chain setup

Mass ratio M₂/M₁0.50

Play speed1.00×

Cryostat & sample rack

MZU-PE / CAL-3 · thermal section

Cryogenic surfaces below 120 K. The calorimeter can only be pulsed once the bath has settled.

Specimen

Bath temperature300 K

T / Θ_D = 0.875

The standard calorimetry specimen. At 300 K copper sits just below the Dulong-Petit value.

Key Equations

Monatomic chain, force constant C and mass M:

\omega = 2\sqrt{\tfrac{C}{M}}\left|\sin\tfrac{ka}{2}\right|

Heat capacity per mole:

C_V^{\text{DP}} = 3R,\qquad C_V^{\text{E}} = 3R\left(\tfrac{\Theta_E}{T}\right)^{2}\frac{e^{\Theta_E/T}}{(e^{\Theta_E/T}-1)^{2}}

C_V^{\text{D}} = 9R\left(\tfrac{T}{\Theta_D}\right)^{3}\!\!\int_0^{\Theta_D/T}\!\frac{x^4e^x}{(e^x-1)^2}dx

At low temperature this reduces to the $T^3$ law, $C_V \approx \tfrac{12\pi^4 R}{5}(T/\Theta_D)^3$ . The calorimeter measures $C = Q/(n\,\Delta T)$ .

Procedure

0/4

AModes of the monatomic chainWith the monatomic chain running, set k to its minimum and describe the motion. Wind k up to the zone boundary and watch what happens to the wavelength, to the displacement pattern of neighbouring atoms, and to the group velocity readout.
BAcoustic and optical branchesSwitch to the diatomic chain. Compare the acoustic and optical branches at small k: which one has the two masses moving together, and which has them moving against each other? Note the frequency gap at the zone boundary and how it changes with the mass ratio.
CMeasure a heat capacityLoad copper, set the bath to 300 K, and fire a heat pulse. Record Q, ΔT and C = Q/nΔT. Repeat at 200, 100, 50 and 20 K, reducing the pulse each time so ΔT stays small, and plot each point.
DWhere the classical model failsRepeat the 300 K measurement on diamond and on lead. Compare each with 3R = 24.94 J/mol·K and explain why one obeys Dulong-Petit at room temperature and the other does not.

Table 3 — calorimetry

0 entries

Keep ΔT below a few per cent of T, or the heat capacity will have changed over the interval you measured.

#	Specimen	Θ_D/ K	T/ K	T/Θ_D	Q/ J	ΔT/ K	C measured/ J/mol·K	C/3R	C Debye/ J/mol·K
No readings yet — set the controls, then press “Record reading”.

Analysis & Reflection

Analysis questions

Plot your measured $C_V$ against $T^3$ for the four lowest copper points. Is the relation linear? What does the gradient give you?
The four specimens have Debye temperatures spanning 105 K to 2230 K. Their molar masses are Pb 207, Cu 63.5, Fe 55.8, Al 27.0 and C 12.0 g/mol. Describe the trend and explain how Θ_D depends on atomic mass and on bond stiffness.
At the zone boundary the group velocity readout falls to zero. What does that mean physically for a phonon with that wavevector?
Explain in physical terms what a phonon is, and why the idea is useful even though no atom travels through the crystal.
Compare the Einstein and Debye curves at low temperature on the plotter. Which one falls too quickly, and what assumption in Einstein's model causes that?

Physics problems

Using $\Theta_D(\text{Cu}) = 343$ K, calculate $C_V/3R$ at 10 K from the $T^3$ law and compare it with the calorimeter.
Lead has Θ_D = 105 K. Predict whether it obeys Dulong-Petit at 300 K, then check it on the bench.
Why does the diatomic chain have a frequency gap between the acoustic and optical branches? Argue it from the restoring forces at the zone boundary.
Define the Debye temperature in terms of the maximum phonon frequency, and explain why a material with a large Θ_D stores less heat at room temperature than one with a small Θ_D.