1. 4-1 supplement : Thermal Expansion of Glass
Thermal expansion determines if a glass will be shock resistant, able to withstand high thermal stresses
Small thermal expansion coefficient leads to high thermal shock resistance
Large thermal expansion leads to low thermal shock resistance
Most materials expand as they are heated
Refractory metals and ceramics-Expand less
Polymers-Expand more
Some materials expand very little
SiO2 glass; b-spodumene, Li2O.Al2O3.4SiO2
Complex systems with more than one material must have matched or compensated thermal expansions

2. Thermal expansion on the atomic scale
sufficient to look at the classic harmonic oscillator.
For zero energy (at 0K) the oscillator is at the bottom, no movement. This is where we define the energy zero.
For finite temperature, the energy is increased by kT.
Quantum: almost the same, non-equidistant levels non-symmetric prob. distr.
again: note that for the harmonic solid which we usually like to look at, there is no expansion

3. • Materials change size when heating.
CTE: coefficient of
thermal expansion (units: 1/K)
Sides symmetry for harmonic approximation

4. Typical Thermal Expansion Coefficients of Materials

5. Polycrystalline materials undergo phase transformations
Thermal expansion changes at each phase transition
c-SiO2 has numerous phase changes and numerous volume changes that must be accounted for during heat up of systems using SiO2

6. For isotropic materials, homogeneous in three directions,…
Volume expansion coefficient is 3 times larger than linear expansion
Glasses are isotropic
Fine grained polycrystals are isotropic

7. Thermal Expansion of Alkali Borate Glasses
Addition of alkali modifier decreases thermal expansion coefficient in alkali borate glasses
Modifier in low alkali borate glasses, cross links glass structure
Creation of tetrahedral borons
Adding bonds to boron, increasing connectivity of network
Strengthening the network
Rigidity of the glassy network increases
Thermal expansion decreases with modifier

8. 4-2 supplement : Morse shows another expression for the potential
Any real crystal resists compression to a smaller volume than its equilibrium value more strongly than expansion due to a larger volume.
Peter Bruesch Phonons：Theory and Experiments Ⅰ P154

9. D deionization， >0 Expand it： Morse potential Defined as 0
0, at equilibrium point
anharmonic term
Harmonic term

10. Only consider the cubic term：
are force constant，
Only consider the cubic term：
Harmonic term
anharmonic term
Based on the Boltzman statistics
With no cubic term
With cubic term, thermal expansion occurs and the coefficient is a constant
Deviation from the equilibrium point

11. Cited from Kittel p89

12. the solution to he anharmonic approximation the equation for a biatomic motion
the reduced mass ① ②
The solution is ③
If we consider the higher term in the Fourier expansion, there will be terms with 3ω
from③ and ①, and assume sA<<1， ⑤ ④ ⑥
For the fact that

13. ⑦
Form the curve slop
equilibrium point , the average kinetics for the biatoms
Substitute it in ⑦
Substitute it in ⑤
Implication：the description of molecules’ motion is very complicated with multi atoms. It not only need the basic frequency but also involved with

14. 4-3 supplement : classical quantization