Thermal expansion (anharmonic effect)

In summary, the thermal expansion parameter \alpha can be calculated using the formula \alpha = \frac{1}{L} \frac{\partial L}{\partial T} = \frac{1}{3B} \sum \frac{\partial(hw)}{\partial V} \frac{\partial n}{\partial T}. When calculating d(hw)/dV, it is important to note that w is related to the lattice constant a, which in turn is related to the volume V according to the lattice. This formula assumes a material with cubic symmetry, but for materials with lower symmetry, the calculation may be more complex. The general formula is \alpha = L^{-1} dL/dT, but for materials with tetragonal
  • #1
cavalier3024
15
0
i have a question regarding calculating the thermal expansion parameter [tex]\alpha[/tex]
it is given by:
[tex]\alpha = \frac{1}{L} \frac{\partial L}{\partial T} = \frac{1}{3B} \sum \frac{\partial(hw)}{\partial V} \frac{\partial n}{\partial T}[/tex]

now, I am not quite sure how i calculate [tex]d(hw)/dV[/tex]. how is [tex]w[/tex] related to [tex]V[/tex]? the only thing i can think of is that [tex]w[/tex] is related to [tex]a[/tex] (the lattice constant) and [tex]a[/tex] is related to [tex]V[/tex] according to the lattice. is that right? and is that the only connection or there are other factors i should take into account?
 
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  • #2
Yes, that's correct. It looks like your equation has assumed a material with cubic symmetry, so that derivatives with respect to lattice constant are just 1/3 of a derivative wrt volume. Things are more complex if the material has lower symmetry.
 
  • #3
thanks alot! i hope this will help me to pass my final exam two days from now

btw this formula is taken from ashcroft, they didnt say there that it applies only to cubic symmetry but I am not sure. do you have the general form of this formula, or can refer me to a site\book?
 
  • #4
The first part [tex]\alpha = L^{-1} dL/dT[/tex] is the general formula. If you have a crystal with tetragonal symmetry (for example), then you will have two expansion coefficients, [tex]\alpha_a = a^{-1} da/dT[/tex] and [tex]\alpha_c = c^{-1} dc/dT[/tex]. The rest of the derivation should be basically the same, but without the substitution that dV = 3a^2 da.
 

What is thermal expansion?

Thermal expansion is the tendency of matter to change in shape, area, and volume in response to a change in temperature. This effect is due to the increase in kinetic energy of particles as temperature increases, causing them to vibrate and take up more space.

What is the difference between linear and volume thermal expansion?

Linear thermal expansion refers to the change in length of a material due to temperature, while volume thermal expansion refers to the change in volume. Linear expansion occurs in one dimension, while volume expansion occurs in three dimensions.

How does anharmonic effect contribute to thermal expansion?

Anharmonic effect refers to the non-linear behavior of atoms in a material, which can cause them to vibrate with different frequencies. This variation in vibration can lead to uneven expansion, resulting in thermal expansion.

What are some practical applications of thermal expansion?

Thermal expansion is used in many everyday objects, such as thermometers, thermostats, and bimetallic strips. It is also important in industries like construction and engineering, where materials need to be able to withstand changes in temperature without significant damage.

How is thermal expansion measured and accounted for in experiments?

Thermal expansion is typically measured using a coefficient of thermal expansion, which quantifies the amount of expansion of a material per unit of temperature change. In experiments, this coefficient is accounted for in calculations and adjustments are made to ensure accurate measurements.

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