# Sound and temperature

1. Jan 10, 2005

### JohanL

How does the velocity of sound depend on the temperarure.
If you set up a simple model of a linear lattice with nearest neighbour
interaction only you get the dispersion relation:

w^2=4c/M * sin^2 (0.5ka)

and v=w/k.
where a is the spacing between atoms, c is the force constant, k the wavevector and M the mass of an atom.
Of these only c dependes on temperature?
How do you model, explain, show how c dependes on temperature?

Maybe you know a better way to show how sound velocity dependes on
temperature.

Thank you.

2. Jan 10, 2005

### Gokul43201

Staff Emeritus
Okay, in a dispersive medium, you have something like

$$c = \sqrt{\frac{E}{\rho}}$$

What follows is my simplistic guess for a crude model.

At low enough temperatures that you do not undergo a phase transformation, E (Young's modulus) decreases almost like the inverse square of the atomic spacing. Also, the density $\rho$ decreases as the inverse cube of the atomic spacing. The atomic spacing is a nearly linear function of temperature, determined by the thermal expansion coefficient, $\alpha$.

So, my guess is that $c_T = A~`c_0~T^{1/2}$ to a first order approximation, at low temperatures.

3. Jan 12, 2005

### JohanL

I have another question.
How do you with a simple model explain the temperature dependence of the
electrical conductivity.
If you use the Drude model you get for the electrical conductivity

sigma = ne^2t / m

where n is the density of mobile electrons and t is the relaxation time.
t is the time between collisions and must be the only variabel here that depends on temperature. How can you estimate t(T).

Maybe there is a better model that describes the temperature dependence of the electrical conductivity.

4. Jan 13, 2005

### broomfieldjay

Practically, one way to deal with temperature and conductivity is through resistivity which is 1/sigma. Resistivity is fairly easy to measure. At room temperature, resistivity is proportional to temperature (rho =constant + A*T) The coefficient, A, is often tabulated and its temperature range of validity is given.
Since rho = m/(n*e^2*t), t is inversely proportional to temperature.

One way to look at this is that at room temperature, the ions vibrate and electrons "collide" with these ions. As the temperature increases, the vibrations increase and more collisions occur and the average time (t) between collisions decreases. So t is inversely proportional to temperature. Likewise, the conductivity declines with temperature for Drude metals.

At extremely low temperatures, the vibrations are so small that the mean free path is dominated by impurities or defects in the material and becomes almost constant with temperature.

Note that all of this applies only to substances that follow the Drude model. For semiconductors, n does depend on temperature (~exp(-1/T)). Thus, for semiconductors conductivity is a rapidly increasing function of temperature. Although collisions still occur, n tends to dominate in semiconductors.