Thermal phonons @ low temperatures

[Liquidxlax]

Homework Statement

a) estimate the number of phonons in a 1mm cube of a crystal at 10K, taking a sound speed of 1000m/s.

b) estimate the frequency ω for which you will find the highest phonon nrg content in the range of ω to ω+dω. (Again, assume low temps). Give your solution in the form hω = αk_BT, finding the value of α.

c) show that the dominant phonon wavelength λ, assuming simple cubic lattice, can be written λ = βaθ_D/T with a being the lattice constant. Find β.


Problem I'm having is am I supposed to find an exact value as he has not stated. (Haven't been able to see my prof)

The Attempt at a Solution

Only thing i can think of for part a is

N_ph = ∫dωD(ω)n

where n is the Planck distribution and D is the density of states. Only problem is that this equation only uses the temperature

debeye frequency is ω³ = 6πv³N/V

I'm assuming N is the number of phonons which is no help.

 

[mark726]

a) To estimate the number of phonons in a 1mm cube of a crystal at 10K, we can use the Debye model which states that the number of phonon modes in a crystal is given by N = 9Vω^3/π^3c^3, where V is the volume of the crystal and c is the speed of sound. Since we are given a cube with a side length of 1mm, the volume V = (1mm)^3 = 10^-9 m^3. The speed of sound in the crystal is given as 1000 m/s. Plugging in these values, we get N = 9(10^-9 m^3)(3.14 x 10^9 m/s)^3/(3.14)^3(1000 m/s)^3 = 2.17 x 10^14 phonons in a 1mm cube of the crystal at 10K.

b) To estimate the frequency ω for which we will find the highest phonon energy content in the range of ω to ω+dω, we can use the Bose-Einstein distribution which gives the average number of phonons in a mode with frequency ω as n(ω) = 1/(e^(hω/kBT)-1). To find the maximum phonon energy content, we need to find the frequency at which n(ω) is maximum. Taking the derivative of n(ω) with respect to ω and setting it equal to 0, we get hω = kBT. Therefore, the frequency ω for which we will find the highest phonon energy content in the range of ω to ω+dω is given by hω = kBT, where α = 1.

c) To show that the dominant phonon wavelength λ can be written as λ = βaθD/T, we first need to define the Debye temperature θD. It is given by θD = hωD/kB, where ωD is the Debye frequency. Using the Debye model, we can write ωD = (6π^2N/V)^1/3c, where N is the number of phonon modes in the crystal and V is the volume of the crystal. Substituting this into the definition of θD, we get θD = (h(6π^2N/V