# Number of acoustical modes

#### aaaa202

In calculating the heat capacity of a solid due to the phonons in the low temperature limit, I am given the impression that the idea is to calculate the amount of standing wave modes available for the phonons in the solid. Is this the correct idea?
But then in calculating the Debye temperature my book says: "for n primitive cells the number of acoustic phonon modes is n." What does it mean by this, what are acoustic phonon modes - are they different standing wave modes of the acoustical branch?

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#### TeethWhitener

Gold Member
Optical modes are intra-unit cell vibrational modes, and acoustic modes are inter-unit cell modes.

I imagine this is a 1D calculation in your book (for 3D, basically just multiply by 3). You can make a (kind of bad) analogy with beads on a string. Each bead represents a unit cell. If you have a string of length $L$ and you vibrate the string, then the possible vibrational modes are ones where the wavelength is a half integer of the string length ($n/L$). Now if you place $N$ beads on that string, then the acoustic modes will have wavelengths of $\{1/L,2/L,\dots , N/L\}$. For $n>N$, the modes are no longer inter-unit cell (because the beads are split over more than one half-wavelength). So the total number of acoustic modes you can have in 1D is equal to the total number of unit cells you have.

#### Henryk

Gold Member
I imagine this is a 1D calculation in your book (for 3D, basically just multiply by 3).
Not quite. In 3D you have three orthogonal phonon propagation directions. For each direction you have two transverse and one longitudinal mode, therefore, you have 9 acoustic modes of vibration in total.

#### TeethWhitener

Gold Member
Not quite. In 3D you have three orthogonal phonon propagation directions. For each direction you have two transverse and one longitudinal mode, therefore, you have 9 acoustic modes of vibration in total.
Where are you getting this from? In an N-atom system, there are 3N degrees of freedom in 3 dimensions. In crystals with 1 atom per unit cell, this means that all the modes are acoustic. Thus the crystal has 3N acoustic modes. This is a standard assumption in both Einstein’s and Debye’s theories of heat capacity.

#### Henryk

Gold Member
TeethWhitener, you are correct. The number of allowed states in the Brillouin zone is equal to the number of primitive unit cells in the entire crystal, that times 3 for 3 polarizations.

"Number of acoustical modes"

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