What Are Acoustic Phonon Modes in Calculating Heat Capacity of a Solid?

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In summary, the conversation discusses the calculation of heat capacity in solids due to phonons. The concept is to calculate the available standing wave modes for the phonons, with acoustic modes being inter-unit cell vibrations and optical modes being intra-unit cell vibrations. In a 1D calculation, the number of acoustic modes is equal to the number of unit cells, while in 3D, there are nine acoustic modes due to three orthogonal directions with three types of vibrations. This is based on the assumption that all modes are acoustic in crystals with 1 atom per unit cell.
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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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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.
 
  • #3
TeethWhitener said:
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.
 
  • #4
Henryk said:
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.
 
  • #5
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.
 

1. What is the meaning of "Number of acoustical modes"?

The number of acoustical modes refers to the different ways in which sound waves can travel through a medium or space. These modes are determined by the physical properties of the medium, such as its density and elasticity.

2. How is the number of acoustical modes calculated?

The number of acoustical modes is calculated using the formula N = 2L/λ, where N is the number of modes, L is the length of the medium, and λ is the wavelength of the sound wave. This formula is based on the principle that the length of the medium must be equal to an integer multiple of half the wavelength for a standing wave to form.

3. What factors affect the number of acoustical modes?

The number of acoustical modes is affected by the physical properties of the medium, such as its length, density, and elasticity. It is also influenced by external factors such as temperature, pressure, and the presence of obstacles or boundaries in the medium.

4. How does the number of acoustical modes impact the transmission of sound?

The number of acoustical modes can impact the transmission of sound in various ways. In a medium with a higher number of modes, there will be more paths for sound to travel, resulting in a higher transmission rate. However, in certain situations, a higher number of modes can also lead to interference and reduced transmission of sound.

5. Can the number of acoustical modes be changed?

The number of acoustical modes is a property of the medium and cannot be changed. However, it can be influenced by external factors such as temperature, pressure, and obstacles, which can alter the physical properties of the medium and thus impact the number of modes.

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