Analyzing Phonon Collisions in a Cubic Lattice

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Homework Statement

In a simple cubic lattice of spacing 0.2nm a phonon traveling in the {1 0 0} direction with wavelength 0.42nm collides with another phonon of the same wavelength which is traveling in the {1 1 0} direction. Draw a reciprocal space diagram to show the magnitude and direction of the resultant phonon and discuss how the resulting phonon should be designated in the First Brillouin Zone.

Explain how such processes can be responsible for the finding that certain crystals have a maximum in their thermal conductivity as a function of temperature?

The Attempt at a Solution

So I just draw the first phonon going along the positive h-direction, and the second phonon is at an angle of 45 degrees to that? So the resultant phonon is at an angle of 22.5 degrees to the horizontal? But how do I work out what the wavelength of the resultant phonon is? Is there going to be constructive or destructive inteference? Do I just add the wavelengths of the two phonons?

What does it mean by 'discuss how the resulting phonon should be designated in the First Brillouin Zone'? Please help, since my notes and textbooks tell me hardly anything.

 

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Regarding the second part of the question, I think it is related to the fact that phonon-phonon collisions can cause energy to be transferred between phonons, which in turn affects the thermal conductivity of the crystal. When the temperature is low, there is less energy in the system, so there are fewer phonon-phonon collisions and thus lower thermal conductivity. As the temperature increases, more energy is available and thus more phonon-phonon collisions occur, leading to higher thermal conductivity. At some point, however, the phonon-phonon collisions become so frequent that they start interfering with each other, leading to a decrease in thermal conductivity again.