PHZ3400-09 Problem Set 5: Difference between revisions

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(New page: '''Problem 1''' a. Consider a chain of atoms with alternating masses <math>m_1\;</math> and <math>m_1\;</math>, connected with elastic springs with constant <math>K\;</math>. Derive the d...)
 
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'''Problem 1'''
'''Diatomic harmonic chain'''


a. Consider a chain of atoms with alternating masses <math>m_1\;</math> and <math>m_1\;</math>, connected with elastic springs with constant <math>K\;</math>. Derive the dispersion relation <math>\omega^{\alpha} (k)\;</math> for this chain, with the index <math>\alpha = 1,2\;</math> coresponding to the acoustic and the optical branch, respectively.


b. Determine the speed of sound for this chain. Is there sound corresponding to the optical branch?
1. Consider a chain of atoms with alternating masses <math>m_1\;</math> and <math>m_2\;</math>, connected with elastic springs with constant <math>K\;</math>, moving only in the x-direction. Derive the dispersion relation <math>\omega^{\alpha} (k)\;</math> for this chain, with the index <math>\alpha = 1,2\;</math> corresponding to the acoustic and the optical branch, respectively.


c. Sketch the motion of the atoms corresponding to the edge of the Brillouin zone, both for the optical and the acoustic branch.


d. Determine the Debye temperature for this system, and determine the form of the specific heat <math>c_V (T)</math> in the limits of high and low temperatures.
2. Determine the speed of sound for this chain. What is the lowest frequency of long-wavelength sound corresponding to the optical branch?




'''Problem 2'''
3. Sketch the motion of the atoms corresponding to the edge of the Brillouin zone, both for the optical and the acoustic branch.


a. Consider a two-dimensional square lattice of atoms interacting with springs. Determine the the dispersion relation <math>\omega^{\alpha} (k_x, k_y)\;</math> for this system.


b. Determine the speed of sound, the Debye temperature, and the low-temperature specific heat for this system.  
4. Determine the Debye temperature for this system, and determine the form of the specific heat <math>c_V (T)</math> in the limits of high and low temperatures.  


c. Consider low temperatures (<math> T \ll T_D\;</math>) and determine the wavelength of the most abundant phonons <math>\lambda_{max}</math> (Hint: note the analogy with Wien's Law!)
 
5. Consider low temperatures (<math> T \ll T_D\;</math>) and determine the wavelength of the most abundant phonons <math>\lambda_{max}</math> (Hint: note the analogy with Wien's Law!)

Latest revision as of 19:26, 24 February 2009

Diatomic harmonic chain


1. Consider a chain of atoms with alternating masses and , connected with elastic springs with constant , moving only in the x-direction. Derive the dispersion relation for this chain, with the index corresponding to the acoustic and the optical branch, respectively.


2. Determine the speed of sound for this chain. What is the lowest frequency of long-wavelength sound corresponding to the optical branch?


3. Sketch the motion of the atoms corresponding to the edge of the Brillouin zone, both for the optical and the acoustic branch.


4. Determine the Debye temperature for this system, and determine the form of the specific heat in the limits of high and low temperatures.


5. Consider low temperatures () and determine the wavelength of the most abundant phonons (Hint: note the analogy with Wien's Law!)

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