Solid State Physics - Modes of Oscillation of Atom

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SUMMARY

The discussion focuses on constructing a 3x3 matrix from a 12-element series in the context of solid state physics, specifically regarding the modes of oscillation of atoms. The user seeks clarification on setting the determinant of the matrix to zero, which results in three solutions, two of which are degenerate. Additionally, the user inquires about the meaning of the unit matrix and the diadic formed from unit vectors as referenced in the equations provided.

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  • Understanding of linear algebra concepts, particularly matrix construction and determinants.
  • Familiarity with solid state physics principles, especially atomic interactions and degrees of freedom.
  • Knowledge of mathematical notation used in physics, such as Dirac delta functions.
  • Basic understanding of unit vectors and their applications in physics.
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  • Study linear algebra, focusing on matrix operations and determinants.
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Students and researchers in solid state physics, particularly those dealing with atomic models and matrix formulations in physical systems.

MightyDogg
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Homework Statement


upload_2016-5-10_14-9-42.png
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My question is more about the math of the problem. For equation 22.97, how do I construct a 3x3 matrix from a 12 element series? After constructing the matrix should I set the determinant equal to 0 where I will find the determinant has 3 solutions (2 being degenerate)? Also, under 22.98, what does "1 is the unit matrix ((1)uv=diracdelta_uv), and RR is the diadic formed from the unit vectors R=R/R..." mean?
I've never taken linear algebra before so I am unsure how to go from series to matrix, but I know the basics of how to solve them. If it would be easier feel free to send me a link on relevant information regarding matrices.

I don't need much help, but I would appreciate a push in the right direction.

Homework Equations

The Attempt at a Solution

 

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Your unit cell is monoatomic, so this atom has only 3 degrees of freedom, hence you get a 3x3 matrix. However, this atom interacts with 12 neighbours. This determines the form of the matrix elements, but not its dimension.
 

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