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Unformatted text preview: Homework # 5 - Solution Problem #1 The dispersion relation for the 1D monoatomic lattice is 4 sin sin 2 2 m C qa qa M ϖ ϖ = = , where ϖ m is the maximum frequency. The density of vibrational modes is given by 1 ( ) / L D d dq ϖ π ϖ = . By differentiating we obtain 1 2 2 2 2 2 1 2 1 1 2 1 ( ) cos 2 2 1 sin 1 2 m m m m m L a qa L L N D a a qa ϖ ϖ π π ϖ ϖ π π ϖ ϖ ϖ ϖ- & ¡ = = = = ¢ £ ¤ ¥- & ¡-- ¢ £ ¤ ¥ . The density of states has singularity at ϖ = ϖ m . Problem #2 First, we find the density of modes for the 2D solid. We need to calculate the number of modes lying inside the ring of radius q shown in the figure. Each point in the figure determines one vibration mode. The area of this ring is 2 π qdq , and since the volume per point is (2 π / L ) 2 , it follows that the number we seek is 2 2 2 2 2 qdq L q dq L π π π = & ¡ ¢ £ ¤ ¥ . By definition of the density of modes, this is equal to D ( ϖ ) d ϖ . We arrive at 2 1 ( ) 2 / L q D d dq ϖ π ϖ = . ....
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This note was uploaded on 09/21/2011 for the course PHYSICS 101 taught by Professor Wormer during the Spring '08 term at Aarhus Universitet.
- Spring '08