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Homework Statement
Consider a solid of compressibility ##\kappa##. Assume that the atoms in this solid are arranged on a regular cubic lattice, the distance between their nearest neighbors being ##a##. Assume further that a restoring force ##-k_0 \Delta a## acts on a given atom when it is displaced by a distance ##\Delta a## from its nearest neighbor. Use simple reasoning to find an approximate relation between the spring constant ##k_0## and the compressibility ##\kappa## of this solid.
Homework Equations
##\kappa = \frac{1}{V}\frac{\partial V}{\partial p}|_{T}##.
The Attempt at a Solution
Consider a cubic lattice of ##N+1## atoms on each edge. Take ##N \gg 1## so that ##N+1 \approx N##. The area of any given face of the lattice is ##A = N^2 a^2##. Imagine now that the face is pushed inward by the amount ##\Delta a## where ##\frac{\Delta a}{a} \ll 1##. Then there is an increase in the total force exerted on this face due to the restoring force ##F = -k_0 \Delta a## on each of the ##(N+1)^2## atoms in this face; the increase in total force on this face will be given in magnitude by ##\Delta F = (N+1)^2 \Delta a k_0 \approx N^2 \Delta a k_0##.
Then ##\kappa = \frac{1}{V}\frac{\partial V}{\partial p}|_T = \frac{1}{N a}\frac{\partial a}{\partial p}|_T \approx \frac{1}{a}\frac{\Delta a}{\Delta p} = \frac{1}{a}\frac{N^2 a^2 \Delta a}{N^2 \Delta a k_0} = \frac{a}{N k_0}##.
According to the book the correct answer is ##\kappa = \frac{a}{k_0}## which is not what I got. However I am not sure of how to fix my solution. I could I suppose take a parallelipiped with sides of length ##Na## whose faces are to be pushed inwards, and sides of length ##a## which are parallel to the force applied to push the aforementioned faces inwards. However this looks to be a very contrived escape in fixing my solution on top of the assumptions I already made that make my solution above look quite non-rigorous. For example I took the solid to be a cubic lattice as opposed to any crystal lattice and I also took ##N \gg 1## to make ##(N+1)^2 \approx N^2##, neither of which are assumptions made in the problem statement. Could anyone help me in fixing my solution. Also, could any comment on whether or not I could make the calculation more rigorous? Any hints on how to do so? Thanks!
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