Editing Bulk modulus (section)

== Microscopic origin ==
=== Interatomic potential and linear elasticity ===
[[File:Interatomic potentual.png|alt=The left one shows the interatomic potential and equilibrium position, while the right one shows the force|thumb|440x440px|Interatomic potential and force]]
Since linear elasticity is a direct result of interatomic interaction, it is related to the extension/compression of bonds. It can then be derived from the [[interatomic potential]] for crystalline materials.<ref>{{Cite book|title=Mechanical Behavior of Materials|last=H.|first=Courtney, Thomas|date=2013|publisher=McGraw Hill Education (India)|isbn=978-1259027512|edition=2nd ed. Reimp|location=New Delhi| oclc=929663641}}</ref> First, let us examine the potential energy of two interacting atoms. Starting from very far points, they will feel an attraction towards each other. As they approach each other, their potential energy will decrease. On the other hand, when two atoms are very close to each other, their total energy will be very high due to repulsive interaction. Together, these potentials guarantee an interatomic distance that achieves a minimal energy state. This occurs at some distance r<sub>0</sub>, where the total force is zero:

:<math>F=-{\partial U \over \partial r}=0</math>

Where U is interatomic potential and r is the interatomic distance. This means the atoms are in equilibrium.

To extend the two atoms approach into solid, consider a simple model, say, a 1-D array of one element with interatomic distance of r, and the equilibrium distance is ''r''<sub>0</sub>. Its potential energy-interatomic distance relationship has similar form as the two atoms case, which reaches minimal at ''r''<sub>0</sub>, The Taylor expansion for this is:

:<math>u(r)=u(r_0)+ \left({\partial u \over \partial r} \right )_{r=r_0}(r-r_0)+{1 \over 2} \left ({\partial^2\over\partial r^2}u \right )_{r=r_0}(r-r_0)^2+O \left ((r-r_0)^3 \right )</math>

At equilibrium, the first derivative is 0, so the dominant term is the quadratic one. When displacement is small, the higher order terms should be omitted. The expression becomes:

:<math>u(r)=u(r_0)+{1 \over 2} \left ({\partial^2\over\partial r^2}u \right )_{r=r_0} (r-r_0)^2</math>

:<math>F(a)=-{\partial u \over \partial r}= \left ({\partial^2\over\partial r^2}u \right )_{r=r_0}(r-r_0)</math>

Which is clearly linear elasticity.

Note that the derivation is done considering two neighboring atoms, so the Hook's coefficient is:

:<math>K=r_0{dF \over dr}=r_0 \left ({\partial^2\over\partial r^2}u \right )_{r=r_0}</math>

This form can be easily extended to 3-D case, with volume per atom(Ω) in place of interatomic distance.

:<math>K=\Omega_0 \left ({\partial^2\over\partial \Omega^2}u \right )_{\Omega=\Omega_0}</math>