a fluid which shows its ability to change its volume under its pressure. Selected values A material with a bulk modulus of 35 GPa loses one percent of its volume when subjected to an external pressure of 0.35 GPa (~). Microscopic origin Interatomic potential and linear elasticity 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. 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 a0, where the total force is zero: 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 a, and the equilibrium distance is a0. Its potential energy-interatomic distance relationship has similar form as the two atoms case, which reaches minimal at a0, The Taylor expansion for this is: At equilibrium, the first derivative is 0, so the dominate term is the quadratic one. When displacement is small, the higher order terms should be omitted. The expression becomes: Which is clearly linear elasticity. Note that the derivation is done considering two neighboring atoms, so the Hook's coefficient is: This form can be easily extended to 3-D case, with volume per atom(Ω) in place of interatomic distance. Relationship with atomic radius As derived above, the bulk modulus is directly related the interatomic potential and volume per atoms. We can further evaluate the interatomic potential to connect K with other properties. Usually, the interatomic potential can be expressed as a function of distance that has two terms, one term for attraction and another term for repulsion. Where A > 0 represents the attraction term and B > 0 represents repulsion. n and m are usually integral, and m is usually larger than n, which represents short range nature of repulsion. At equilibrium position, u is at its minimal, so