Density

Water crystal (ice).

A basic formula for density of a solid is to weigh its mass m for a given reference volume V and then to divide the mass with the reference volume to yield density ρ. Because a crystal is constructible from many unit cells due to periodic arrangements of atoms, we can use the volume of a unit cell as the reference volume.

There are different unit cells for different crystals, depending on how the crystal's atoms organize themselves spatially. The number of atoms per unit cell, thus the mass of the unit cell, and the unit cell volume are difficult to predict. Experimentally, they are determined mainly from x-ray diffraction, which is also not straightforward.

Let us consider a simpler situation. Suppose a solid has two atoms, A and B, i.e., an alloy AB, then the density can be thought of as equal to

ρ = (m_A + m_B) ⁄(V_A + V_B).

It assumes that we know the unit cell volume V_i for atom i and the number of atoms i within each unit cell volume. It further assumes that the alloy AB adds the individual volumes from atoms A and atoms B to produce the alloy. This is not always the case, which is often surprising for many students. If atom A is much smaller than atom B, then atom A can occupy a tiny space–interstitial site–available between nearest neighboring atoms B. In this case, the density of alloy AB is better approximated by

ρ = (m_A + m_B) ⁄V_B

since atoms A only increases the unit cell's mass. Unfortunately, the formula

ρ = (m_A + m_B) ⁄(V_A + V_B)

and its equivalent expression

ρ = (C_A ⁄ρ_A + C_B ⁄ρ_B)^-1,

where C_A = m_A ⁄(m_A + m_B) and C_B = m_B ⁄(m_A + m_B), are often used indiscriminately. The unit cell volume additive property should work well only for atoms A and B that have the same crystal structures.

The formula

 ρ = (m_A + m_B) ⁄(V_A + V_B),

however, can work well when A and B refer not to atoms, but to phases. Two phases can coexist and each contributes its mass and volume. Both mass and volume are thus additive. The equivalent expression

ρ = (C_A ⁄ρ_A + C_B ⁄ρ_B)^-1

then requires us to determine the density of each phase separately. C_A and C_B are now determined by the lever rule.