Relation between Coefficients of Expansion

Because $\alpha$, $\beta$ and $\gamma$ all describe the same underlying phenomenon (the same linear stretching of interatomic spacing) viewed along one, two, or three dimensions respectively, they are not independent -- they are fixed multiples of each other.

Relation between $\beta$ and $\alpha$: consider a square plate of side $l_0$ at 0 °C, becoming $l_T$ at $T\,°\text{C}$, so $l_T = l_0(1+\alpha T)$ from Eq. (7.11). Its area at 0 °C is $A_0 = l_0^2$, and at $T\,°\text{C}$, $A_T = l_T^2 = l_0^2(1+\alpha T)^2 = A_0(1+\alpha T)^2$ --- (7.19). But also, directly from the areal-expansion definition, $A_T = A_0(1+\beta T)$ --- (7.20). Equating the two and expanding: $1 + 2\alpha T + \alpha^2 T^2 = 1 + \beta T$. Since $\alpha$ is very small, the $\alpha^2T^2$ term is negligible, leaving

This holds generally, since any flat solid can be thought of as built from many small squares.

Relation between $\gamma$ and $\alpha$: by an identical argument for a cube of side $l_0$, volume at 0 °C is $V_0 = l_0^3$, and at $T\,°\text{C}$, $V_T = l_T^3 = l_0^3(1+\alpha T)^3 = V_0(1+\alpha T)^3$ --- (7.22); also directly, $V_T = V_0(1+\gamma T)$ --- (7.23). Equating and expanding, and again neglecting the tiny higher-power-of-$\alpha$ terms, gives …