White Dwarfs Mass-radius relation for different chemical compositions

The standard non-relativistic mass-radius relation of low mass white dwarfs is this : \begin{equation}\tag{1} R(M) = \frac{\mathcal{C}}{M^{\frac{1}{3}}}, \end{equation} where $\mathcal{C}$ is a constant that may (AFAIK) depend on the chemical composition of the star (He, C, N, O, ...).

I would like to know that constant (empirical or theoretical value) for several chemical compositions (especially the He, C, N, O white dwarfs), so I could plot the curves on a $R-M$ diagram.

The relationship you are looking for is $$\left(\frac{R}{R_{\odot}}\right) = 0.013\left(\frac{\mu_e}{2}\right)^{-5/3}\left(\frac{M}{M_{\odot}}\right)^{-1/3},$$ where $\mu_e$ is the number of electrons per mass unit in the gas, which of course is composition dependent.
For "standard" white dwarf stars made of carbon or oxygen, then $\mu_e=2$ (e.g. an ionised carbon atom has 12 mass units and 6 electrons). For He $\mu_e=2$, for H $\mu_e=1$ and if such a thing as an iron white dwarf existed, then $\mu_e =56/26$ and it would be smaller than a "standard" white dwarf of the same mass, because there are fewer electrons to provide the degeneracy pressure.