I am reading Blundell's book of Magnetism in Condensed Matter, and I couldn't convince myself why a $\mu_0$ term pops out in the potential energy expression in Stoner ferromagnetism, i.e. spontaneously spin-split bands, explanation. Below you can see the screenshot of the page that bothers me:
To give more detail, I believe infinitesimal energy change due to magnetic field is given by $-\mu.\delta B$. Magnetization, $M$, is actually magnetic moment, $\mu$, per volume. There is no external magnetic field, hence we only have molecular or mean field magnetic field, $B_{mf}$ which is assumed to be linear and given by $\lambda.M$. It doesn't explicitly have $\mu_0$. But surprisingly when integrating this infinitesimal energy change over $dM$ we see a $\mu_0$ term inside the integral. What am I missing?
