# For the transition metals, how does counting the number of up-spins and down-spins still give you a non-integer magnetic moment?

The transition metals like Fe, Co and Ni have magnetic moments of 2.2, 1.7 and 0.6 Bohr magnetons, respectively.

The band theory says that you get this when you calculate the density-of-states of the 3d band and you subtract the number of spin-up electrons from spin-down electrons.

How does this subtraction nonetheless net you a non-integer value for the magnetic moments? Shouldn't there be nothing but integer values of available states?

The references where I found that the magnetic moment can be calculated by the difference of spin-up and spin-down electrons in the 3d band of the density of states, are:

• Stöhr, J. and H. C. Siegmann. “Magnetism: From Fundamentals to Nanoscale Dynamics” (2006) <WCat>.
• Chikazumi, S. “Physics of Magnetism” (1964) <WCat>.

The magnitude $M$ of the magnetic moments in a real material (like a simple or transition metal) is usually obtained by measuring or computing the static magnetic susceptibility and comparing the result to the Curie susceptibility $\chi_C$ (which describes uncorrelated magnetic moments): $$\chi_{C} = \frac{M^2}{3 k_B T} \quad ; \quad M^2 = (g_J \mu_B)^2 \, J(J+1) ~,$$ where $T$ denotes the temperature, $g_J$ is the gyromagnetic ratio (Landé $g$-factor) which depends on the total angular momentum $J$, $\mu_B$ is the Bohr magneton, and $k_B$ is the Boltzmann constant.
Therefore, $$\frac{M}{\mu_B} = g_J \sqrt{J(J+1)}$$ which is not guaranteed to be an integer, regarding the complicated dependence of the $g$-factor on $J$ and the square root. Even in the simple metals, the lattice structure affects $J$ and the $g$-factor in a non-trivial manner. The simple counting procedure (subtracting the number of spin-up electrons from spin-down electrons) only works as a first (crude) estimate to begin with; actually, in a paramagnetic metallic system (away from magnetic order), one would find that number of up- and down-spins are the same.
For a more detailed discussion, I suggest consulting the following lecture note (esp. $\S$ 2.2 and Tables 1 & 2):