Free electron gas in magnetic field I have a question regarding the calculation of "spin up" number of particles in a free electron gas, when placed in a uniform magnetic field $\textbf B$. In my lecture notes it's said that the spin up number of particles equals: $N_\uparrow=\int g(\epsilon-\textbf B\mu_m)f(\epsilon)\space d\epsilon $, where $g$ is the states density function $g=\frac{dN}{d\epsilon}$ and $f$ is the Fermi-Dirac distribution function $f(\epsilon)=\frac{1}{e^{(\epsilon-\mu)/k T}+1}$. 
My question: If the density of states function changes: $g(\epsilon)\rightarrow g(\epsilon-\textbf B \mu_m)$, why doesn't the F-D function change as well, so that $f(\epsilon)\rightarrow f(\epsilon-\textbf B \mu_m)$ ?
This is connected to the calculation of the Pauli paramagnetism.
 A: Let $\varepsilon$ be the total energy of an electron. Then you can divide it into the energy of the spatial part and the energy of the spin part as follows
$$\varepsilon=\varepsilon_{\boldsymbol{k}}\pm\mu_{\rm B}\cdot\boldsymbol{B}$$
Now comes the key part: the Fermi-Dirac distribution is related to the total energy of the state $\varepsilon$, and so you should use $f\left(\varepsilon\right)$. On the other hand, the density of states is related to the $\boldsymbol{k}$ number of states, so this time the correct expression you need to use is $g\left(\varepsilon_{\boldsymbol{k}}\right)=g\left(\varepsilon\mp\mu_{\rm B}\cdot\boldsymbol{B}\right)$.
A: In this picture of the Pauli paramagnetism, all energy states with "spin-up" electrons are shifted by an energy $$\Delta \epsilon= B \mu_m$$ Thus the density of states of the "spin-up" electrons as a function of energy $g(\epsilon)$ is shifted to higher energies by $\Delta \epsilon$ while the density of states of the "spin-down" electrons is shifted down by $-\Delta \epsilon$. The chemical potential $\mu$, which  is the only system parameter in the FD distribution function of the electron, gas stays the same. Thus explaining the paramagnetism by a reduction in the total number of "spin-up" electrons and an increase in total number of the "spin-down" electrons in the electron gas in the magnetic field.   
