Explanation of de Haas-van Alphen effect from Kittel, Chapter 9 I am reading de Haas van Alphen effect from Kittel's Introduction to Solid State Physics.
In presence of a magnetic field $B$, the total energy of $N$ free electrons, in $s+1$ Landau levels is given by $$U=\frac{1}{2}\rho B\hslash\omega_c s^2+(N-\rho Bs)\hslash\omega_c\left(s+\frac{1}{2}\right)$$ where $\omega_c=eB/m^*c$ is the cyclotron frequency, $D$ is the degeneracy of each Landau level given by $D=\rho B$ and $\rho=eL^2/2\pi\hslash c$ is a geometric factor.
We have assumed that the lowest $s$ levels are completely filled and the $(s+1)$-th Landau level is partly filled. See equations $(35)$ and $(36)$ of Chapter 9, Fermi Surfaces and Metals.
Simplifying the expression above, we get,
$$U=\frac{eB\hslash}{2m^*c}\left[N+\left(2N-\rho B\right)s-\rho Bs^2\right].$$ It can be shown that $U$ has local maxima at $$B=\frac{N(s+1/2)}{\rho s(s+1)}, ~{\rm or}~\frac{1}{B}=\frac{\rho s(s+1)}{N(s+1/2)}.$$
From there, how can we show that $U$ is periodic in $1/B$ and how do we find the periodicity $\Delta(1/B)$?
 A: The crucial point to notice here is that the degeneracy of each Landau level is a function of the external magnetic field $B$. Let me ignore the electronic spin for the sake of simplicity. More precisely, from the solution of the Schrodinger equation, one obtains that the number of electrons that can be accomodated in every Landau level is $N B/B_0 \equiv g$, where $B_0 = 2\pi \hbar n/e$, $n=N/L^2$ being the electronic density.
So how many Landau levels do the $N$ electrons fill? Again neglecting the spin, you put $g$ electrons per level up to some level labeled by $s$, and the remaining $N-sg$ partially occupy the level labeled by s+1. So you can see that $s$ is given by the first integer which is smaller than $N/g$: i.e. $s=\mathrm{floor}(N/g)=\mathrm{floor}(B_0/B)$. The function $s(B)$ is flat in intervals of length $B_0$ and jumps from integer to integer every $B_0$.
This is precisely the physical mechanism responsible for the De Haas - Van Alphen effect.
About the "periodicity" of the magnetization vs. $1/B$ (which is by the way not a perfect periodicity), from the arguments above you can immediately guess that the "period" is $\sim 1/B_0$. You can for instance plot the magnetization $M \propto \partial U/\partial B \propto [N + 2Ns - 2\rho s(1+s)B ]$ as function of $1/B$: you will find hyperbolas defined in intervals of $1/B_0$ with discontinuities once every $1/B_0$, and in every interval $s$ has a different value, so the function changes a little from interval to interval.
Hope this helps, sorry for being a little sloppy, but I wanted to avoid lots of technicalities which in this case might confuse rather than explain. If you want I can detail my answer more, let me know!
