# The Landau-Zener transition & spin-flipping in an external field

Hello gentlemen,

I am trying to understand better processes that can cause "spin-flipping" to occur for magnetic moments in a bias field along the z-axis, Bz.

The application of a oscillating field in the xy-plane is well known to me. If the oscillating frequency is at the Larmor frequency, $$\omega_L = \gamma B$$ then the probability of a spin-flip will be given by $$sin(\omega_R t)$$, where $$\omega_R$$ is the Rabi frequency and t is the duration time the field is applied for.

I am trying to understand the Landau-Zener transitions. When I try to search google, I just get lots of hits for quantum qubit transitions. If I am interpreting correctly, the LZ transition is almost just a special case of the Rabi transition if the perturbation is just for a short time? The excitation is normally treated as a perturbing field in the x-axis B_x which varies linearly. The spin-flip probability is something like $$exp(-2 \pi \Gamma)$$, where $$\Gamma = \frac{B_x^2}{d\DeltaE/dt}$$. $$\DeltaE$$ is the size of the energy gap and for the magnetic moment example, it is $$2 \mu B$$. Am I right?

I am confused by this since why does the energy gap change if you're only applying a small perturbation in Bx?

Also, how can one treat a problem where the variations is in the z-axis? Something like a $$\delta_{B_Z}$$? Can this induce spin-flips? Can I use the Landau-Zener transition for this since $$\Delta_E$$ changes? What if this extra $$\delta_{B_Z}$$ causes the total BZ to go through zero?