In page 188 of Christopher Foot's book on atomic physics, he talks about the Doppler cooling limit, and how it comes from a random walk caused by spontaneous emission. He claims that each spontaneous photon gives a recoil kick in the z-direction of $\hbar k$, where $k$ is the wave number of the laser light.
I have trouble proving that is true. I know that the momentum from a photon spontaneously decaying should be $\hbar \omega_0 /c$, where $\omega_0$ is the energy level difference. I also know that because the atom is moving we might have to make some adjustment to the momentum in the lab frame. Can someone show me how I can make this adjustment to get to the result of $\hbar k$? Or am I missing something else?