Zeeman - Paschen-Back effect I've recently came to know that there is two effects of an atom in external magnetic field (assume the magnetic field is constant in time and direction).
One of them is the Zeeman effect the second is Paschen-Back effect. The effects differ only by how strong the magnetic field is.   
So my question is how to distinguish between them? On what basis we identify a strong or weak magnetic field? Mathematically , How do I know if I have some $B_0$ is considered Weak or Strong field?
Thanks.
 A: We say strong or weak magnetic field in the sense of it's effect in the coupling of the orbital and spin angular momenta. 
If the applied magnetic field strength is low, we mean that it leaves the orbital and spin angular momentum in the coupled state (i.e., the $L-S$ coupling dominates than the interaction of the magnetic field with the magnetic dipole), we state that the field strength is weak and in such cases, the splitting of spectral lines is explained using Zeeman effect.  
But, when the applied magnetic field is so intense, such that it breaks down the coupling between the spin and orbital angular momentum, the splitting of spectral lines is described using Paschen-Back effect.  

Update: 

Zeeman effect: 
Here the spin-orbit interaction dominates over the effect of the external magnetic field, $\vec {L}$ and $\vec {S}$ are not separately conserved, only the total angular momentum $\vec {J}=\vec {L}+\vec {S}$ is.  
The Hamiltonian of the system is :  
$$H=H_0+V_M$$
 where $H_0$ is the unperturbed Hamiltonian and $V_M$ is the perturbed hamiltonain due to the presence of a magnetic fielf $\vec{B}$.   
$$V_M=-\vec{\mu}\cdot\vec{B}$$  
which represents the interaction energy of the magnetic dipole. Here we can approximate $\vec{\mu}$ is solely due to the spin and orbital angular moment:
$$\vec{\mu}\approx-\frac{\mu_Bg\vec{J}}{\bar{h}}=-\frac{\mu_B(g_l\vec{L}+g_s\vec{S})}{\bar{h}}$$
In the case of $L-S$ coupling, one can sum over all electrons in the atom:  
$$g\vec{J}=\langle(g_l\vec{L}+g_s\vec{S})\rangle$$  
In the case of Zeeman effect, the perturbation $V_M$ is very less. In such weak magnetic fields, the spin and orbital angular momentum vectors precesses around the fixed (since it is  conserved) total angular momentum vector $\vec{J}$. In such case, we can represent the time-averaged orbital and spin angular momenta as a projection onto the direction of $\vec{J}$: 
\begin{align}
\vec{S}_{avg}&=\frac{\vec{S}\cdot\vec{J}}{J^2}  & \vec{L}_{avg}&=\frac{\vec{L}\cdot\vec{J}}{J^2}
\end{align}
Hence  
$$\langle V_M\rangle=\frac{\mu_B}{\bar{h}}\vec{J}\left(g_l\frac{\vec{L}\cdot\vec{J}}{J^2}+g_s\frac{\vec{L}\cdot\vec{J}}{J^2}\right)$$ 
After making some manipulations, we finally gt  
$$V_M=\mu_B B_{ext}m_jg_j$$  
where $g_j$ is the Lande $g$-factor given by 
$$g_j=\left[1+(g_s-1)\frac{j(j+1)-l(l+1)+s(s+1)}{2j(j+1)}\right]$$  
$m_j$ is the $z$- component (assuming the external magnetic field points in the $z$-direction) of total angular momentum, $\mu_B$ is the Bohr magneton and $B_{ext}$ is the strength of the external magnetic field.   
The Zeeman first order correction to the energy is  
$$\Delta E_z=\mu_Bg_jB_{ext}m_j$$  
From this, the magnetic field strength can be written as: 
$$ B_{ext}=\frac{\Delta E_z}{\mu_Bg_jm_j}$$   
i.e., if we know the energy involved in the transition giving rise to Zeeman effect, we can find out the external magnetic field required for that transition (even though it is unusual to write the equation as above).    
Paschen_Back effect: 
Here the splitting of atomic energy levels take place in a strong magnetic field such that the $L-S$ coupling is broken. Hence the effect is a strong field limit to the Zeeman effect. When the magnetic-field perturbation significantly exceeds the spin-orbit interaction, one can safely assume $[H_{0},S]=0$. In such a case,  
$$\Delta E_z=E_z-E_0=B_z\mu_B(m_l+g_sm_s)$$  
where $E_0$ corresponds to the unperturbed part of the energy eigen value of the Hamiltonian (or the ground state if you prefer). This gives  
$$B_z=\frac{\Delta E_z}{\mu_B(m_l+g_sm_s)}$$
