# Spin Orbit Coupling Hamiltonians

I am really struggling with something fundamental.

I keep coming across two versions of the hamiltonian for spin orbit coupling:

$H_{soc}=\frac{\mu_B}{2c^2}(v \times E) \cdot \sigma$

$\mu_B =$ bohr magnetron

$v =$ velocity

$E =$ Electric Field

$\sigma =$ pauli matrices

and

$H_{soc} = \alpha L\cdot S$

$\alpha =$ constant

$L =$ Orbital Angular Momentum

$S =$ Spin angular momentum

Are these equivalent? If not what situations are they referring to.