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. 
 A: No, these are not always the same thing.
Spin-orbit coupling in atoms
Spin-orbit coupling can be derived by reduction of the Dirac equation to non-relativistic limit, as one of several relativistic corrections. It is then generally expressed as
$$
H_{so}\propto \left[\mathbf{p}\times\nabla U(\mathbf{r})\right]\cdot\mathbf{S}
$$
where $\mathbf{S}=\hbar\mathbf{\sigma}/2$. (See this answer and references therein.)
In case of a spherically symmetric potential, such as that of an atom, this degenerates into
$$
H_{so}\propto\mathbf{L}\cdot\mathbf{S},
$$
which is the form appearing in quantum mechanics books.
Spin-orbit coupling in crystals
In solid state physics one usually works with an effective mass Hamiltonian, which is similar to that of a free particle, but which in practice is the expansion of the band energy near the band minimum,
$$
E(\mathbf{k}) = E_0 + \sum_{\alpha,\beta=x,y,z}\frac{1}{2}\frac{\partial^2E(\mathbf{k})}{\partial k_\alpha\partial k_\beta}k_\alpha k_\beta + 
\sum_{\alpha,\beta,\gamma=x,y,z}\frac{1}{6}\frac{\partial^3E(\mathbf{k})}{\partial k_\alpha\partial k_\beta\partial k_\gamma}k_\alpha k_\beta k_\gamma + ...
$$
In principle, one could derive the spin-orbit coupling in effective mass approximation from the first principles, by performing the band-structure calculations with the account of the spin-orbit interaction, mentioned above. However, it is more common to guess the form of the coupling from the time-reversal symmetry considerations, which suggest that it should appear in the third order terms in the expansion, and that only certain coefficients are not zero (and only for certain types of lattices, such as those not having inversion symmetry). The magnitude of the coupling constant is then deduced experimentally.
One well-known type of coupling deduced in this way is Dresselhaus coupling (occasionally called Dyakonov-Perel coupling, due to the associated spin relaxation mechanism) - this is typical of Zinc-Blende materials, such as the ubiquitous GaAs/GaAlAs.
Spin-orbit coupling in nanostructures
In nanostructures the energy dispersion is modified and the minimum of the electron energy may shift from the minimum of the Brillouin zone of the host crystal (i.e., from the point that is usually taken as $\mathbf{k}=0$. This results in linear terms in $\mathbf{k}$ in the dispersion relation. Emmanuel Rashba and co-workers pointed out that, due to the time-reversal symmetry, such terms must be coupled to the electron spin. The Hamiltonian is then often written in the form
$$
H_{so}=\alpha \left[\mathbf{p}\times\mathbf{E})\right]\cdot\mathbf{S},
$$
where $\mathbf{E}$ is now the electric field due to the nanostructure confining potential, whereas coefficient $\alpha$ is much bigger than the prefactor in the relativistic corrections obtained for free electrons.
Since nanostructures are often formed on the basis of a free electron gas, formed at the surface of pn-junction or metal-semiconductor junction (and then possibly using metallic gates to confine the electron motion in other directions), the resulting asymmetry is usually important only for the direction along the junction, and the Hamiltonian can be written in a simplified form:
$$
H_{R}=\alpha(\mathbf{\sigma}\times\mathbf{E})\cdot\mathbf{z}_0,
$$
where $\mathbf{z}_0$ is the unit vector across the junction.
To summarize: although the Rashba Hamiltonian has the form very similar to the spin-orbit coupling deduced from the Dirac equation, they are not the same thing. Notably, Rashba coupling is characterized by a much stronger coupling constant, making it useful in nanostructure device engineering (notably in spintronic applications).
A: If $\mathbf{E}$ is a radial function, $\mathbf{E}(\mathbf{r})=\mathbf{E}(r)$, then both equations are true.
A: The first formula is generally valid while the second applies to spherical symmetry.
