Stern-Gerlach's exercise

[fluidistic]

Homework Statement

In an experiment of Stern-Gerlach, a beam of hydrogen atoms exit an oven with a temperature of 500 K and goes through a 0.5m region in which there's a gradiant of magnetic field of $20 Tm^{-1}$ whose direction is perpendicular to the beam. Calculate the separation between the beams when they leave the magnetic field.
Why is it valid to assume that the hidrogen atoms are in ground state?

Homework Equations

$F= -\mu \cdot \text{grad}B$.
$E=\frac{3}{2}k_{B}T$.

The Attempt at a Solution

I don't understand the formula given. How can you take the gradiant of a field rather than a scalar? Do they mean the divergence?
I calculated the thermal energy for an atom and it's about $0.065 eV$. I know that in the ground state, the absolute value of the energy of the atom is about 13.6 eV. But I have no idea why a very small thermal energy means that the atoms are in ground state. Maybe because the atom requires about 4 eV to get into the excited state? That must be this.
I'd like some tip on how to solve the problem. Should I include the spin somehwere?

 

[fluidistic]

Ok so I checked out in http://hyperphysics.phy-astr.gsu.edu/hbase/spin.html.
I used the formula (and understand it fully I think) that gives z and reached 0.25 mm. So overall the distance between the 2 beams is about 0.5 mm. It seems a bit small but I don't really have a good intuition on this. I know that the spin isn't so easy to detect so this might be right. After all 0.5 m for the applied field region is very small and 20 T is somehow near of the strongest magnetic field one can reach in a lab I think. So 0.5 mm could be right, could someone verify this?