# Stern-Gerlach and Hund's second rule

According to Hund's second rule, the spin tends to be maximal. That would, in my understanding, imply that, regarding the Stern-Gerlach experiment, the important electron in a silver atom has spin 1/2 (or "up"). This should hold for all atoms.

Every atom would have the same spin magnetic moment (smm) - because the smm points in direction of the spin - and thus be affected by the B-field just like any other atom.

But, if every electron has the same smm w.r.t. the magnetic field, why do we get two "blobs" on the screen and not just one "blob" above or below the z-axis?

Where is the error in my thought process?

When we say that the spin of a silver atom (in a magnetic field) is $+1/2$ or $-1/2$ we mean its component in the direction of the magnetic field (referred to as $S_z$) is $+1/2$ or $-1/2$. The magnitude of the spin is the same in both cases, it's just the direction that's different.

Hund's rule just tells you what the magnitude of the total spin is, and doesn't say anything about the direction that spin is pointing. For example with two unpaired electrons Hund's rule tells us the total spin will be $S = 1$. However put that atom in a magnetic field and you can have $S_z = 1$, $0$ or $-1$.

The two blobs in the Stern-Gerlach experiment correspond to the electrons with $S_z = +1/2$ and $-1/2$, but all the electrons have the same total spin $S = 1/2$. The two unpaired electron system with $S = 1$ would give us three blobs corresponding to $S_z = 1$, $0$ and $-1$.

• So the spin points in a certain direction, it could be up or down, but we don't know it yet. And only after putting the atom in a magnetic field we know whether it's up or down? And, as for Stern-Gerlach: Why do some atoms have total spin 1/2 and others -1/2 w.r.t. B_z? Is that just a statistical distribution? – Jo Be Oct 2 '13 at 15:19
• Yes, the spins in the vapour of At are randomly distributed. In the magnetic field roughly equal numbers end up pointing up and down. – John Rennie Oct 2 '13 at 15:20
• Let's say we look at Nitrogen. It's total spin (w/out nucleus) should be 3/2. But S_z could be +-3/2 or +-1/2? So the total spin is the sum of the absolute values of the spin corresponding to each triplet (n,l,m_l)? – Jo Be Oct 2 '13 at 15:45
• @JoBe: Nitrogen has a pair of electrons in the $1s$ orbital (total spin zero), a pair of electrons in the $2s$ orbital (total spin zero) and three electrons in the 2p orbitals. Hund's rule tells us we get one electron in each of the $2p_x$, $2p_y$ and $2p_z$ orbitals, and they align to give total spin $S = 3/2$. Put the nitrogen atom in a magnetic field and the component of spin along the field can be $3/2$, $1/2$, $-1/2$ or $-3/2$. A Steen-Gerlach experiment with nitrogen atoms would give four blobs. – John Rennie Oct 2 '13 at 16:46
• @JoBe: PS I think it's Hund's first rule not the second that tells us the total spin. IIRC the second rule tells you how to calculate the orbital angular momentum not the spin. – John Rennie Oct 2 '13 at 17:09

Although an electron is referred to as a spin 1/2 particle, it doesn't mean it is in a spin 1/2 state (confused yet?). A spin 1/2 particle has two states it can be in. It can be +1/2 in which case it goes up Stern-Gerlach experiment, or it can be in the -1/2 state in which case it goes down.

If there are two unpaired electrons in a molecule, then it is called a spin 1 molecule and has three states (+1, 0 and -1). If the 2 electrons are both spin +1/2, the molecule will go up, if 1 electron is spin +1/2 and the other spin -1/2, the molecule will go straight. Finally, if both electrons are spin -1/2, the molecule will go down in a Stern-Gerlach experiment.

• But if we have two unpaired electrons, how can one have spin 1/2 and the other one -1/2? Doesn't that violate Hund's rule, i.e. the total spin is not maximal? – Jo Be Oct 2 '13 at 15:30
• Due to the configuration of the molecule, you may have 2 electrons at different energy levels. These electrons cannot pair up and both will contribute their random direction of spin to the overall molecule. – AnimatedPhysics Oct 2 '13 at 15:59
• I guess my error was in thinking the total spin is the sum of the the spin quantum numbers of each electron, whereas it's really the sum of the spins. And the spin of an electron is, as you stated, 1/2. I confused spin with state. – Jo Be Oct 2 '13 at 16:37