# Total magnetic moment

The total magnetic moment is a combination of spin (S) and orbital magnetic moment (L) which is given by

J=L+S

If there are n number of electrons then the spin angular momentum is given by

S=s1+s2+...+sn

The orbital angular momentum is given by

L=l1+l2+...+ln

The resultant total angular momentum J is given by

J=L+S

The orbital magnetic moment is given by

µLBL

and the spin magnetic moment is

µS=2µBS

The resultant magnetic moment is given by

µRLS

B(L+2S)

=gJµB

where g is given by

g={1+J(J+1)+S(S+1)-L(L+1)}/{2J(J+1)}

When S=0

J=L    and g=1 which gives the orbital contribution

When L=0

J=S and g=2 which gives the spin contribution

Hund's Rule in magnetic moment

In an atom, the set of values for S and L are governed by Hund's rule given according to the following

1) The spin of the electron follows Pauli's principle. Electrons fill vacant shells first. Each spin  will contribute to 1 Bohr magneton. Therefore 5 electrons in 3d shell will give raise to 5µB

After all the electrons have occupied all the vacant shells the extra electron will pair up with the available electrons with opposite spins. The opposite spins will give raise to negative values. The resultant spin magnetic moment in such case with 6 electrons in the d shell will be 4µB since 5µB in the positive and 1µB in the negative direction will give a net value of 4µB. Completely filled d orbital will give a net moment of zero.

2) The orbital magnetic moment arises from the electrons occupying the shells according to the values given by

ml,ml-1,....mo

For instance for 5 3d electrons

mL=+2 +1 + 0 -1 -2  = 0

for 7 3d electrons

mL=+2 +1 +0 -1 -2 +2 +1 = +3

3) The total quantum number is given by

J=L+S

If the shell is more than half filled then

J=L+S

If the shell is less than half filled then

J=L - S

If the shell is half filled

J=S

If the shell is completely filled

J=0