Adding angular momentum of 2 electrons

[Andrusko]

1. The problem statement, all variables and given/known data

Use the rules for addition of angular momentum vectors to show that there are 12 possible values of (s',l',j') for two electrons with l1=1, l2=3, s1=1/2, s2=1/2.

2. Relevant equations

Total Spin S'=S1 + S2
Total orbital angular momentum L'=L1 + L2

Total angular momentum of system J'=L'+S'

3. The attempt at a solution

the spin vector S' can have eigenvalues s'=0,1 (I don't know why it just written in the textbook)

the L' vector apparentely has the rule that l' = abs(l1-l2) .... l1+l2

so that gives l' = 2, 3, 4

and apparentely j'= l'+1/2,l'-1/2

So two j' for each l' means a total of 6 states. Where's the other 6?

My only thought is that you have to somehow take m_{l} quantum numbers into account but I'm totally confused about that because I don't understand anything about m_{l} quantum numbers.

[vela]

1. The problem statement, all variables and given/known data

Use the rules for addition of angular momentum vectors to show that there are 12 possible values of (s',l',j') for two electrons with l1=1, l2=3, s1=1/2, s2=1/2.

2. Relevant equations

Total Spin S'=S1 + S2
Total orbital angular momentum L'=L1 + L2

Total angular momentum of system J'=L'+S'

3. The attempt at a solution

the spin vector S' can have eigenvalues s'=0,1 (I don't know why it just written in the textbook)

the L' vector apparentely has the rule that l' = abs(l1-l2) .... l1+l2
The same rule applies for s', so s' ranges from |1/2-1/2|=0 to 1/2+1/2=1.
so that gives l' = 2, 3, 4

and apparentely j'= l'+1/2,l'-1/2
You want to use the values for s', which as you noted above are 0 and 1, not 1/2.
So two j' for each l' means a total of 6 states. Where's the other 6?

My only thought is that you have to somehow take m_{l} quantum numbers into account but I'm totally confused about that because I don't understand anything about m_{l} quantum numbers.