Statement about Forbidden Alpha decay transitions.

[Silversonic]

Hi, I'm confused about a statement about the change in final/initial states of the daughter/parent atom in an alpha decay. It is the following;

"The spin between the parent (I_i) and daughter (I_f) can change by lh (h being h-bar, l is the orbital angular quantum number of the alpha particle), where;

\vec{I_i} = \vec{I_f} + \vec{l}

and the parity changes by (-1)^l "

I'm confused because if, for example, we take the initial state of the parent to be 0^+, then there are the following cases;

\vec{I_i} = \vec{I_f} + \vec{l} means

|\vec{I_i}| = |\vec{I_f} + \vec{l}|

Coupling angular momentum together would surely mean that the total orbital quantum number would be of multiple values;

L = I_f + l, I_f + l - 1, I_f + l - 2, ... I_f - l

So if I_i = 0^+ we could have multiple I_f for a given l. For example l = 1, then

L = I_f + 1, I_f , I_f - 1 = I_i = 0

Meaning I_f could take on values 0 or 1. My notes seem to suggest only the I_f = 1 state is possible. Am I looking at this in completely the wrong way? I think I don't fully understand what it means by "can change by lh", what is the signficance of the "can" change?

I looked on the internet and in my textbook, not much to a detail that I can understand. Beta decay forbidden decays seemed to be the closest I could find which might explain it but I don't know how applicable it is to this situation.

 

[mfb]

$I_i=0$ gives $|\vec{I_f}+\vec{l}|=0$, which has the solution $\vec{I_f}=-\vec{l}$ only.

 

[Silversonic]

$I_i=0$ gives $|\vec{I_f}+\vec{l}|=0$, which has the solution $\vec{I_f}=-\vec{l}$ only.

Ah yes clearly. I should've realized that. Here's a better example which might help me understand it for general l and I_f.

Say the initial state was I_i = 1^+. We must have the coupled orbital angular momentum quantum number L equal to 1. Then say the alpha particle had l = 2

The coupled orbital angular momentum quantum number of

\vec{I_i} = \vec{I_f} + \vec{l}

Would be

L = I_f + l, |I_f + l - 1|, |I_f + l - 2|, ... |I_f - l|

So if we had I_f = 3

L = 5,4,3,2,1

Corresponding to a possibility to have L = 1

If we had I_f = 2

L = 4,3,2,1,0

Corresponding to another possibility to have L = 1

And lastly; If we had I_f = 1

L = 3,2,1

Corresponding to another possibility to have L = 1So in a transition from a state 1^+ with the emission of an alpha with l = 2, it's possible to have 3 final states for the daughter (3^-, 2^-, 1^-)? Is this correct?

 

[mfb]

Looks correct.

$|l-I_i| \leq I_f \leq |I+I_i|$