# Beta Decay Questions: Help Understanding Angular Momentum

• Starbug
In summary, the conversation discusses the selection rules for beta decay, which include conservation of angular momentum and parity. The transition probability decreases with increasing angular momentum, and the spin of the lepton system must couple to 0 or 1. The conversation also touches on the allowed values for total angular momentum change and the classification of transitions as Fermi or Gamow-Teller. The justification for these rules is debated and various articles are referenced for further reading.
Starbug
Hello,

I'm having a hard time understanding some aspects of beta decay and I wondered if someone could help. (Perhaps this post belongs in the homework forum, but i don't have a specific question to do as such.) I'm not being helped by the fact that my general understanding of angular momentum is so poor, but anyway, as I understand it the selection rules for beta decay are

conservation of angular momentum:

$\vec{J}_P = \vec{J}_D + \vec{L}_\beta + \vec{S}_\beta$

and parity

$\pi_P = \pi_{D} (-1)^{L_\beta}$

Where $L_\beta$ is the orbital angular momentum carried away by the lepton system. The transistion probability decreases rapidly with increasing L, and measurements of the comparitive half-life will allow us to classify a transition as (super-)allowed, first forbidden etc depending on L=0,1,... with log of the comparitive half life scaling about 4 units with each change in L.

$S_\beta$ is the spin of the lepton system which must couple to 0 or 1. (Something I'm not quite sure about). Anyway if S=0 the transition is classified as Fermi, if S=1 is called Gamow-Teller.

What I'm struggling with is the quoted allowed values for the total angular momentum change. For an allowed Fermi transition the $$\Delta J$$ is zero, and the 0+ to 0+ transition is called superallowed. For the allowed Gamow-Teller my book says that $$\Delta J$$ can be zero or one, yet 0+ to 0+ can't be Gamow-Teller. I don't understand why the change can be zero, or for a zero change why it can't be Gamow-Teller if the initial or final state is 0. If I was asked to classify a 1+ to 1+ transition as Fermi or Gamow how would I do it?

Similarly for the first forbidden, Fermi transitions can be zero or one, but only one if it's from or to a zero state. Gamow transitions can be 0,1,2 but with a couple of disallowed possibilities, like 0- to 0+, 1/2+ to 1/2-, 1+ to 0-. I'm sure I've just missed something obvious but I can't make much sense of this at all.

Does your book "invent" (give without any jusitification) those seletion rules,or it computes them using Wigner-Eckart's theorem...?In the latter case,i think there's not too much for debating.

Daniel.

It's very much an introductory course I'm doing so much of the rigorous maths was skated over, although some arguments were made for the reasonableness of the selection rules. The set of rules for total angular momentum change for allowed, first forbidden etc, though were just quoted, I got the impression they followed straightforwardly but can't really see it.

Here's the article (i don't have access into Prola,maybe u have).

http://prola.aps.org/abstract/PR/v59/i11/p908_1

And another one (maybe u have access into this one,too)

http://www.iop.org/EJ/abstract/0370-1298/65/10/303

and more here http://hepwww.ph.qmul.ac.uk/~rizvi/npa/week04.htm

Daniel.

Last edited by a moderator:
Thank you, that was most helpful.

## 1. What is beta decay?

Beta decay is a type of radioactive decay in which an unstable atomic nucleus emits a beta particle (either an electron or positron) in order to become more stable.

## 2. How does beta decay relate to angular momentum?

Beta decay involves a change in the nuclear spin of an atom, which is directly related to its angular momentum. The emission of a beta particle causes a change in the spin of the nucleus, resulting in a change in its overall angular momentum.

## 3. Can you explain the concept of angular momentum in more detail?

Angular momentum is a physical quantity that describes the rotational motion of an object. It is calculated by multiplying an object's moment of inertia (a measure of its resistance to rotation) by its angular velocity (a measure of how fast it is rotating). In the case of beta decay, the emission of a beta particle causes a change in the spin of the nucleus, which affects its overall angular momentum.

## 4. How does beta decay impact the stability of an atom?

Beta decay is a process that occurs in unstable atoms in order for them to become more stable. When an atom undergoes beta decay, it releases energy and changes its nuclear spin, which in turn affects its overall stability.

## 5. What types of particles are emitted during beta decay?

In beta decay, either an electron or positron is emitted from the nucleus. This emission of a beta particle is accompanied by the emission of a neutrino or antineutrino, which are neutral particles with very little mass. The type of beta particle emitted depends on the type of radioactive isotope undergoing decay.

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