# How are branching ratios determined? by experiment only?

1. Jan 8, 2010

### WarpedWatch

Greetings,

when I look up how nuclear reactions take place, I often see branching ratios listed as percentages. So I'm wondering: are those ratios determined only by experiment or is there some fundamental theory that can predict what those percentages will be?

many thanks,
Mark

2. Jan 8, 2010

### blechman

If you are studying a process X -> Y, the "rate" at which this process occurs is called $\Gamma(X\rightarrow Y)$. This is computable. If we denote the TOTAL width (that is, the inverse lifetime, the sum of all possible decay widths for all possible Y, also computable) as $\Gamma(X)$, then a "branching ratio" is defined as:

$$\frac{\Gamma(X\rightarrow Y)}{\Gamma(X)}$$

That is, it is the fraction of total decays of X that go into Y. So all this is computable, and it is always a fraction (percentage), since

$$\Gamma(X)=\sum_Y \Gamma(X\rightarrow Y)$$.

These things are all computable from quantum mechanics/quantum field theory.

3. Jan 8, 2010

### WarpedWatch

Yes, that is what I wanted to know. I was wondering if this kind of phenomenon is something that could have been figured out without running direct experiments.

many, many thanks for your help,
Mark

4. Jan 8, 2010

### blechman

yes, one can calculate partial widths directly from Fermi's Golden Rule in Quantum Mechanics (and its relativistic generalizations in Quantum Field Theory) and compare to experiment.

Happy I was able to help!