# A Equivalence equation between cross section & half life?

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1. Nov 21, 2016

### kiwaho

Is there equivalence equation between cross section & half life?
For beta decay, we usually use half life to describe how fast or slow the decay undergo.
For nuclear reaction, we use cross section to describe the possibility of reaction.
In a sense, they reflect the same root physics spirit.
Is there equation to associate half life and cross section? So as to assess nuclear reaction how fast or slow.
For example, Co-60 half life about 5 years, if other reaction can consume half reactants in 5 years, then its cross section can be regarded as equivalent to half life 5 years.

2. Nov 21, 2016

### Staff: Mentor

The half-life depends on the material only. There is no equivalent material property for reactions (where a cross section would appear) that have a unit of time.

3. Nov 21, 2016

### kiwaho

I think beta decay is a special nuclear reaction: (e,neutrino) for electron capture beta+, or (neutrino, e) for beta-, so it may need concept of cross section.

4. Nov 21, 2016

### kiwaho

or given cross section 5b at perfect high temperature for D+T = n + alpha, how to calculate the half life of D+T mix, i.e. half fuel consumed.

5. Nov 21, 2016

### ChrisVer

you cannot calculate it because it's an interaction and not a decay.
For decays of resonances the half life can be determined by the width of the resonance (the $\Gamma$) by $\tau = \frac{1}{\Gamma}$.

6. Nov 21, 2016

### Staff: Mentor

No. You can have a cross section if you shoot an electron beam on a target, but then the reaction rate will depend on the beam intensity.
You'll need the density and temperature of your fusion plasma to predict how its density goes down. It does not go down exponentially, so a half life doesn't make sense even if you fix those free parameters.

7. Nov 21, 2016

### ohwilleke

In the Standard Model, the way that you determine half-life from first principles is to determine the probability of a decay into every possible physically permitted decay path (a data set from which it is trivial to determine the cross-section data), which are added in a consistent and correct manner, to determine the half-life of the particle.

Normally the decays are determined by doing the calculation of the "decay width" for each possible path, something that in principle can involve cross-sections as part of the determination, but that is not the complete story. The decay width for any particular path can be converted to a probability of decay by that path within a particular time period and the total decay width can be converted to a half life.

Half-life and mean lifetime can be converted to each other by a factor of roughly 1.44.

The probability of decay for each path is computed independently of the probability of decay for every other path based upon Standard Model constants such as the relevant coupling constants. In general, the more ways a particle can decay, and the more probable each such possibility is, the shorter the half-life of the particle. A physics stack exchange answer on the topic is http://physics.stackexchange.com/qu...cay-width-and-why-is-it-given-in-energy-units

Decays via the strong force are generally more rapid than decays via the weak force.

A cross-section would often be quoted as a fraction of 100% which normalizes the results for individual decay paths against the total probability of decay for all decay paths.