Equivalence equation between cross section & half life?

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• kiwaho
In summary, there is no equivalence equation between half life and cross section. They reflect the same root physics spirit, but there is no equation that associates one with the other.
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.

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.

mfb said:
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.
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.

mfb said:
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.
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.

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 $\tau = \frac{1}{\Gamma}$.

MisterX
kiwaho said:
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.
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.
kiwaho said:
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.
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.

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.

1. What is the equivalence equation between cross section and half life?

The equivalence equation between cross section and half life is: σ = ln(2)/λ, where σ represents the cross section and λ represents the decay constant.

2. How is the cross section related to the half life?

The cross section is inversely proportional to the half life. This means that as the cross section increases, the half life decreases, and vice versa.

3. Can the equivalence equation be used for all types of decay?

No, the equivalence equation between cross section and half life is specific to radioactive decay processes. It cannot be applied to other types of decay, such as chemical reactions.

4. Why is the equivalence equation important in nuclear physics?

The equivalence equation allows scientists to calculate the cross section of a radioactive nucleus based on its half life, and vice versa. This is important in understanding and predicting the behavior of radioactive materials, as well as in the design and operation of nuclear reactors.

5. How is the half life related to the stability of a radioactive isotope?

The half life is an indicator of the stability of a radioactive isotope. A longer half life generally indicates a more stable isotope, while a shorter half life indicates a less stable isotope. This is because a longer half life means the isotope decays at a slower rate, while a shorter half life means it decays at a faster rate.

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