# B Radioactive decay, Stability and Halflife

#### A M

Summary
I have 3 main questions:
As it's written in the following article, nuclear binding energy is always a positive number; thus it takes energy to disassemble a nucleus into its nucleons.
...The binding energy is always a positive number, as we need to spend energy in moving these nucleons, attracted to each other by the strong nuclear force, away from each other.

And according to this diagram, some unstable nuclei like U_235 and U_238 have positive binding energy per nucleon (about 7.5 Mev) even greater than many stable ones'.

So, why should a radioactive nucleus (with BE/N higher than many stable nucleus) decay?

What does the stability of nuclei depend on?

And I also have a question about half life;
If decay energy of the nucleus -or anything else that makes it decay- is greater than nuclear binding energy, the unstable nucleus is expected to suddenly get disassembled into its component NUCLEONS. Or it shouldn't be created at all...
But they are stable for a consistent period of time and then converse to other NUCLEI.
Would you please explain the reason?

I would be grateful if anyone could elucidate these problems.
A M

A person who never made a mistake, never tried anything new.
-Albert Einstein

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#### mfb

Mentor
The decays go to nuclei with an even higher binding energy per nucleon. They never go into "all free protons and neutrons".
A nuclei is stable if there is no possible decay mode that releases energy. Some decay modes are so unlikely that nuclei can appear stable even if they should be able to decay in theory. This happens for many of the heavier nuclei we consider stable, e.g. some lead nuclei: Fission of lead could release energy, but it is so unlikely that it won't happen over any realistic timescale.
If decay energy of the nucleus -or anything else that makes it decay- is greater than nuclear binding energy
It isn't.

#### A M

Hmm... I'm still a little confused.
First of all:
If
It isn't.
How can decay energy dominate binding energy?
What exactly makes a radioactive nucleus decay?

- And I've read there are two main parameters on which binding energy depends:
Nuclear strong force & repulsive Coulomb force.
But I guess there should be a great force to overcome binding energy (in particular for heavier elements)
Right?
A M
A person who never made a mistake, never tried anything new.
-Albert Einstein

#### mfb

Mentor
Toy example: An uranium-235 nucleus with 7.5 MeV binding energy per nucleon fissions to two nuclei with 8.5 MeV binding energy per nucleon for both of them. Released energy: 235 MeV.
- And I've read there are two main parameters on which binding energy depends:
Nuclear strong force & repulsive Coulomb force.
But I guess there should be a great force to overcome binding energy (in particular for heavier elements)
For heavier elements the repulsive Coulomb force becomes larger fast, lowering their binding energy per nucleon relative to previous elements.

#### A M

But even for heaviest nuclei, Coulomb force is smaller than strong nuclear force. Thus there should be another influencing factor...
A M

Staff Emeritus
But even for heaviest nuclei, Coulomb force is smaller than strong nuclear force. Thus there should be another influencing factor...
That's already included in the binding energy.

#### A M

That's already included in the binding energy.
So increasing Coulomb force can't be the reason for instability of heavy nuclei.

#### A M

My main question is:
Why are heavy elements mostly unstable?
Lower binding energy per nucleon, Greater Coulomb force,...

Staff Emeritus
So increasing Coulomb force can't be the reason for instability of heavy nuclei.
I said no such thing.

Last edited:

Hmm...

Staff Emeritus

#### PeterDonis

Mentor
Binding energy is always a positive number; so we need energy to disassemble such nuclei.
Radioactive decay is not "disassembling" the nucleus. As has already been said, radioactive decay takes a nucleus with a particular binding energy per nucleon to another nucleus with even higher binding energy per nucleon. So "binding energy is always a positive number" is not a reason for nuclei not to decay.

Binding energy/nucleon of many heavy unstable nuclei is greater than some table ones
That's because there are only two kinds of radioactive decay that take one nucleus to another: alpha decay and beta decay. If there is not another nucleus with higher binding energy per nucleon that is reachable by one of those decays, a nucleus will be stable. The nucleus doesn't "care" that there are other unstable nuclei with higher binding energy per nucleon; those other unstable nuclei just happen to have a nucleus with even higher binding energy per nucleon reachable by alpha or beta decay (for heavy nuclei it's almost always alpha).

Increasing repulsive Coulomb force can't be the reason; because even for the heaviest nuclei, strong nuclear force is greater
The balance of forces is involved in determining what the binding energy per nucleon is for a given nucleus; but since we already know the binding energy per nucleon, we don't care how the particular balance of forces in a nucleus produced it if all we're interested in is stability.

why are heavy nuclei (Z>82) unstable?
Because every nucleus with Z>82 has at least one other nucleus with higher binding energy per nucleon reachable by alpha or beta decay, while at least some nuclei with Z <= 82 don't.

A M

#### A M

That makes so much more sense!

#### A M

there are only two kinds of radioactive decay that take one nucleus to another: alpha decay and beta decay.
You mean Isomeric Transition doesn't take one nucleus to another?!
Therefore they have some differences....

#### PeterDonis

Mentor
Isomeric Transition doesn't take one nucleus to another?!
It doesn't change the isotope, no. The atomic number and mass number are the same after the transition as before.

#### PeterDonis

Mentor
That would count as beta decay, which I included.

#### A M

Oops... yes, yes.
Good luck!

#### A M

The decays go to nuclei with an even higher binding energy per nucleon.
radioactive decay takes a nucleus with a particular binding energy per nucleon to another nucleus with even higher binding energy per nucleon.
unstable nuclei just happen to have a nucleus with even higher binding energy per nucleon reachable by alpha or beta decay
What do you mean "even"?
You mean there are some decays that go to "lower" binding energy?
In that case the final binding energy is higher, therefore energy is "needed"!

#### PeterDonis

Mentor
What do you mean "even"?
You mean there are some decays that go to "lower" binding energy?
In that case the final binding energy is higher, therefore energy is "needed"!
You're making this way harder than it needs to be.

Say we have three nuclei, A, B, and C, with binding energies per nucleon $a$, $b$, and $c$, such that $a < b < c$.

Suppose there is a decay path (alpha or beta) from A to C, but no decay path from B to any nucleus of higher binding energy per nucleon (C or any other).

Then A will be unstable but B will be stable, even though B has higher binding energy per nucleon than A. And B will be stable even though there is at least one nucleus, C, with higher binding energy per nucleon than B, because B has no decay path to any such nucleus.

#### A M

I see!
there are only two kinds of radioactive decay that take one nucleus to another: alpha decay and beta decay.
But how about proton emission, neutron emission, spontaneous fission and cluster decay?

#### PeterDonis

Mentor
how about proton emission, neutron emission, spontaneous fission and cluster decay?
These are all rare, but sure, if you want to throw them into the mix, that's fine. Everything I've said would apply to them as well.

#### A M

As you said (or I've understood!) if there is not another nucleus with higher binding energy per nucleon that is reachable by α, β, ... decays, a nucleus will be stable. But unstable nuclei have a nucleus with higher binding energy per nucleon reachable by alpha, beta... decays.
Now, we know why a nucleus is stable or unstable.
And we also know that among unstable nuclei, those that have shorter half lives are more radioactive and more unstable.
But why do some unstable nuclei decay faster than the others?
For example:
 Decay mode Half life Bismuth_209 alpha ≈2×10^19 years Beryllium_8 alpha ≈8×10^-17 seconds

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