What is the meaning of unbound in the context of half-lives of nuclei?

[Mark lamorey]

I have been looking at half-life of various isotopes and many of the tables / information I am finding is giving energies (MeV, KeV) for certain half lifes.

This does not make sense to me, am I missing something ?

For example :
http://ie.lbl.gov/toi/listnuc.asp?sql=&A1=5&A2=5

⁵Li and ⁵He give energies ?

 

[Vanadium 50]

These half-lives are so short that it makes more sense to give them in terms of the width of the nucleus (in units of energy).

 

[Mark lamorey]

I am not sure I understand what you mean by width of the nucleus -link ?
How can I convert from this energy unit to time or time to this energy unit if I need a constant reference across isotopes ?

I appreciate any advice / direction.

 

[Vanadium 50]

Do you know any quantum mechanics? I am trying to figure out the right level to explain this.

 

[Mark lamorey]

Do you know any quantum mechanics? I am trying to figure out the right level to explain this.

Yes, I have a certain amount of quantum mechanics background. Worked with it through various solid state classes, learning, work experiences.

 

[e.bar.goum]

These half-lives are so short that it makes more sense to give them in terms of the width of the nucleus (in units of energy).

I think you mean "width of the resonance" rather than "width of the nucleus" here? I've never heard anyone use the term "width of the nucleus" to mean this.

But otherwise, Mark, V50 is right on the money. The half-life for 5Li, for instance, is on the order of a zeptosecond (which is conveniently about the same as the nuclear reaction timescale. This makes the reaction dynamics of 5Li (and similar nuclei) rather interesting.

You can think of nuclei like 5Li as an α+p resonance, as it's unbound, and as such, appears as a peak in an a+p scattering cross section. But anyway, you can relate the lifetime to the width by the relation

$\Gamma = \frac{\hbar}{\tau}$

Where $\Gamma$ is the width, $\tau$ the lifetime, and $\hbar$ the reduced Planck constant. This relationship will be familiar to you if you've done quantum mechanics (think uncertainty principle).

EDIT: mixed up my gammas and my lambdas, how embarrassing.

 

[Vanadium 50]

OK, so you know there is an uncertainty relationship between energy and time. The nuclear energy level has a width, measured in terms of energy, and that can be converted to time. It's more practical in this case to use the width, and that has units of energy.

 

[Mark lamorey]

I think you mean "width of the resonance" rather than "width of the nucleus" here? I've never heard anyone use the term "width of the nucleus" to mean this.

But otherwise, Mark, V50 is right on the money. The half-life for 5Li, for instance, is on the order of a zeptosecond (which is conveniently about the same as the nuclear reaction timescale. This makes the reaction dynamics of 5Li (and similar nuclei) rather interesting.

You can think of nuclei like 5Li as an α+p resonance, as it's unbound, and as such, appears as a peak in an a+p scattering cross section. But anyway, you can relate the lifetime to the width by the relation

$\Gamma = \frac{\hbar}{\tau}$

Where $\Gamma$ is the width, $\tau$ the lifetime, and $\hbar$ the reduced Planck constant. This relationship will be familiar to you if you've done quantum mechanics (think uncertainty principle).

EDIT: mixed up my gammas and my lambdas, how embarrassing.

Thanks for the equation, that helps a lot.
Still a little confused on width of the nucleus.

I understand that we might be happier dealing in Mev, and Kev rather than zepto seconds.
Similarly, reduced Planck or Dirac have units of (energy*time), so the equation results in time.
My thought on why we say nucleus width is in relation to c (speed of light - distance/time) - but this is likely not the case.

On the quantum side - is the width term in relation to the width of the "well" that the 'energy wave' is in?
This idea of well width makes some sense but perhaps well height would be more correct, and we can simplify by say the nucleus width rather than the well height?

 

[e.bar.goum]

Thanks for the equation, that helps a lot.
Still a little confused on width of the nucleus.

I understand that we might be happier dealing in Mev, and Kev rather than zepto seconds.
Similarly, reduced Planck or Dirac have units of (energy*time), so the equation results in time.
My thought on why we say nucleus width is in relation to c (speed of light - distance/time) - but this is likely not the case.

On the quantum side - is the width term in relation to the width of the "well" that the 'energy wave' is in?
This idea of well width makes some sense but perhaps well height would be more correct, and we can simplify by say the nucleus width rather than the well height?

I'd like to emphasize that to the best of my knowledge, the "width of the nucleus" is a term that is not used to mean what V50 said it means. (And I'm someone who thinks about the lifetimes of particles like 5Li very frequently). I think that using that term is highly misleading, because of basically what you've thought of above. What we're talking about here doesn't have anything to do with the physical size of the nucleus.

It's to do with the width of the quantum state of the nucleus. Like atoms, nuclei have discrete energy levels. These energy levels have some uncertainty associated with the lifetime. If the state is long lived, we can say with precision what the energy is. As the lifetime becomes shorter, the uncertainty in the width increases. When the nucleus is unbound, like 5Li, we don't use the phrase "energy level", but "resonance".

Look at:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/parlif.html
http://quantummechanics.ucsd.edu/ph130a/130_notes/node428.html

 

[Vanadium 50]

Right. By width I meant line width, not physical width in meters.

 

[Mark lamorey]

I'd like to emphasize that to the best of my knowledge, the "width of the nucleus" is a term that is not used to mean what V50 said it means. (And I'm someone who thinks about the lifetimes of particles like 5Li very frequently). I think that using that term is highly misleading, because of basically what you've thought of above. What we're talking about here doesn't have anything to do with the physical size of the nucleus.

It's to do with the width of the quantum state of the nucleus. Like atoms, nuclei have discrete energy levels. These energy levels have some uncertainty associated with the lifetime. If the state is long lived, we can say with precision what the energy is. As the lifetime becomes shorter, the uncertainty in the width increases. When the nucleus is unbound, like 5Li, we don't use the phrase "energy level", but "resonance".

Look at:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/parlif.html
http://quantummechanics.ucsd.edu/ph130a/130_notes/node428.html

Super, THANKS !

Resonance makes a lot of sense to me in this case.
A further question: You say unbound. I am guessing that is more than just not stable or is the term 'unbound' specific to all nuclei that have half lives ?