Interpreting energy terms in the Breit-Wigner resonance formula

[rabbit44]

(urgent) Can someone help me understand the factors in the Breit-Wigner formula?

Hi, I have the BW formula as:

$$\sigma = \frac{\lambda^2 (2J+1)}{\pi (2S_a+1)(2S_b+1)} \frac{\Gamma^2 / 4}{(E-E_R)^2 + \Gamma^2/4}$$

So $$E_R$$: this is described as the 'resonance energy'. I'm pretty sure this is the energy of the resonant state (i.e. of the compound particle) - is this the compound particle's mass as well? Or does the compound particle have kinetic energy?

Then E: This is described as 'the centre of mass energy of the initial state'. Does this include both rest energy and kinetic energy? And say we had a nucleus incident on a stationary nucleus, how would we calculate E?

The other symbols are fine.

Thanks very much.

EDIT:

Oh something else confuses me. Often you get plots of cross section vs. energy (e.g. incident neutron energy), and there are multiple peaks. Is $$E_R$$ different for all these peaks?

 

[Meir Achuz]

E_R is the mass of the resonant state. It is the center of mass energy W, which equals
$$[\sum_i E_i]^2-[\sum_i {\vec p_i}]^2$$ for all the decay particles
(or two incident particles).

 

[rabbit44]

E_R is the mass of the resonant state. It is the center of mass energy W, which equals
$$[\sum_i E_i]^2-[\sum_i {\vec p_i}]^2$$ for all the decay particles
(or two incident particles).

Thanks for replying, but I don't get how $$E_R$$ is the mass of the resonant state. Because $$E_R$$ is the kinetic energy in the centre of mass frame at which resonance occurs (I think). So for a reaction

a + B --> X -->

where X is the compound nucleus, shouldn't conservation of energy give:

$$(M_a + M_B)c^2 + E_R = M_X c^2 +$$excitation energy of compound nucleus

? So unless the mass of the resonant state means something different to the mass of the compound nucleus, the above must be wrong?

*so confused*

 

[blechman]

E is the CM energy of A + B (and therefore by conservation of energy, the rest energy of the resonant state X), while E_R is the "mass" of the resonance. The idea being that if this state is a resonance with width $\Gamma\neq 0$, then it can be produced at rest with energy $E\neq E_R$ due to quantum uncertainty.

One would say that "the resonance is produced off-shell." This is fine, because you don't actually "observe" X, but only X's decay products, so there's nothing wrong with it being off-shell.

In practice: E is a kinematic variable, while E_R is a number you compute in your theory (or treat as a parameter).

Hope that helps.

 

[rabbit44]

E is the CM energy of A + B (and therefore by conservation of energy, the rest energy of the resonant state X), while E_R is the "mass" of the resonance. The idea being that if this state is a resonance with width $\Gamma\neq 0$, then it can be produced at rest with energy $E\neq E_R$ due to quantum uncertainty.

One would say that "the resonance is produced off-shell." This is fine, because you don't actually "observe" X, but only X's decay products, so there's nothing wrong with it being off-shell.

In practice: E is a kinematic variable, while E_R is a number you compute in your theory (or treat as a parameter).

Hope that helps.

Thanks very much for replying.

So does E, the CM energy, include the rest energies of the incident particles?

I just realized also that a large part of my confusion was thinking that the mass of the compound particle is the mass equivalent of its ground state energy. But an excited nucleus is heavier because of a reduced binding energy, bringing it closer to being the sum of the unbound masses of the nucleons.

 

[blechman]

So does E, the CM energy, include the rest energies of the incident particles?

yes.

 

[Rajini]

Hi, in short:
E_R is the position of peak. E is the complete range (i mean x-axis).
when you draw a Lorentzian, E in x-axis and for your formula E_R means the position of the peak.
If you have 2 peaks then there will be 2 E_R.
If you dealing with cross section, then the first term in your formula is just the nuclear cross section (or it is just the height of the peak).

 

[Rajini]

see picture

it is better so see this picture