Calculating Max/Min Energies of Particle P1 in Decay of A

In summary, Daniel determined that the maximum energy that a particle can have is the rest energy of the particle, which is just created. If the rest mass of the particle is smaller than the sum of the rest mass of the other particles, then the decay should not occur. However, this is false because two particles must have nonzero momentum. If the rest mass of the particle is greater than the sum of the rest mass of the other particles, then the decay may occur.
  • #1
crit
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Particle A, at rest, decays into three or more particles: P1, P2, ..., Pn.
Determine the maximum and minimum energies that P1 can have in such a decay, in terms of various masses.

My solution:

First of all, the decay should not occur if the rest mass of A would be smaller than the sum of the rest mass of P1, P2, ..., Pn.

energy conservation. E1 minimum is the rest energy of P1 which is just created, with zero momentum. The rest of the energy is taken by the rest energies and momentum of P2, P3, etc.

E1 max if E2+E3+...En is minimum, that is if P2 is at rest, P3 is at rest, ..., Pn is at rest. But this is false, because then, by conservation of momentum, P1 should also be at rest and would have E1 min, not maximum. At least two particles should have non zero momentum. E1 and another one. But then? I tried an analytical approach, but the calculus would need Mathematica.

Any suggestions for a straighforward solution? Thanks.
 
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  • #2
The first part is okay.Indeed,simple logics and virtually no computations will lead to the result that the minimum energy of the particle is the rest one.

For the second...I'm not really sure it is a min/max problem,one which would require Lagrange multipliers...It's more about logics.

Indeed,the same logics tells u that at least 2 particles must have nonzero momentum.The logics tells u that this number MUST BE EXACTLY 2.I hope u see why...If there are at least three,then the energy of the one u're interested in will be smaller.(the sum of the three would be constant...)
My logics says that form the N-1 other particles,the one with the smallest rest mass will fulfil the requirement.Let's label it with "2" and the particle of interest with "1".Think about it.The energies of the 2 particles are constant and equal to the difference between the initial particle's rest energy ([itex] m_{A} [/itex]) and the sum of the rest energies of the other (N-2) particles...

So
[tex] E_{1}+E_{2}=constant=m_{A}-\sum_{j=3}^{n} m_{j} [/tex](1)

Okay??
But u know that the 2 particles must have nonzero momenta,however,of opposite sign and same modulus (this,as u may have noticed,results from the conservation of total momentum in the decay process)

[tex] E_{1}=\sqrt{\vec{p}_{1}^{2}+m_{1}^{2}} [/tex](2)
[tex] E_{2}=\sqrt{\vec{p}_{2}^{2}+m_{2}^{2}} [/tex](3)

[tex] \vec{p}_{1}+\vec{p}_{2}=\vec{0} [/tex] (4)


From (4) u find [itex] \vec{p}_{2} [/itex] in terms of [itex] \vec{p}_{1} [/itex] and then take square and substitute in (3).Use the relation (2),the "new version" of (3) and the condition (1) to get one ugly equation in terms of [itex] \vec{p}_{1}^{2} [/itex] .

Then substitute in (2) and find the maximum energy...

Daniel.

P.S.Bai,Adita,ce mama ma-sii,nu putea sa-ti imaginezi si tu chestia asta??Si cu "crit-clit" ce e?? :tongue2:
P.P.S.Asta e mai simpla decit o problema de la Judeteana intr-a 12-a... :tongue:
 
  • #3
I have discovered the same reasoning as yours during my last class and I have calculated that momentum of both P2 and P3. But I want mathematical result to tell me that P3 with be the one with the smallest rest mass and not "logic", which in this case is just an intuition (that was also my intuition). I am annalizing that and I will write again here.
 
  • #4
What P_{3} are u talking about??I didn't mention any P_{3}...I have chosen P_{2} as the momentum of the particle with smallest rest mass...The charcateristics of a third (why not a 4-th,5-th and so on) particle do not interviene whatsoever...

I don't know what u find against logics...Mathematics is founded on logics...

Waiting for an answer and "nu-mi mai trimite meiluri !" :grumpy: .Sint tot timpul online.

Daniel.
 

1. What is the formula for calculating the maximum energy of particle P1 in the decay of A?

The formula for calculating the maximum energy of particle P1 in the decay of A is given by:
Emax = (mA - mP1)c2,
where mA and mP1 are the rest masses of particles A and P1, respectively, and c is the speed of light.

2. How do I determine the minimum energy of particle P1 in the decay of A?

The minimum energy of particle P1 can be determined by using the formula:
Emin = (mA - mP1 - mP2)c2,
where mA, mP1, and mP2 are the rest masses of particles A, P1, and P2, respectively.

3. How do I calculate the energy difference between the maximum and minimum energies of particle P1 in the decay of A?

The energy difference between the maximum and minimum energies of particle P1 can be calculated by subtracting the minimum energy from the maximum energy. This will give you the range of energies that particle P1 can have in the decay of A.

4. What factors can affect the maximum and minimum energies of particle P1 in the decay of A?

The maximum and minimum energies of particle P1 can be affected by the masses of particles A, P1, and P2, as well as the speed of light. Other factors such as external forces or interactions can also play a role in determining the energies of particle P1 in the decay of A.

5. Can the maximum and minimum energies of particle P1 in the decay of A be experimentally measured?

Yes, the maximum and minimum energies of particle P1 can be experimentally measured using techniques such as particle accelerators and detectors. By measuring the energies of the particles produced in the decay of A, the maximum and minimum energies of particle P1 can be inferred and compared to the calculated values.

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