Particle Disintegration: Equation trouble

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In summary, the conversation revolves around the equation for particle disintegration and the confusion regarding the resulting equation for cos θ0. It is mentioned that the equation is valid for non-relativistic cases and that the angles are measured relative to the direction of V. After some discussion and reworking of the equations, the correct equation for cos θ0 is derived.
  • #1
Piano man
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Hi, I'm reading about particle disintegration at the moment and there's a step I don't follow.

I've got the following equation:

[tex]\tan\theta=\frac{v_0\sin\theta_0}{v_0\cos\theta_0+V} [/tex] where [tex] \theta [/tex] is the resultant angle in the Laboratory system and [tex] \theta_0 [/tex] is the resultant angle in the Centre of Mass system.

Also given is [tex]v=V+v_0[/tex] which are respectively the velocity of a resulting particle in the L system, the velocity of the primary particle in the L system, and the velocity of the resulting particle in the C system.

Solving for [tex]\cos\theta_0[/tex] one should obtain

[tex]\cos\theta_0=-\frac{V}{v_0}\sin^2\theta \pm \cos\theta\sqrt{1-\frac{V^2\sin^2\theta}{v_0^2}}[/tex]

but I've gotten [tex]\cos\theta_0=\frac{V}{v_0}(\cos\theta-1)+\cos\theta[/tex]

from the substitution [tex]\sin\theta_0=\sin\theta\left(\frac{V+v_0}{v_0}\right)[/tex] which seems geometrically sound.

Can anyone see where that other equation comes from for [tex]\cos\theta_0[/tex]?

Thanks.
 
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  • #2
Are you sure the equation you started with applies to this situation? It's valid for non-relativistic cases, but particle decay typically involves relativistic speeds.
 
  • #3
Yes, in the book, it follows on immediately, with no mention of relativistic effects.
I think it's just some algebraic reworking of the equation that I'm not seeing.
 
  • #4
From the equation

[tex]\tan\theta=\frac{v_0\sin\theta_0}{v_0\cos\theta_0+V}[/tex]

I'm inferring that the angles are measured relative to the direction of V, the velocity of the primary particle in the lab frame. Your second equation from equating the y-components in the two frames is wrong. It should be

[tex]v_0\sin\theta_0=\sqrt{(v_0\cos\theta_0+V)^2+(v_0\sin\theta_0)^2}\sin\theta[/tex]

The equation relating the velocities should be

[tex]\vec{v}=\vec{v}_0+\vec{V}[/tex]

which is a vector equation, so you can't just add the magnitudes of v0 and V to get the magnitude of v.

To derive the other equation, start with the tan θ equation, square it, and rewrite sin2 θ0 in terms of cos θ0. You'll get a quadratic equation in cos θ0.
 
  • #5
Excellent! I got it!

That was some marathon of a reworking - three pages and an hour later...

Thanks vela for your help. :)
 
  • #6
Excellent! I got it!

That was some marathon of a reworking - three pages and an hour later...

Thanks vela for your help.
 

1. What is particle disintegration?

Particle disintegration is the process of breaking down a larger particle into smaller, more stable particles. This can occur naturally or artificially through various methods such as nuclear reactions or collisions with other particles.

2. What causes particle disintegration?

Particle disintegration can be caused by a variety of factors, including changes in energy levels, collisions with other particles, or the presence of unstable particles. In nuclear reactions, it is often caused by the release of excess energy.

3. What is the equation for particle disintegration?

The equation for particle disintegration varies depending on the specific process and particles involved. Generally, it involves the release of energy and the transformation of particles into more stable forms. One example is the equation for alpha decay, which can be represented as A → B + α, where A is the original particle, B is the resulting particle, and α is the alpha particle that is released.

4. How is particle disintegration measured?

The measurement of particle disintegration can vary depending on the specific process and particles involved. In nuclear reactions, it is often measured through the detection of emitted particles or changes in energy levels. In particle physics, it can be measured through the detection of subatomic particles or changes in particle properties.

5. What are the real-world applications of particle disintegration?

Particle disintegration has many real-world applications, including energy production through nuclear reactions, medical imaging and treatments using radiation, and material analysis using particle accelerators. It is also used in fundamental research to study the properties and behavior of subatomic particles.

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