Special relativity - scattering angle

In summary, the conversation discusses finding the maximum scattering angle of a particle with mass m1 and relativistic velocity v, hitting a stationary particle with mass m2. The attempt at a solution involves using conservation of 4-momentum and taking the derivative to find the maximum angle. After some corrections, the final result is found to be cosθmax=√(1−(m2/m1)^2), which is the same as the nonrelativistic result.
  • #1
Aleolomorfo
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4

Homework Statement


Finding the maximum scattering angle of a particle whose mass in ##m_1## which hits with relativistic velocity ##v## a particle at rest with mass ##m_2<m_1##.

The Attempt at a Solution


I've written the 4-momenta (p before the collision, k after the collision and the z-axis is along the direction of the incident particle):
$$p_1=(m_1\gamma,0,0,m_1\gamma v)$$
$$p_2=(m_2,0,0,0)$$
$$k_1=(E,0,\sqrt{E^2-m^2_1}\sin{\theta},\sqrt{E^2-m^2_1}\cos{\theta})$$
For ##k_2## the components are not important, it's important that ##k^2_2=m^2_2##
Then I've used the conservation of 4-momentum ##p_1+p_2=k_1+k_2##, then ##k_2=p_1+p_2-k_1##, then ##k^2_2=p^2_1+p^2_2+k^2_1+2p_1p_2-2p_1k_1-2p_2k_1##. After calculations I've found:
$$\cos{\theta}=\frac{m^2_1+m_1m_2\gamma-2E(m_1\gamma+m_2)}{m_1\gamma v\sqrt{E^2-m^2_1}}$$.
Then I've taken the derivative ##\frac{d(\cos{\theta})}{dE}## and put it equal to 0. However, I've found an equation quite difficult to solve and I think it's wrong.
I think the way I set up the problem is not incorrect, but maybe there is a easier way or some trick to reduce calcus.
 
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  • #2
Aleolomorfo said:
Then I've used the conservation of 4-momentum ##p_1+p_2=k_1+k_2##, then ##k_2=p_1+p_2-k_1##, then ##k^2_2=p^2_1+p^2_2+k^2_1+2p_1p_2-2p_1k_1-2p_2k_1##. After calculations I've found:
$$\cos{\theta}=\frac{m^2_1+m_1m_2\gamma-2E(m_1\gamma+m_2)}{m_1\gamma v\sqrt{E^2-m^2_1}}$$.
Check the overall sign of the right hand side of the equation for ##\cos \theta## and also check if the factor of 2 in the numerator is correct.

Then I've taken the derivative ##\frac{d(\cos{\theta})}{dE}## and put it equal to 0. However, I've found an equation quite difficult to solve and I think it's wrong.
I think the way I set up the problem is not incorrect, but maybe there is a easier way or some trick to reduce calcus.
With the corrections mentioned above, it will work out if you slog through it. I don't know of a clever trick to reduce the algebra. Maybe someone else can show us a better way.
 
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  • #3
TSny said:
Check the overall sign of the right hand side of the equation for ##\cos \theta## and also check if the factor of 2 in the numerator is correct.

With the corrections mentioned above, it will work out if you slog through it. I don't know of a clever trick to reduce the algebra. Maybe someone else can show us a better way.

Thank you for your help. The worst is when you make a mistake at the beginning. My result is ##\cos{\theta_{max}}=\sqrt{1-\frac{m^2_2}{m^2_1}}##, I think it's correct or at least is plausible.
 
  • #4
Aleolomorfo said:
Thank you for your help. The worst is when you make a mistake at the beginning. My result is ##\cos{\theta_{max}}=\sqrt{1-\frac{m^2_2}{m^2_1}}##, I think it's correct or at least is plausible.
Yes, that's the correct answer. It's even nicer when expressed in terms of ##\sin \theta##. Since the answer is independent of the speed of the incoming particle, the result is the same as the nonrelativistic, Newtonian result.
 

Related to Special relativity - scattering angle

1. What is special relativity?

Special relativity is a theory in physics that describes the relationship between space and time, and how they are affected by the motion of objects. It was developed by Albert Einstein in 1905 and is based on two main principles: the constancy of the speed of light and the principle of relativity.

2. What is the scattering angle in special relativity?

The scattering angle in special relativity refers to the angle at which particles are deflected or scattered after interacting with each other. It is an important concept in the study of high-energy particle collisions, such as those that occur in particle accelerators.

3. How is the scattering angle calculated in special relativity?

The scattering angle is calculated using the laws of conservation of energy and momentum in special relativity. This involves considering the initial and final velocities of the particles, as well as their masses and energies.

4. What is the significance of the scattering angle in special relativity?

The scattering angle is significant because it provides information about the fundamental forces and interactions between particles. It can also help us understand the structure of matter and the behavior of particles at high energies.

5. How does special relativity affect the scattering angle in particle collisions?

Special relativity predicts that the scattering angle will be larger at higher energies due to the increase in the speed of particles. It also predicts that the scattering angle will approach 180 degrees as the energy of the particles approaches infinity, indicating a more head-on collision.

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