# Scattering angle in relativistic kinematics

• kalok87
In summary, the conversation discusses the 4-momentum relation in the context of elastic collisions between two particles with momenta. It also touches on the derivation of an equation for the scattering angle, taking into account both relativity and non-relativistic cases. The final result for the maximum scattering angle is derived and the question of resolving the equation when considering relativity is raised.
kalok87

## Homework Statement

Considering 2 scattering particles with momenta ##p_{1}, p_{2}##, where ##p_{2} = 0## in the Lab reference. The momenta of these 2 particles after elastic collision are ##p_{1}', p_{2}'##, respectively. Due to the 4-momentum relation, we have ##p_{1i}p_{1}^{'i} = e_{1}e_{1}' - p_{1} \cdot p_{1}' = e_{1}e_{1}' - p_{1} \cdot p_{1}' \cos{\theta_{1}}##, here ##\theta_{1}## is the angle of scattering of the incident particle ##m_{1}##, and the ##e_{1}## is the energy of particle ##m_{1}##. Finally we can derive the equation for scattering angle ##\theta_{1}##:

$$\cos{\theta_{1}} = \frac{e_{1}'(e_{1} + m_{2}) - e_{1}m_{2} - m_{1}^{2}}{p_{1}p_{1}'}$$

In Landau's book The Classical Theory of Fields, section 13, a function of ##\theta_{1}## is derived as follows if the rest mass of incident particle ##m_{1}## is larger than the mass of another particle ##m_{2}##:

$$\sin{\theta_{1\, max}} = \frac{m_{2}}{m_{1}}$$

But how could we derive this result?

## The Attempt at a Solution

If we don't consider the relativity, an equation for ##\theta_{1}## can be derived:

$$\cos{\theta_{1}} = \frac{(1 + \alpha) \beta^{2} + 1 - \alpha}{2\beta}$$

where ##\alpha = \frac{m_{2}}{m_{1}}## and ##\beta = \frac{v_{1}'}{v_{1}}##. Since ##\alpha## is a constant, we can find the extreme of ##\theta_{1}## by treating ##\beta## as an independent variable.:

$$\frac{\mathrm d}{\mathrm d \beta}\cos{\theta_{1}} = 0$$

From above equation we obtain ##\cos{\theta_{1\, min}} = \frac{2\beta}{1 + \beta^{2}}## for ##\alpha = \frac{1 - \beta^{2}}{1 + \beta^{2}}##. Finally we get ##\sin{\theta_{1}}_{max} = \frac{m_{2}}{m_{1}}##.

But if we are considering the relativity, the equation for ##\theta_{1}## is hard to simplify. How can I resolve this? Thank you for the advice.

kalok87 said:
$$\cos{\theta_{1}} = \frac{e_{1}'(e_{1} + m_{2}) - e_{1}m_{2} - m_{1}^{2}}{p_{1}p_{1}'}$$
But if we are considering the relativity, the equation for ##\theta_{1}## is hard to simplify. How can I resolve this? Thank you for the advice.
It is messy, but it will work out. It might be helpful to write the above equation as $$\cos{\theta_{1}} = \frac{ae_{1}' -b}{p_{1}\sqrt{e_{1}'^2-m_1^2}}$$where ##a## and ##b## are certain constants. Show that the derivative of the right hand side equals zero when ##e_{1}' = \large \frac{a m_1^2}{b}##.

## 1. What is scattering angle in relativistic kinematics?

Scattering angle in relativistic kinematics is the angle between the initial and final directions of a particle after it has undergone a scattering event.

## 2. How is scattering angle calculated in relativistic kinematics?

Scattering angle is calculated using the Lorentz transformation equations, which take into account the relativistic effects of time dilation and length contraction.

## 3. What is the importance of scattering angle in relativistic kinematics?

Scattering angle is important in understanding the dynamics of particles in high-energy collisions. It can provide information about the properties of the particles involved and the nature of the interaction.

## 4. How does scattering angle differ in classical and relativistic kinematics?

In classical kinematics, scattering angle is calculated using Newtonian mechanics and does not take into account the effects of special relativity. In relativistic kinematics, the scattering angle is calculated using the Lorentz transformation equations, which account for the relativistic effects of high speeds.

## 5. Can scattering angle be measured experimentally?

Yes, scattering angle can be measured experimentally using detectors such as particle accelerators and detectors. These experiments allow for the precise measurement of scattering angle and can provide valuable information for understanding the fundamental properties of matter.

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