Centre of mass frame in nuclear reactions?

• AntiElephant
In summary, we can transfer to the centre of mass frame before a nuclear reaction using the formula v_c = \frac{\sum m_i v_i}{\sum m_i} and note that the total energy before the reaction is E_{CM} = E_{masses~before} + E_{kinetic~before~in~CM}. After the reaction, the total energy is E_{CM} = E_{masses~after} + E_{kinetic~after~in~CM}. This is possible because momentum is conserved and the relativistic formula for the speed of the center of mass takes into account changes in mass. This means that the frame in which the momentum is zero remains the same before and after the reaction
AntiElephant
I'm slightly confused as to how we discuss centre of mass frames in nuclear reactions.

We have the reaction a(A,B)b. Before the reaction we can transfer to the centre of mass frame

$v_c = \frac{\sum m_i v_i}{\sum m_i}$

and note that the total energy before the reaction is

$E_{CM} = E_{masses~before} + E_{kinetic~before~in~CM}$

And the total energy after is

$E_{CM} = E_{masses~after} + E_{kinetic~after~in~CM}$

The problem I'm having to understand is that these two before/after energies are often equated, but the centre of mass frame is not a constant since the masses are not constant in a nuclear reaction, so $v_c$ changes since the denominator $\sum m_i$ changes (numerator remains the same due to conservation of momentum).

Am I wrong here? If I were to guess why we can do this, it's because the change in masses is so small that we can assume the centre of mass frame is constant before/after the reaction. But then this discussion breaks down when we're considering only light particles and the fractional mass change is larger, right?

Am I wrong here? If I were to guess why we can do this, it's because the change in masses is so small that we can assume the centre of mass frame is constant before/after the reaction. But then this discussion breaks down when we're considering only light particles and the fractional mass change is larger, right?

Momentum is conserved. If you work in a frame where total momentum is zero before the interaction then total momentum will also be zero after the interaction.

The formula:

$v_c = \frac{\sum m_i v_i}{\sum m_i}$

is a non-relativistic approximation. It will not, in general, correctly tell you the relative velocity of the frame in which total momentum is zero.

The relativistic formula for the speed of the center of mass is

$v_c = \frac{\sum m_i \gamma_i v_i}{\sum m_i \gamma_i},$

where $$\gamma_i = \frac{1}{\sqrt{1-\frac{v_i^2}{c^2}}}$$

dauto said:
The relativistic formula for the speed of the center of mass is

$v_c = \frac{\sum m_i \gamma_i v_i}{\sum m_i \gamma_i},$

where $$\gamma_i = \frac{1}{\sqrt{1-\frac{v_i^2}{c^2}}}$$

That seems not to account for the possibility of massless particles moving at light speed. May I assume that one would simply divide total momentum by total energy (using units in which c=1) to cover that case?

jbriggs444 said:
That seems not to account for the possibility of massless particles moving at light speed. May I assume that one would simply divide total momentum by total energy (using units in which c=1) to cover that case?

Yes.

So the frame in which the momentum is zero, is the same before and after a nuclear reaction, because the actual formula takes relativistic mass changes into account?

Yes.

What is the Centre of Mass (COM) frame in nuclear reactions?

The Centre of Mass (COM) frame in nuclear reactions is a reference frame where the total momentum of the system is zero. It is a frame of reference that takes into account the motion of all the particles involved in the reaction.

Why is the COM frame important in nuclear reactions?

The COM frame is important in nuclear reactions because it simplifies the calculations and allows for a better understanding of the underlying physics. In this frame, the motion of the particles is described in a more straightforward manner, making it easier to analyze and interpret the results of the reaction.

How is the COM frame determined in nuclear reactions?

The COM frame in nuclear reactions is determined by considering the total momentum of all the particles involved. The frame is chosen such that the total momentum of the system is equal to zero. This means that the velocities of the particles are adjusted in such a way that their combined momentum is zero.

What is the role of the COM frame in studying nuclear structure?

The COM frame plays a crucial role in studying nuclear structure as it allows for a more accurate description of the internal motion of the particles. By taking into account the motion of the particles in the COM frame, scientists can better understand the forces that hold the nucleus together and the properties of the particles within it.

How does the COM frame affect the cross-section of nuclear reactions?

The COM frame affects the cross-section of nuclear reactions by influencing the energy and angle at which the particles collide. In the COM frame, the relative velocity of the particles is reduced, resulting in a lower collision energy and a narrower range of angles. This can affect the probability of a reaction occurring and the types of products that are formed.

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