Of Particular Significance

Invariant Mass

Temporary version:

If a heavy particle A decays to two very lightweight particles B and C, what you need to do to find the mass of A is measure the energy E_B and E_C, and the angle theta_BC between the directions of motion of the two particles; then compute

2 E_B E_C (1 – cos theta_BC),

take the square root, and divide by the speed of light squared.  The answer is called the invariant mass of particles B and C, and it equals the mass of particle A.  Thus you can determine the mass of particle A simply by knowing E_B, E_C and theta_BC.

If particles B and C aren’t that lightweight, a more precise answer is to replace 2 E_B E_C (1 – cos theta_BC) with

(m_B)^2 c^4 + (m_C)^2 c^4 + 2 E_B E_C (1 – v_B v_C/c^2 cos theta_BC),

where m_B and v_B are the mass and velocity of particle B (and similarly for C), and c is the speed of light.  Again the square root of this quantity, divided by c^2, is the invariant mass, and this will be the mass of the parent particle A.

5 Responses

1. mohsen lutephy says:

Referring to my book is suitable. “Pythagorean physics: fields are not energy carrier and force is transformer”.
In this book we see that invariant mass is a reason to know that force is not energy carrier suppose it is transformer to transfer static mass to kinetic energy and inverse.

2. Ah, here it is:

$2 E_B E_C (1-\cos{\theta}_{BC})$

This was rendered using the following HTML source:

latex 2 E_B E_C (1-\cos{\theta}_{BC})\$

3. And it looks like it tried to parse, but I had a typo. Oops….

4. FYI, you might find it useful to know that WordPress supports inline LaTeX. For example, the first expression above could be rendered by coding it as

$2 E_B E_C (1 – cos \Theta_BC)$

(Frankly, I won’t know until I click the post button whether the above LaTeX code will get formatted with the LaTeX engine or if the source will be shown. Let’s see….)

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