When 2 particles of size \(n=1\) coalesce 2 photons \(E_{1\rightarrow2}\) are emitted, and when 2 particles of sizes \(n=1\) and \(n=2\) coalesce 2 different photons \(E_{1\rightarrow 3}\) and \(E_{2\rightarrow 3}\) are emitted.
This is not the simple scheme where emissions as the result of hops from energy level to energy level has equal intensity, but in the example above, \(E_{1\rightarrow 2}\) has twice the intensity of \(E_{1\rightarrow 3}\) and \(E_{2\rightarrow 3}\).
The plot below shows ((x+1)^(1/3)-x^(1/3))/(((x+1)*x)^1/3), ((x+2)^(1/3)-x^(1/3))/(((x+2)*x)^1/3) and ((x+3)^(1/3)-x^(1/3))/(((x+3)*x)^1/3).
where particle of size \(n=1\), \(n=2\) and \(n=3\) coalesce with particle of size \(n=x\).
We see that the highest energy transition occurs with \(E_{1\rightarrow n=large}\), when a basic particle \(a_{\psi\,c}\) (ie. \(n=1\)) coalesces with a large particle \(n\rightarrow 77\). Two photons are released \(E_{1\rightarrow n=large}\) and \(E_{large\rightarrow large+1}\).
Emission occurs in bands as \(n=x\) increases and such bands narrows with increasing \(n=x\).
In the graph above, the top most three horizontal lines maroon, red and blue correspond to \(x=1\). The next band of maroon, red and blue correspond to \(x=2\). Each band progressively narrows as \(x\) increases.
As \(x\) increases, all graph approaches asymptotically to zero, ie as \(x\) increases all emissions due to the coalescence of particles of various sizes, \(n\) approaches zero.
Good night.
