Finding the recoil energy of an excited atom when it de-excites

[sadhu]

I know that i am commiting some serious errors in this , but just can't notice them
suppose we have to find the recoil energy of an excited atom when it de excites

suppose the total energy of the process is E

an atom will emit a photon on returning to normal state . energy of the photon is... e

recoil velocity of the atom is v;
mass of the atom is M

Mv+e/c=0....momentum conservation...c=velocity of light.initialy the atomis at rest

Mv*v/2 + e=E...energy conservation


from first equation

v=-e/(Mc)...(i)

thus recoil energy must come out by substituting v

Me*e/(2*M*M*c*c)

e*e/(2Mc*c)

substituting this in energy conservation equation

e+e*e/(2*Mc*c)=E
e+e*e/(2*Mc*c)-E=0

solving for e;
$$\Delta$$=1+2E/(M*c*c)

whats is the solution ??

 

[chbeatbama]

The recoil energy (K) is given as:

$$K = E^{2}_{photon}/2mc^{2}$$

where mc² describes the rest mass of the particle. mc² will be of the order 10⁹ to 10¹¹ eV for atoms and nuclei.