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problems statement:
1. a nucleus of mass m initially at rest absorbs a gamma ray (photon) and is excited to a higher energy state such that its mass now is 1.01m, find the energy of the incoming photon needed to carry out this excitation.
2. A moving radioactive nucleus of known mass M emits a gamma ray in the forward direction and drops to its stable nonradiactive state of known mass m.
Find the energy E_A of the incoming nucleus such that the resulting mass m nucleus is at rest. The unknown energy E_c of the outgoing gamma ray should not appear in the answer.
attempt at solution
1.well, for the first question i think this is fairly simple:
from conservation of 4-momentum we have before 4-momentum is:(mc,0) after
(E/c+E_ph/c,P) so we have : (mc)^2=(E/c+E_ph/c)^2-P^2=(E/c)^2-P^2+2EE_ph/c^2+(E_ph/c)^2=(1.01mc)^2+2EE_ph/c^2+(E_ph/c)^2 where (E/c)^2-P^2=(1.01mc)^2, here I am kind of stuck with E which is not given, any hints?
2.for the second the answer in the book is E_A=((M^2+m^2)/2m)c^2
but i don't get it, here's my attempt to solve it:
the before 4 momentum is (E_A/c,P) after: (E_c/c,0)+(mc,0)=(E_c/c+mc,0)
which by the square of the momentums we get that:
(E_A/c)^2-P^2=(E_c/c+mc)^2=(Mc)^2 but I am not given P so I am kind of stuck here again, i thought perhaps calculate it in the rest frame of M which means that the before is:
(E_A/c,0) the after is (E_c/c,0)+(E/c,-P) but still don't get far with it, any help is appreciated, thanks in advance.
1. a nucleus of mass m initially at rest absorbs a gamma ray (photon) and is excited to a higher energy state such that its mass now is 1.01m, find the energy of the incoming photon needed to carry out this excitation.
2. A moving radioactive nucleus of known mass M emits a gamma ray in the forward direction and drops to its stable nonradiactive state of known mass m.
Find the energy E_A of the incoming nucleus such that the resulting mass m nucleus is at rest. The unknown energy E_c of the outgoing gamma ray should not appear in the answer.
attempt at solution
1.well, for the first question i think this is fairly simple:
from conservation of 4-momentum we have before 4-momentum is:(mc,0) after
(E/c+E_ph/c,P) so we have : (mc)^2=(E/c+E_ph/c)^2-P^2=(E/c)^2-P^2+2EE_ph/c^2+(E_ph/c)^2=(1.01mc)^2+2EE_ph/c^2+(E_ph/c)^2 where (E/c)^2-P^2=(1.01mc)^2, here I am kind of stuck with E which is not given, any hints?
2.for the second the answer in the book is E_A=((M^2+m^2)/2m)c^2
but i don't get it, here's my attempt to solve it:
the before 4 momentum is (E_A/c,P) after: (E_c/c,0)+(mc,0)=(E_c/c+mc,0)
which by the square of the momentums we get that:
(E_A/c)^2-P^2=(E_c/c+mc)^2=(Mc)^2 but I am not given P so I am kind of stuck here again, i thought perhaps calculate it in the rest frame of M which means that the before is:
(E_A/c,0) the after is (E_c/c,0)+(E/c,-P) but still don't get far with it, any help is appreciated, thanks in advance.
