Let us take the example of 89Y(n,n')89mY reaction. In the calculation of Q value we require mass of reactants and products. Both are same here. Then how to calculate the Q value of this reaction or how to calculate the threshold energy of this type of reactions?
According to Einstein's mass-energy equivalence, 89Y and 89mY are not the same, as 89mY is in an excited state.
The threshold (in the center-of-mass frame) of the inelastic scattering reaction corresponds to the energy of the nuclear level on which the neutron is scattered. For example 89Y(n,n')89Y* threshold is 909 keV in the center-of-mass (and about 919 keV = 909 keV *(89+1)/89 in the lab frame) for neutron scattering on the first level of 89Y* (where * means excited). See the link below for more information on nuclear levels.
In the case of 89Y, it happens that the first level at 909 keV is a metastable state with a half life of 15 sec. Hence, the threshold is the same for both reactions 89Y(n,n')89mY and 89Y(n,n')89Y*.
http://www.nndc.bnl.gov/nudat2/
To add to Mr. Dupont's point, strictly speaking, the incoming neutron needs to have energy equivalent to the difference between the group state and excited state of the resultant metastable state.
There are a number of these meta states depending on the isotopes, and the threshold is the energy difference between the first excited state and the group state (in the CM frame).
Respected Sir,
The above reaction is an example of inelastic scattering and these reactions are characterized by the fact that even though there is no change in particles (a=b) there is a net decrease in the kinetic energy . The outgoing particle is same as the incident particle but it leaves with different kinetic energies.
So, in a nuclear reaction any net decrease in the rest mass is balanced by equal increase in the kinetic energy of the system. Consider a reaction:
a + X = Y + b
To conserve mass and energy we can write:
(Ka + mac2) + (KX + mXc2) = (KY + mYc2) + (Kb + mbc2)
So, Q value of the reaction is:
Q = (KY + Kb) - (KX + Ka)
= final K.E - initial K.E
or
Q = [(mX + ma) - (mY + mb)]c2
= initial rest mass energy - final rest mass energy
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