Question 12.9.Q3: Several nuclear reactions, all endothermic and based on part......

(a) Q value of a nuclear reaction is defined as the difference between \sum_{i, \text { before }} M_i c^2, the sum of rest energies M_i c^2 of reactants (typically, the projectile and target) before the reaction and \sum_{i, \text { after }} M_i c^2, the sum of rest energies of reaction products after the reaction. In short, Q value with the rest energy method is determined as follows

Q=\sum_{i, \text { before }} M_i c^2 \sum_{i, \text { after }} M_i c^2         (12.303)

Alternatively, Q value of a nuclear reaction is defined as the difference between \sum_{i, \text { after }} E_{\mathrm{B}}(i), the sum of binding energies E_B(i) of reaction products after the reaction and \sum_{i, \text { before }} E_{\mathrm{B}}(i), the sum of binding energies of reactants (typically, the projectile and target) before the reaction. In short, Q value with the binding energy method is given as

Q=\sum_{i, \text { after }} E_{\mathrm{B}}(i)-\sum_{i, \text { before }} E_{\mathrm{B}}(i) .           (12.304)

Thus, two methods are in use for determining a nuclear reaction Q value: (1) rest energy method and (2) binding energy method, and both methods should provide identical result. Each nuclear reaction possesses a characteristic Q value that can be either positive (Q > 0), zero (Q = 0), or negative (Q < 0).

For Q > 0, the nuclear reaction is called exothermic (or exoergic) and results in release of energy.
For Q = 0, the nuclear reaction is termed elastic and no energy is released or
absorbed.
For Q < 0, the nuclear reaction is called endothermic (or endoergic) and, to take place, it requires an energy transfer from the projectile to the target.

An exothermic reaction can proceed spontaneously; an endothermic reaction, on the other hand, cannot take place unless the projectile has a kinetic energy exceeding a minimum energy referred to as threshold energy E_{thr}. In general, threshold kinetic energy \left(E_K\right)_{thr} is related to Q value by the following expression (T5.15)

\left(E_{\mathrm{K}}\right)_{\mathrm{thr}}=-Q\left[1+\frac{m_{\text {projectile }} c^2}{M_{\text {target }} c^2}\right]          (12.305)

where m_{projectile}\ and\ M_{target} are rest masses of the projectile and target, respectively. As a result of photon rest mass m_γ being zero, we note that threshold energy \left(E_\gamma^{\mathrm{PN}}\right)_{\mathrm{thr}} for a photonuclear reaction in which a photon plays the role of projectile is equal to −Q or we can say that the absolute value of Q is equal to threshold energy, i.e., \left(E_\gamma^{\mathrm{PN}}\right)_{\mathrm{thr}}=|Q| .
Alternatively, threshold kinetic energy \left(E_K\right)_{thr} of an endothermic nuclear reaction (or minimum energy that a projectile must possess in order to trigger an endothermic nuclear reaction) can be derived using the relativistic invariant E^2-p^2 c^2=\operatorname{inv} (T5.9) to get the following expression (T5.13)

\left(E_{\mathrm{K}}\right)_{\mathrm{thr}}=\frac{\left[\sum_{i, \text { after }} M_i c^2\right]^2-\left[M_{\text {target }} c^2+m_{\text {projectile }} c^2\right]^2}{2 M_{\text {target }} c^2},         (12.306)

where \sum_{i, \text { after }} M_i c^2 stands for a sum of rest energies of reaction products after the nuclear reaction. Note that (12.305) is derived from (12.306) making an assumption that Q \ll M_{\text {target }} c^2 \text {. }

(b) We now determine Q value of the { }_{42}^{100} \mathrm{Mo}(\gamma, \mathrm{n}){ }_{42}^{99} \mathrm{Mo} photonuclear reaction, first using the rest energy method of (12.303) and then the binding energy method of (12.304):

(1) Rest energy method:

(2) Binding energy method:

Components i of the photonuclear reaction are defined in Fig. 12.17 and the parameters M_ic^2 of (12.307) and E_B(i) of (12.308) are listed in Appendix A. As evident from (12.307) and (12.308), the two methods for calculation of Q value of photonuclear reaction { }_{42}^{100} \mathrm{Mo}(\gamma, \mathrm{n}){ }_{42}^{99} \mathrm{Mo} yield identical results of −8.29 MeV. A negative Q value indicates that the reaction is endothermic, which means that the reaction cannot run spontaneously, rather, energy must be supplied for reaction to occur. Usually energy for endothermic reactions is supplied in the form of kinetic energy of the projectile when the projectile is a particle. In the case of photonuclear reactions the photon plays the role of projectile and energy is supplied in the form of photon energy hν.

Threshold energy \left(E_\gamma^{\mathrm{PN}}\right)_{\mathrm{thr}} \text { of } 8.29 \mathrm{MeV} \text { for the }{ }_{42}^{100} \mathrm{Mo}(\gamma, \mathrm{n}){ }_{42}^{99} \mathrm{Mo} photonuclear reaction is relatively high and to get a photon spectrum with such a high maximum energy requires a high-energy electron accelerator that generates high-energy bremsstrahlung x rays through bombarding a suitable thick target with electrons of kinetic energy exceeding 8.29 MeV. The electron linac also should produce a high intensity bremsstrahlung beam, because the cross section for the photonuclear reaction is relatively small.

(c) Q value of nuclear reaction { }_{40}^{96} \mathrm{Zr}(\alpha, \mathrm{n}){ }_{42}^{99} \mathrm{Mo} is determined using: (1) rest energy method of (12.303) and (2) binding energy method of (12.304):

(1) Rest energy method:

(2) Binding energy method:

Since Q value is negative, the nuclear reaction is endothermic and is triggered by energy supplied in the form of kinetic energy of the α particle projectile. Minimum energy \left(E_{\mathrm{K}}^\alpha\right)_{\mathrm{thr}} to trigger the reaction is referred to as threshold energy and is from (12.305) calculated as

We can also calculate threshold energy of the α particle projectile directly from (12.306) as follows

As expected, (12.311) and (12.312) give identical result, confirming 5.34 MeV as threshold kinetic energy \left(E_{\mathrm{K}}^\alpha\right)_{\mathrm{thr}} that a α particle must exceed to trigger the nuclear reaction { }_{40}^{96} \mathrm{Zr}(\alpha, \mathrm{n}){ }_{42}^{99} \mathrm{Mo} .

(d) Q value of nuclear reaction { }_{42}^{100} \mathrm{Mo}(\mathrm{p}, 2 \mathrm{n}){ }_{43}^{99 \mathrm{~m}} \mathrm{Tc} is determined using: (1) rest energy method of (12.303) and (2) binding energy method of (12.304):

(1) Rest energy method:

(2) Binding energy method:

Since Q value is negative, the nuclear reaction is endothermic and is triggered by energy supplied in the form of kinetic energy of the proton projectile. Both the rest energy method and the binding energy method give almost identical results with an average Q value of −7.7129 MeV. Minimum energy \left(E_{\mathrm{K}}^{\mathrm{p}}\right)_{\mathrm{thr}} to trigger the reaction is referred to as threshold energy and is from (12.305) calculated as

We can also calculate threshold energy of the proton projectile \left(E_{\mathrm{K}}^{\mathrm{p}}\right)_{\mathrm{th}} directly from (12.316)

As expected, (12.314) and (12.316) give identical result, confirming 7.79 MeV as threshold kinetic energy \left(E_{\mathrm{K}}^{\mathrm{p}}\right)_{\mathrm{thr}} that a proton must exceed to trigger the nuclear reaction { }_{42}^{100} \mathrm{Mo}(\mathrm{p}, 2 \mathrm{n}){ }_{43}^{99 \mathrm{~m}} \mathrm{Tc} .

(e) Q value of neutron activation reaction { }_{42}^{100} \mathrm{Mo}(\mathrm{n}, 2 \mathrm{n}){ }_{42}^{99} \mathrm{Mo} calculated with: (1) the rest energy method of (12.303) and (2) the binding energy method of (12.304) is obtained as follows

(1) Rest energy method:

(2) Binding energy method:

Since Q value is negative, the neutron activation reaction is endothermic and is triggered by energy supplied in the form of kinetic energy of the neutron projectile. This means that thermal neutrons cannot be used to trigger this nuclear reaction; however, fast neutrons with kinetic energy exceeding threshold kinetic energy will be suitable. We now use (12.305) to determine the threshold kinetic energy \left(E_{\mathrm{K}}^{\mathrm{n}}\right)_{\mathrm{thr}} that a fast neutron must posses to be able to trigger the nuclear reaction

We can also calculate threshold energy of the neutron projectile \left(E_{\mathrm{K}}^{\mathrm{n}}\right)_{\mathrm{thr}} directly from (12.306)

Both (12.319) and (12.320) give the same result, confirming that threshold kinetic energy the neutron projectile in neutron activation reaction { }_{42}^{100} \mathrm{Mo}(\mathrm{n}, 2 \mathrm{n}){ }_{42}^{99} \mathrm{Mo} must possess is 8.37 MeV and suggesting that fast neutrons from a machine accelerating deuterons that bombard a light nuclear target, such as tritium or carbon, may be suitable for this purpose.