– Binding enery of a nucleus:
\[
E_{\mathrm{b}}=\mathbf{Z}_{\mathrm{n}}+(\boldsymbol{A}-\mathbf{Z}) \mathbf{m}_{\mathrm{n}}-\mathbf{M} \text {, }
\]
where $\boldsymbol{Z}$ is the charge of the necless (in units of $\boldsymbol{f}$ ), $\boldsymbol{A}$ is the mas number, $\mathrm{F}_{\mathrm{H}}$. $\mathrm{m}$, and $M$ are the mases of a bydrogen atom, a neutros, and an atom correr ponding to the civen nucleas.
In calculations the following formula is more cenvenient to ose:
\[
E_{\mathrm{b}}=2 \mathrm{~s}_{\mathrm{n}}+(A-z) \Delta_{\mathrm{s}}-\Delta_{\mathrm{s}}
\]
shere $\Delta_{\mathrm{H}} \cdot \Delta_{\mathrm{s}}$, and $\mathrm{s}^{\mathrm{are}}$ the mase surplase of a bydroges atem, a neutron, and an atom corresponding to the given nacleus.
– Energy diagran of a nuclear raction
\[
=+M \rightarrow M^{*} \rightarrow \mathbf{n}^{\prime}+M^{*}+Q
\]
is illustrated in Fie. 6.12, where $m+M$ and $\mathbf{n}^{\prime \prime}+M^{\prime}$ are the wums of rest mases of particles belors and after the raction, $\bar{F}$ and $P^{\prime}$ are the total kinetic enercies of particles belors and after the maction (in the lrame of the centre of inertia), $E^{*}$ is the excitation energy of the transitional nueleas, $Q$ is the energy of the reaction, $\boldsymbol{E}$ and $E^{\prime}$ are the binding energies of the portieles $m$ and $m$ ‘ in the transitional nueleus. J, 2, 3 are the energy levels of the trans: tional neeleus.
– Threshold (minimum) kinetic energy of as incoming particle at which as ebdoercie nuclear reaction
\[
T_{\mathrm{a}}=\frac{\mathrm{m}+\mathrm{M}}{M}|0|
\]
becomes posible; bere $\mathbf{m}$ and $\boldsymbol{M}$ are the manses of the incoming partlete and the target socleus.
Fig. 6.12.
6.269. An alpha-particle with kiaetic energy $T_{\mathrm{g}}=7.0 \mathrm{MeV}$ is scattered elastically by an initially stationary Li nucleus. Find the kinetic energy of the recoll nucleus if the angle of divergence of the two particles is $\theta=60^{\circ}$.
6.250. A neutron collides elastically with an initially stationary deuteron. Pind the fraction of the kinetic energy lost by the neutron
(a) in a head-on collision;
(b) in scattering at right angles.
6.251. Find the greatest possible angle through which a deuteron is scattered as a result of elastic collision with an initially stationary proton.
6.252. Asuming the radius of a nucleus to be equal to $R=$ $=0.13 \sqrt{A} p m$, where $A$ is its mass number, evaluate the density of nuclei and the number of nucleons per unit volume of the nucleus.
6.253. Write missing symbols, denoted by $x$, in the following nuclear reactions:
(a) $\mathrm{B}^{\text {in }}(x, a) B e^{*}$;
(b) $O^{n t}(d, n) x$;
(c) $\mathrm{Na}^{2}(p, x) \mathrm{Ne}^{\mathrm{an}}$;
(d) $x(p, n) \mathrm{Ar}^{3+}$.
6.254. Demonstrate that the binding energy of a nucleus with mass number $A$ and charge $Z$ can be found from $\mathrm{Eq}$. (6.6b).
6.255. Find the binding energy of a nucleus consisting of equal numbers of protons and neutrons and having the radius one and a half times smaller than that of $\mathrm{Al}^{\mathrm{x}}$ nucleus.
6.256. Making use of the tables of atomic masses, find:
(a) the mean binding energy per one aucleon in $0^{\text {is }}$ nucleus;
(b) the binding energy of a neutron and an alpha-particle in a $\mathrm{B}^{\text {in }}$ nucleus:
(c) the energy required for separation of an $0^{14}$ necleus inte four identical particles.
6.257. Find the difference in binding energies of a neutren and * proton in a $\mathbf{B}^{\mathrm{u}}$ nucleus. Explain why there is the difference.
6.258. Find the energy required for separation of a $\mathrm{Ne}^{20}$ nucleus Inte two alpha-particles and a $\mathrm{C}^{n}$ nucleus if it is known that the bisding energies per one nucleon in $\mathrm{Ne}^{\mathrm{H}}, \mathrm{He}^{4}$, and $\mathrm{C}^{\mathrm{H}}$ nuclei are equal to $8.00,7.67$, and $7.68 \mathrm{MeV}$ respectively.
6.259. Calculate in atemic mass units the mass of
(a) a L.l\” atom whose nucleus has the binding energy $41.3 \mathrm{MeV}$;
(b) a $\mathrm{C}^{\mathrm{i}}$ nucleus whese binding energy per nuelees is equal to $6.04 \mathrm{MeV}$.
6.250. The nuclei involved in the nelear reaction $A_{1}+A_{1} \rightarrow$ $\rightarrow A_{3}+A_{4}$ have the binding energles $E_{1}, E_{2}, E_{3}$ and $E_{6}$. Find the energy of this reaction.
6.251. Assuming that the splitting of a $U^{\text {2as }}$ nucleus liberates the energy of $200 \mathrm{MeV}$, find:
(a) the energy liberated in the fissien of one kilogram of $U^{\text {ra }}$ isotope, and the mass of coal with calorific value of $30 \mathrm{~kJ} / \mathrm{g}$ which is equivalent to that for one $\mathrm{kg}$ of $\mathrm{U}^{\mathrm{rs}}$;
(b) the mass of $\mathrm{U}^{\mathrm{ms}}$ isotope split during the explosion of the atomic bomb with $30 \mathrm{kt}$ trotyl equivalent if the calorific value of trotyl is $4.1 \mathrm{~kJ} / \mathrm{g}$.
6.262. What amount of heat is liberated during the formation of one gram of $\mathrm{He}^{4}$ from deuterium $\mathrm{H}^{2}$ ? What mass of coal with calorific value of $30 \mathrm{~kJ} / \mathrm{g}$ is thermally equivalent to the magnitude obtained?
6.253. Taking the values of atomic masses from the tables, calculate the energy per nucleon which is liberated in the nuclear reaction $\mathrm{Li}^{+}+\mathrm{H}^{3} \rightarrow 2 \mathrm{He}$. Compare the obtained magnitude with the energy per nucleon liberated in the fission of $U^{\text {tas }}$ nucleus.
6.284. Find the energy of the reaction $\mathrm{LI}^{7}+\mathrm{P} \rightarrow 2 \mathrm{He}^{4}$ If the binding energies per nucleon in $\mathrm{Li}^{7}$ and $\mathrm{He}^{4}$ nuclei are known to be equal to 5.60 and $7.06 \mathrm{MeV}$ respectively.
6.265. Find the energy of the reaction $\mathrm{N}^{14}(\alpha, p) \mathrm{O}^{11}$ if the kinetic eaergy of the incoming alpha-particle is $T_{\mathrm{o}}=4.0 \mathrm{MeV}$ and the proton outgoing at an angle $\theta=60^{\circ}$ to the motion direction of the alpha-particle has a kinetic energy $T_{p}=2.09 \mathrm{MeV}$.
6.266. Making use of the tables of atomic inasses, determine the energies of the following reactions:
(a) $\mathrm{Li}^{\prime}(p, n) \mathrm{Be}^{?}$ :
(b) $\mathrm{Be}^{*}(n, \gamma) \mathrm{Be}^{\mathrm{in}}$;
(c) L.f’ $(a, n) B^{10}$;
(d) $\mathrm{O}^{14}(d, a) \mathrm{N}^{14}$.
6.257. Making use of the tables of atomic masses, find the velocity with which the products of the reaction $\mathrm{B}^{-10}(\mathrm{n}, a) \mathrm{Li}^{\prime}$ come apart; the reaction proceeds via interaction of very slow neutrons with stationary boron nuclei.
6.268. Protons striking a stationary lithium target activate a reaction Li’ $(p, n) \mathrm{Be}^{\prime}$. At what value of the proton’s kinetic energy can the resulting neutron be stationary?
6.359. An alpha particle with kisetic eaergy $T=5.3 \mathrm{MeV}$ initiates a nuclear reaction $\mathrm{Be}^{*}(a, n) \mathrm{C}^{\mathrm{u}}$ with energy yield $Q=$ $=+5.7 \mathrm{MeV}$. Find the kinetic energy of the neutron outgoing at right angles to the motion direction of the alpha-particle.
6.270. Protons with kinetic energy $T=1.0 \mathrm{MeV}$ striking a lithium target induce a nuclear reaction $\mathrm{p}+\mathrm{Li}^{+} \rightarrow 2 \mathrm{He} \mathrm{e}^{4}$. Find the kinetic energy of each alpha-particle and the angle of their divergence provided their motion directions are symmetrical with respect to that of incoming protons.
6.271. A particle of mass $m$ strikes a stationary nuclens of mass $M$ and activates an endoergic reaction. Demonstrate that the threshold (minimal) kinetic energy required to initiate this reaction is defined by $\mathrm{Eg}_{\text {. }}$ (6.6d).
6.272. What kinetic energy must a proton possess to split a deuteron $\mathrm{H}^{\mathrm{i}}$ whose binding energy is $E_{\mathrm{b}}=2.2 \mathrm{MeV}$ ?
6.273. The irradiation of lithium and beryllium targets by a monoergic stream of protons reveals that the reaction $\mathrm{Li}^{i}(p, n) \mathrm{Be}^{7}-$ $-1.65 \mathrm{MeV}$ is initiated whereas the reaction $\mathrm{Be}^{*}(p, n) \mathrm{B}^{*}-1.85 \mathrm{MeV}$ does not take place. Find the possible values of kinetic energy of the protons.
6.274. To activate the reaction $(n, a)$ with stationary $\mathrm{B}^{\mathrm{H}}$ nuclei, neutrons must have the threshold kinetic energy $T_{t h}=4.0 \mathrm{MeV}$. Find the energy of this reaction.
6.275. Calculate the threshold kinetic energies of protons required to activate the reactions $(p, n)$ and $(p, d)$ with $\mathrm{Li}^{\prime}$ nuclei.
6.276. Using the tabular values of atomic masses, find the threshold kinetic energy of an alpha particle required to activate the auclear reaction $\mathrm{Li}^{7}(a, n) \mathrm{B}^{10}$. What is the velocity of the $\mathrm{B}^{10}$ nueleus in this case?
6.277. A neutron with kinetie energy $T=10 \mathrm{MeV}$ activates a nuelear reaction $\mathrm{Cu}^{\mathrm{a}}(\mathrm{n}, \mathrm{a}) \mathrm{Be}$ whose threshold is $T_{\mathrm{s}}=6.17 \mathrm{MeV}$. Find the kinetic energy of the alpha-particles outgoing at right angles to the incoming neutrons’ direction.
276
6.278. How much, in per cent, does the threshold energy of gamma quantem exceed the binding energy of a deuteron $\left(E_{\mathrm{b}}=2.2 \mathrm{MeV}\right)$ in the reaction $\gamma+\mathrm{H}^{\prime} \rightarrow n+p$ ?
6.279. A proton with kinetic energy $T=1.5 \mathrm{MeV}$ is captured by a deuteron $\mathrm{H}^{2}$. Find the excitation energy of the formed nucleus.
6.280. The yield of the nuelear reaction $\mathrm{C}^{-1}(d, n) \mathrm{N}^{44}$ has maximum magnitudes at the following values of kinetic energy $T_{\text {, }}$ of bombarding deuterons: $0.60,0.90,1.55$, and $1.80 \mathrm{MeV}$. Making ase of the table of atomic masses, find the corresponding energy levels of the transitional nucleus through which this reaction proceeds.
6.281. A narrow beam of thermal neutrons is attenuated $\eta=$ – 360 times after passing through a cadmium plate of thickness $\boldsymbol{d}=0.50 \mathrm{~mm}$. Determine the eflective cross-section of interaction of these neutrons with cadmium nuclei.
6.282. Determine how many times the intensity of a narrow beam of thermal neutrons will decrease after passing through the heavy water layer of thickness $d=5.0 \mathrm{~cm}$. The eflective cross-sections of interaction of deuterium and oxygen nuclei with thermal neutrons are equal to $\sigma_{1}=7.0 \mathrm{~b}$ and $\sigma_{1}=4.2 \mathrm{~b}$ respectively.
6.283. A narrow beam of thermal neutrons passes through a plate of iron whose absorption and scattering effective cross-sections are equal to $\sigma_{s}=2.5 \mathrm{~b}$ and $\sigma_{s}-11 \mathrm{~b}$ respectively. Find the fraction of neutrons quitting the beam due to scattering if the thickness of the plate is $d=0.50 \mathrm{~cm}$.
6.284. The yield of a nuclear reaction producing radionuclides may be described in two ways: either by the ratio of the number of nuelear reactions to the number of bombarding particles, or by the quantity $k$, the ratio of the activity of the formed radionuclide to the number of bombarding particles. Find:
(a) the half-life of the formed radionuclide, assuming $w$ and $k$ to be known: a lithium target by a beam of protons (over $t=2.0$ hours and with beam current $\boldsymbol{I}=10 \mu \mathrm{A}$ ) the activity of Be’ became equal to $\boldsymbol{A}=$ $=1.35-10^{\circ}$ dis/s and its hall-life to $T=53$ days.
6.285. Thermal neutrons fall normally on the surface of a thin zold foil consisting of stable $\mathrm{Au}^{\text {in }}$ nuclide. The neutron flux density is $J=1.0 \cdot 10^{10}$ part. $/\left(\mathrm{s} \cdot \mathrm{cm}^{2}\right)$. The mass of the foil is $m=10 \mathrm{mg}$. The neutron capture produces beta-active $\mathrm{Au}^{\text {tin }}$ nuclei with half-life $\boldsymbol{r}=2.7$ days. The effective capture cross-section is $\sigma=98 \mathrm{~b}$. Find:
(a) the irradiation time after which the number of $\mathrm{Au}^{\mathrm{m}}$ nuclei decreases by $\eta=1.0 \%$;
(b) the maximum aumber of $\mathrm{Au}^{\mathrm{in}}$ nuclei that cas be formed during protracted irradiation.
6.286. A thin foil of certain stable isotope is irradiated by thermal neutrons falling normally on its surface. Due to the capture of neutrons a radionuclide with decay constant $\lambda$ appears. Find the law describing accumulation of that radionuclide $N(f)$ per unit area of the foil’s surface. The neutron fax density is $J$, the number of nelel per unit area of the foil’s surface is $n$, and the effective crosssection of formation of active nuclei is $\sigma$.
6.287. A gold foil of mass $m=0.20 \mathrm{~g}$ was irradiated during $t=6.0$ hours by a thermal neutron fux falling normally on its surface. Following $\tau=12$ hours after the completion of irradiation the activity of the foil became equal to $A=1.9 \cdot 10^{\circ} \mathrm{dis} / \mathrm{s}$. Find the aeutron flax density if the effective eross-section of formation of a radioactive nueleus is $a-96 \mathrm{~b}$, and the half-life is equal to $T=2.7$ days.
6.289. How many neutrons are there in the hundredth generation if the fission process starts with $N_{0}=1000$ neutrons and takes place in a medium with multiplication constant $k=1.05$ ?
6.289. Find the number of neutrons generated per unit time in a uranium reactor whose thermal power is $P=100 \mathrm{MW}$ if the average number of neutrons liberated in each nuclear splitting is $v=2.5$. Each splitting is assumed to release an energy $E$ $=200 \mathrm{MeV}$.
6.290. In a thermal reactor the mean lifetime of one generation of thermal neutrons is $\tau=0.10 \mathrm{~s}$. Assuming the multiplication constant to be equal to $k=1.010$, find:
(a) how many times the number of neutrons in the reactor, and consequently its power, will increase over $t=1.0$ min;
(b) the period $T$ of the reactor, i.e. the time period over which its power increases e-told.