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6.6. NUCLEAR REACTIONS
- Binding energy of a nucleus: Eb = ZmH ± (A Z) mn— M, (6.6a)
where Z is the charge of the nucleus (in units of e), A is the mass number, mil, mn, and M are the masses of a hydrogen atom, a neutron, and an (^) atom corres- ponding to the given nucleus. In calculations the following formula is more convenient to use: Eb = ZAH -F (^) (A — Z)A, — A, (6.6b)
where AH, An , and A are the mass surpluses of a hydrogen atom, a neutron, and an atom corresponding to the given nucleus.
- Energy diagram of a nuclear reaction m M M* m' M' Q (6.6c) is illustrated in Fig. 6.12, where m--+M^ and^ m'+M'^ are the sums of rest masses of particles before and after the reaction, - f• and f- are the total kinetic ener- gies of particles before and after the reaction (in the frame of the centre of inertia), E* is the excitation energy of the transitional nucleus, Q is the energy of the reaction,^ E^ A^2 and E' are the binding energies of the par- ticles m and m' in the transitional nucleus,
T
1, 2, 3 are the energy levels of the transi- tional nucleus. filtM
- Threshold (minimum) kinetic energy of an incoming particle at which an endoer- gic nuclear reaction /77W
Tth —
m+M IQI (6.6d)
becomes possible; here m and M^ are the masses of the incoming particle and the target nucleus. 6.249. An alpha-particle with kinetic energy Ta= 7.0 MeV is scattered elastically by an initially stationary Li6nucleus. Find the kinetic energy of the recoil nucleus if the angle of divergence of the two particles is 0 = 60°. 6.250. A neutron collides elastically with an initially stationary deuteron. Find 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. Assuming the radius of a nucleus to be equal to (^) R =
= 0.13 VA pm, 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) 131° ( x, a) Be;
Fig. 6.12.
(b) 017 (d,^ n)^ x; (c) Na23 (p, x) Ne20; (d) x (p, n) Ar37. 6.254. Demonstrate that the binding energy of a nucleus with mass number A and charge Z can be found from 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 A127nucleus. 6.256. Making use of the tables of atomic masses, find: (a) the mean binding energy per one nucleon in 016nucleus; (b) the binding energy of a neutron and an alpha-particle in a B11nucleus; (c) the energy required for separation of an 016nucleus into four identical particles. 6.257. Find the difference in binding energies of a neutron and a proton in a B"nucleus. Explain why there is the difference. 6.258. Find the energy required for separation of a Ne20nucleus into two alpha-particles and a C12nucleus if it is known that the binding energies per one nucleon in Ne20, He4, and C12nuclei are equal to 8.03, 7.07, and 7.68 MeV respectively. 6.259. Calculate in atomic mass units the mass of (a) a Lib atom whose nucleus has the binding energy 41.3 MeV; (b) a C1° nucleus whose binding energy per nucleon is equal to 6.04 MeV. 6.260. The nuclei involved in the nuclear reaction A l + A -* —*A 3 + A 4 have the binding energies El , E,, E3, and E4. Find the energy of this reaction. 6.261. Assuming that the splitting of a U236nucleus liberates the energy of 200 MeV, find: (a) the energy liberated in the fission of one kilogram of U isotope, and the mass of coal with calorific value of 30 kJ/g which is equivalent to that for one kg of U235; (b) the mass of U235isotope split during the explosion of the atomic bomb with 30 kt trotyl equivalent if the calorific value of trotyl is 4.1 kJ/g. 6.262. What amount of heat is liberated during the formation of one gram of He4from deuterium H2? What mass of coal with calo- rific value of 30 kJ/g is thermally equivalent to the magnitude obtained? 6.263. Taking the values of atomic masses from the tables, calcu- late the energy per nucleon which is liberated in the nuclear reaction Lib -I- H2—4- 2He4. Compare the obtained magnitude with the energy per nucleon liberated in the fission of U235nucleus. 6.264. Find the energy of the reaction Li7 p -÷2He4 if the binding energies per nucleon in Li7and He4nuclei are known to be equal to 5.60 and 7.06 MeV respectively. 6.265. Find the energy of the reaction 1\114 (a, p) On if the kinetic energy of the incoming alpha-particle is T = 4.0 MeV and the
18*
6.278. How much, in per cent, does the threshold energy of gam- ma quantum exceed the binding energy of a deuteron (Eb = 2.2 MeV) in the reaction Y + H2 n p? 6.279. A proton with kinetic energy T = 1.5 MeV is captured by a deuteron H2. Find the excitation energy of the formed nucleus. 6.280. The yield of the nuclear reaction C'3(d, n)N" has maximum magnitudes at the following values of kinetic energy T, (^) of bombard- ing deuterons: 0.60, 0.90, 1.55, and 1.80 MeV. Making use 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 = 360 times after passing through a cadmium plate of thickness d = 0.50 mm. Determine the effective 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 cm. The effective cross-sections of interaction of deuterium and oxygen nuclei with thermal neutrons are equal to al= 7.0 b and a2--- 4.2 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 o- c, = 2.5 b and a8= 11 b respectively. Find the fraction of neutrons quitting the beam due to scattering if the thickness of the plate is (^) d = 0.50 cm. 6.284. The yield of a nuclear reaction producing radionuclides may be described in two ways: either by the ratio w of the number of nuclear 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; (b) the yield w of the reaction Li7(p, n)Be7if after irradiation of a lithium target by a beam of protons (over t = 2.0 hours and with beam current I = 10 p,A) the activity of Bel became equal to A = = 1.35.108dis/s and its half-life to T = 53 days. 6.285. Thermal neutrons fall normally on the surface of a thin gold foil consisting of stable Au197nuclide. The neutron flux density is J = 1.0.1010part./(s- cm2). The mass of the foil is in = 10 mg. The neutron capture produces beta-active Au188nuclei with half-life T = 2.7 days. The effective capture cross-section is a = 98 b. Find: (a) the irradiation time after which the number of Au187nuclei decreases by = 1.0%; (b) the maximum number of Aul" nuclei that can be formed dur- ing 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 k appears. Find the law
describing accumulation of that radionuclide N (t)^ per unit area
of the foil's surface. The neutron flux density is J,^ the number of
nuclei per unit area of the foil's surface is n, and the effective cross- section of formation of active nuclei is a. 6.287. A gold foil of mass m = 0.20 g was irradiated during = 6.0 hours by a thermal neutron flux falling normally on its
surface. Following i = 12 hours after the completion of irradiation
the activity of the foil became equal to A =^ 1.9.107dis/s. Find
the neutron flux density if the effective cross-section of formation of a radioactive nucleus is a --= 96 b, and the half-life is equal
to T = 2.7 days.
6.288. How many neutrons are there in the hundredth generation if the fission process starts with No= 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 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 MeV. 6.290. In a thermal reactor the mean lifetime of one generation
of thermal neutrons is ti = 0.10 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-fold.
6.7. ELEMENTARY PARTICLES
- Total energy and momentum of a relativistic particle: E = moc2 (^) T,pc =ITT (T 2rnoc2), (6.7a)
where T is the kinetic energy of the particle.
- When examining collisions of particles it pays to use the invariant: E2 —p2c2 =m8c4, (6.7b) where E and p are the total energy and the total momentum of the system prior to collision, mo is the rest mass of the formed particle.
- Threshold (minimal) kinetic energy of a particle m striking a stationary particle M and activating the endoergic reaction in M m1+ m2 + ... :
Tth= (rni+m2+ • • •)2— (m+M)2 c2,^ (6.7c) 2M where in, M, m1, m 2,... are the rest masses of the respective particles.
- (^) Quantum numbers classifying elementary particles: Q, electric charge, L, (^) lepton charge, B, baryon charge, T, (^) isotopic spin, T2, its projection, S, strangeness, S = 2(Q) — B, Y, hypercharge, Y = B + S.
