Problem book for the course Nuclear Physics. Educational edition

d)− + → + ;

e)4He + 3H → 7Li + ;

f)4He + 7Li → 10В + ;

j)4He + 222Rn → 226Ra.

2.Properties of Nuclei

The binding energy of a nucleus is given by

mass, ( , ) is the mass of an atom with an ordinal numberand a mass number .

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where a.m.u. is the atomic mass unit.

The radius of the nucleus (in the approximation of a spherical nucleus of uniform density) is

where 0 ≈ 1.4 fm.

2.1.Estimate the density of matter in the nucleus.

2.2.Estimate the volume density of electric charge in the nucleus.

2.3.Evaluate the concentration of nucleons in the nucleus.

2.4.Estimate the average distance between nucleons in the nucleus.

2.5.Based on the definition of the atomic mass unit, find the value a.m.u. 2 in MeV.

2.6.Derive the formula for the calculation of the energy that is necessary – (or that will be released – ) to divide the

nucleus ( , ) into two fragments ( , ) and ( − , − ), if the mass excesses of original and daughter nuclei are given.

2.7. Find the binding energy and binding energy per nucleon for the following nuclei: 4He, 5He, 9B, 12C, 27Al, 56Fe, 107Ag,

236U.

2.8. Calculate the binding energy of a neutron in the nuclei: 4He, 5He, 9B, 12C, 27Al, 56Fe, 107Ag, 236U.

2.9. Calculate the binding energy of a proton in the nuclei: 4He, 5He, 9B, 12C, 27Al, 56Fe, 107Ag, 236U.

2.10. Calculate the binding energy of an -particle in the nuclei: 12C, 27Al, 56Fe, 107Ag, 236U, 210Po, 226Ra.

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2.11. Calculate the energy that will be released in the decay of 236U to 143Ba, 90Kr and 3 neutrons. Estimate the

amount of heat that will be released in the decay of 1 kg of uranium. Determine the amount of anthracite (specific heat of combustion is 34 MJ/kg) that must be burned to obtain the same amount of heat.

2.12.Calculate the energy required to split the nucleus 16O into two identical fragments.

2.13.Calculate the energy required to split the nucleus 8Be into two identical fragments.

2.14.Calculate the energy required to split the nucleus 12С into three -particles.

2.15.Calculate the excitation energy that the 235U nucleus

will get after the absorption of a neutron with negligible kinetic energy. Calculate the excitation energy that the 6Li

nucleus will get after the absorption of a thermal neutron with negligible kinetic energy.

2.16.What is the size of objects that Rutherford could study if he scattered -particles with energy = 5 MeV on them?

2.17.During the scattering of electrons with = 750 MeV on 40Ca nuclei, in the diffraction pattern minima are observed

at the angles min = 18, 31 and 48 . Determine the radius of the 40Ca nucleus.

2.18. By the difference in the binding energies of the mirror 3H and 3He nuclei estimate the distance between the protons

in the 3He nucleus.

2.19. Using the table with binding energies, calculate the energy released in each proton-proton cycle reaction:

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1)+ → 2H + + + ;

2)2H + → 3He;

3)3He + 3He → 4He + 2 .

3.Models of nuclei

The binding energy of a nucleus is given by the formula of Weizsacker

( , ) = − 2/3 − ( − 1)−

1/3

− ( − 2 )2 + −3/4. (3.1)

Parameter values: = 15.6 MeV, = 17.2 MeV,

= 0.72 MeV, = 23.6 MeV,

+34 MeV – even - even,

= 0 – odd,

−34 MeV – odd - odd.

The intrinsic quadrupole moment of a uniformly charged ellipsoid is

(3.2)

where and are the long and short ellipsoid semiaxes;

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The relation between experimental and intrinsic quadrupole moment of a nucleus is given by

Fig. 3.1. Schematic representation of single-particle levels in a spherically symmetric potential

The shell model of the nucleus is given in Fig. 3.1. The rules for spins and parities in the ground state of the

nucleus are

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where , , , , , refer to the total and orbital angular momentum of the odd nucleon (neutron, proton).

The magnetic moment of the nucleus (maximal projection)

is

where is the gyromagnetic factor, is the nuclear spin, is the nuclear magneton.

The gyromagnetic factor of a nucleon in the state , is

= − 1/2; , are spin and orbital gyromagnetic factors. For the proton = 2 · 2.793, = 1. For the neutron

= 2 · (−1.913), = 0.

3.1.Find the relation between and for -stable nuclei.

3.2.Using the formula derived in 3.1, predict the character of the activity of the following -active nuclei: 107Cd, 133Xe,

141Ce, 163Er.

3.3. Find the opt for the nuclei with = 20, 21, 116, 117,

123, 124, 128 and 129, at which the mass of the nucleus will be minimal.

3.4.Plot the dependence 2( ) for the nuclei with = 116 and 117 in the region [ opt − 5; opt + 5].

3.5.Using the Weizs¨acker formula, calculate the energy released in the fission of 238U into two equal fragments.

3.6.Find the critical value of 2/ at which the fission of the

nucleus into two equal fragments will be possible. In the third term in equation (3.1) approximate ( − 1) ≈ 2.

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3.7. Find the critical value of 2/ at which the nucleus

becomes unstable. In the third term in equation (3.1) approximate ( − 1) ≈ 2.

3.8. Compare the binding energy and the binding energy per nucleon calculated by using the liquid drop model for 4He,

12C, 27Al, 56Fe, 107Ag, 236U with tabular values (cf. the answer

for 2.7).

3.9. Derive a formula for calculating the energy of the Coulomb repulsion of protons ( , ), where is the nucleus

charge, is the nucleus radius. Consider the distribution of

charge in the volume of the nucleus as uniform and the nucleus itself spherical.

3.10.Assuming that the difference in the binding energies of mirror nuclei is determined only by the difference in the

Coulomb repulsion energies in these nuclei (cf. the answer for 3.9), calculate the radii of mirror nuclei 23Na, 23Mg.

3.11.Calculate the spin, parity and magnetic moment of the following nuclei: 3H, 7Li, 13C, 25Mg. Compare the results with

tabular values.

3.12. Calculate the spin, parity and magnetic moment of the nuclei: 29Si, 39K, 45Sc, 63Cu. Compare the results with tabular

values.

3.13. Estimate the possible values of the spins and parity of nuclei: 2D, 6Li, 10B, 14N, 22Na, 26Al. Compare the results with

tabular values.

3.14.Derive the formula 3.2 for the calculation of the electric quadrupole moment of a uniformly charged ellipsoid.

3.15.Observed electrical quadrupole moments of nuclei 36Cl and 55Mn are -0.018 and +0.33 b. What forms do these nuclei

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have? What are the intrinsic electric quadrupole moments of these nuclei?

3.16. Observed electrical quadrupole moments of nuclei 39K

and 63Cu are +0.0585 and -0.211 b. What are the deformation

parameters of these nuclei?

3.17. In the Fermi-gas model, it is assumed that the density of the state of nucleons is / = 2 · 4 2 /(2 ~)3, where

is the momentum of nucleon, = 4 03 /3 is the volume of nucleus. Calculate the maximum possible momentum.

3.18.Calculate the maximum kinetic energy of nucleons and estimate the depth of the potential well for nucleons in the nucleus in the frame of the Fermi gas model.

3.19.Calculate the average kinetic energy of nucleons in the nucleus in the frame of the Fermi gas model .

4.Interaction of charged particles with matter

The Rutherford formula describes the macroscopic differential cross section of scattering

where is the thickness of the target, is the concentration of the target nuclei, is the scattering angle (deviation of the particle from the initial direction), is the total number of incident particles, is the number of particles scattered at an angle in the element of the solid angle , is the kinetic energy of the incident particles, and are the charge

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of the target nuclei and incident particles (in units of electron charge), respectively, and 0 = 8.85 · 10−12 C2/(N·m2) is the

electric constant.

Ionization losses of heavy particles in MeV/(g/cm 2) are described by the formula

where is the Avogadro constant, is the classical electron radius, the particle charge (in units ), is the charge of the nuclei of matter (in units ), is the atomic number of matter, = / , is the velocity of the incoming

√

particle, = 1/ 1 − 2, is the average ionization potential (tab. P. 1.8), max is the maximum energy that can be transferred to a free electron in a single collision, is the a term that takes into account the density effect.

where is the mass of the incoming particle.

For the nonrelativistic case ( 2 / 1) max ≈ 2 2 2 2, the second and third terms in square brackets in the formula (4.2) can be neglected and then the calculation is carried out according to the following formula:

where / = 4 2 2/ = 0.307075 MeV/(g/cm2) for

= 1 g/mol.

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Empirical dependences of the path of monoenergetic-particles and protons ′ (cm) in the air on the kinetic energy

(MeV):

The average path of -particles (mg/cm2) in a medium with mass number :

Empirical dependences of the path of monoenergetic electrons (g/cm2) in aluminium (and in almost any

substance, if the electron energy losses are mainly due to ionization) on the kinetic energy (MeV) are described by

The electron path (g/cm2) in a substance with a chargeand a mass number is related to the path in aluminium

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