Fundamental particles and fundamental interactions

In the physics of the microworld, all particles are divided into two classes: fermions and bosons. Fermions are particles with half-integer spins, bosons are particles with integer spins. Spin is the minimum value of the angular momentum that a particle can have. Spins and other moments of impulses are measured in units. For particles with non-zero mass, the spin is equal to the angular momentum of the particle in the coordinate system associated with itself. The particle spin value J, indicated in the tables, is the maximum value of the projection of the angular momentum vector onto the selected axis, divided by .
Fundamental particles are particles that, according to modern concepts, do not have an internal structure. In nature, there are 12 fundamental fermions (with spin 1/2 in units) are given in Table 1. The last column of Table 1 is the electric charges of fundamental fermions in units of the electron charge e.

Fundamental fermions
Interactions				Generations			Charge Q/e
			leptons	v e	ν μ	ν τ	0
				e	μ	τ	-1
			quarks	u	c	t	+2/3
				d	s	b	-1/3

12 fundamental fermions correspond to 12 antifermions.
The interaction of particles is carried out due to 4 types of interactions: strong , electromagnetic , weak and gravitational . The quanta of the corresponding fields are fundamental bosons : gluons; gamma quantum; W + , W - , Z - bosons and graviton .

Fundamental Interactions
Interaction	field quantum	Radius cm	Constant order	Example of manifestation
Strong	gluon	10 -13	1	nucleus, hadrons
electromagnetic	γ	∞	10 -2	atom, gamma transitions
Weak	W,Z	10 -16	10 -6	weak decays of particles, -decay
gravitational	graviton	∞	10 -40	Gravity

Strong interaction quanta are neutral massless gluons. Fundamental fermions, between which a strong interaction is realized - quarks - are characterized by a quantum number "color", which can take 3 values. Gluons have 8 varieties of “color” charges.
Quanta of electromagnetic interaction are gamma quanta . γ-quanta have zero rest mass. Electromagnetic interactions involve fundamental particles occupying the last three rows in Table 1, i.e. charged leptons and quarks. Since quarks in a free state are not observed, but are part of hadrons, i.e. baryons and mesons, all hadrons, along with strong interactions, also participate in electromagnetic interactions.
Weak interaction quanta , in which all leptons and all quarks participate, are W and Z bosons. There are both positive W + bosons and negative W - ; Z-bosons are electrically neutral. The masses of W and Z bosons are large - more than 80 GeV/c 2 . A consequence of the large masses of intermediate bosons of the weak interaction is a small - in comparison with the electromagnetic constant - the constant of the weak interaction. The neutrino participates only in weak interactions.
Gluons, γ-quantum, W and Z bosons are fundamental bosons . The spins of all fundamental bosons are 1.
Gravitational interactions practically do not appear in particle physics. for example, the intensity of the gravitational interaction of two protons is ~10 -38 of the intensity of their electromagnetic interaction.
Table split. 1 on generations justified by the fact that the world around us is almost entirely built from particles of the so-called. first generation (least massive). Particles of the second and, especially, third generations can be detected only at high interaction energies. For example, the t-quark was discovered at the FNAL collider, during the collision of protons and antiprotons with energies of 1000 GeV.
The first two rows in table 5.1 are leptons - fermions that do not participate in strong interactions. Leptons are electrically neutral neutrinos (and antineutrinos) of three types - particles with masses much smaller than the mass of an electron. Neutrinos are only involved in weak interactions. The second row is occupied by the electron, muon and taon - charged structureless particles participating in both weak and electromagnetic interactions.
The third and fourth lines contain 6 quarks(q) - structureless particles with fractional electric charges. In a free state, these particles are not observed, they are part of the observed particles - hadrons .
Natural Phenomena Manifested at Particle Energies<100 МэВ, могут быть практически полностью объяснены взаимодействием фундаментальных частиц 1-го поколения. 2-е поколение фундаментальных частиц проявляется при энергиях порядка сотен МэВ. Для исследования 3-го поколения фундаментальных частиц строят ускорители высоких энергий (E >100 GeV).

Wavelengths and Particle Energies

The objects that are studied by nuclear and particle physics ("subatomic physics") have much smaller characteristic dimensions than atoms and molecules. (This fact is also a consequence of the fact that the structure of objects of subatomic physics is determined by strong interactions)
The study of the structure of any body requires "microscopes" with wavelengths smaller than the dimensions of the objects under study.
The wavelength of both electromagnetic radiation and any particle is related to the momentum by a known relationship (for particles with non-zero rest mass introduced by de Broglie):

where p is the momentum of the particle, h is Planck's constant.
The characteristic linear dimensions of even the "largest" objects of subatomic physics - atomic nuclei with a large number of nucleons A - are of the order of about 10 -12 cm. An experimental study of objects with such dimensions requires the creation of high-energy particle beams.
One of the goals of this workshop is to calculate the energies of accelerated particles, which can be used to study the structure of nuclei and nucleons. Before embarking on such calculations, it is necessary to familiarize yourself with the basic constants that will often be used in further calculations, as well as with the units of measurement of physical quantities accepted in subatomic physics.

Units of subatomic physics

Energy - 1 MeV = 1 MeV = 10 6 eV = 10 -3 GeV = 1.6 . 10 -13 J.
Mass - 1 MeV/c 2 and 1 u\u003d M at (12 C) / 12 \u003d 1.66. 10 -24 years
Length - 1 fm \u003d 1 fm \u003d 10 -13 cm \u003d 10 -15 m.

Important formulas of relativistic physics

In subatomic physics, especially in high energy physics, the system of units ( Heaviside system ) , in which ћ = 1 and c = 1. In this system, the formulas of relativistic physics have a simpler and more convenient form.

Atomic nuclei and their constituent particles are very small, so it is inconvenient to measure them in meters or centimeters. Physicists measure them in femtometers (fm). 1 fm = 10 -15 m, or one quadrillionth of a meter. This is a million times smaller than a nanometer (the typical size of molecules). The size of a proton or neutron is just about 1 fm. There are heavy particles that are even smaller.

The energies in the world of elementary particles are also too small to be measured in Joules. Instead, use the unit of energy electron-volt (eV). 1 eV, by definition, is the energy that an electron will acquire in an electric field when passing through a potential difference of 1 volt. 1 eV is approximately equal to 1.6 10 -19 J. An electron volt is convenient for describing atomic and optical processes. For example, gas molecules at room temperature have a kinetic energy of about 1/40 of an electron volt. Light quanta, photons, in the optical range have an energy of about 1 eV.

Phenomena occurring inside nuclei and inside elementary particles are accompanied by much greater changes in energy. Here, megaelectronvolts are already used ( MeV), gigaelectronvolts ( GeV) and even teraelectronvolts ( TeV). For example, protons and neutrons move inside nuclei with a kinetic energy of several tens of MeV. The energy of proton-proton or electron-proton collisions, in which the internal structure of the proton becomes noticeable, is several GeV. In order to give birth to the heaviest particles known today - top quarks - it is required to push protons with an energy of about 1 TeV.

A correspondence can be established between the distance scale and the energy scale. To do this, we can take a photon with a wavelength L and calculate its energy: E= c h/L. Here c is the speed of light, and h- Planck's constant, a fundamental quantum constant, equal to approximately 6.62 10 -34 J s. This relation can be used not only for the photon, but also more widely, when estimating the energy needed to study matter on a scale L. In "microscopic" units, 1 GeV corresponds to a size of about 1.2 fm.

Einstein's famous formula E 0 = mc 2 , mass and rest energy are closely related. In the world of elementary particles, this relationship manifests itself in the most direct way: when particles with sufficient energy collide, new heavy particles can be born, and when a heavy particle at rest decays, the mass difference passes into the kinetic energy of the resulting particles.

For this reason, particle masses are also commonly expressed in electronvolts (more precisely, in electronvolts divided by the speed of light squared). 1 eV corresponds to a mass of only 1.78 10 -36 kg. An electron in these units weighs 0.511 MeV, and a proton 0.938 GeV. Many even heavier particles have been discovered; the record holder so far is the top quark with a mass of about 170 GeV. The lightest of the known particles with non-zero mass - neutrinos - weigh only a few tens of meV (million electron volts).

So, an electron is an elementary particle charged negatively. Electrons make up the matter that makes up everything that exists. We also note that the electron is a fermion, which indicates its half-integer spin, and also has a dual nature, because it can be both a particle of matter and a wave. If its property is considered as a mass, then its first essence is implied.

The mass of an electron has the same nature as any other macroscopic object, but everything changes when the speeds of movement of material particles become close to the speed of light. In this case, relativistic mechanics comes into force, which is a superset of classical mechanics and extends to cases of motion of bodies at high speeds.

So, in classical mechanics, the concept of "rest mass" does not exist, because it is believed that the mass of a body does not change during its movement. This circumstance is also confirmed by experimental facts. However, this fact is only an approximation for the case of low velocities. Slow speeds here mean speeds that are much smaller than the speed of light. In a situation where the speed of a body is comparable to the speed of light, the mass of any body changes. Electron is no exception. Moreover, this regularity has sufficient significance for microparticles. This is justified by the fact that it is in the microcosm that such high speeds are possible at which mass changes become noticeable. Moreover, on the scale of the microcosm, this effect occurs continuously.

Increase in electron mass

So, when particles (electron) move with relativistic speeds, their mass changes. Moreover, the greater the speed of the particle, the greater its mass. As the value of the speed of the particle tends to the speed of light, its mass tends to infinity. In the case when the velocity of the particle is equal to zero, the mass becomes equal to a constant, which is called the rest mass, including the rest mass of the electron. The reason for this effect lies in the relativistic properties of the particle.

The fact is that the mass of a particle is directly proportional to its energy. The same, in turn, is directly proportional to the sum of the kinetic energy of the particle and its energy at rest, which contains the rest mass. Thus, the first term in this sum causes the mass of the moving particle to increase (as a consequence of the change in energy).

The numerical value of the rest mass of the electron

The rest mass of an electron and other elementary particles is usually measured in electron volts. One electron volt is equal to the energy expended by an elementary charge to overcome a potential difference of one volt. In these units, the rest mass of an electron is 0.511 MeV.

1. The kinetic energy of an electron is 1.02 MeV. Calculate the de Broglie wavelength of this electron.

Given: E k \u003d 1.02 MeV \u003d 16.2 10 -14 J, E 0 \u003d 0.51 MeV \u003d 8.1 10 -14 J.

Find λ.

Solution. The de Broglie wavelength is determined by the formula , (1) where λ is the wavelength corresponding to a particle with momentum ; is Planck's constant. By the condition of the problem, the kinetic energy of an electron is greater than its rest energy: E k = 2E 0 , (2) therefore, a moving electron is a relativistic particle. The momentum of relativistic particles is determined by the formula

or, taking into account relation (2),

Substituting (4) into (1), we obtain

.

Making calculations, we get

Answer: λ = .

2. Using the Heisenberg uncertainty relation, show that the nuclei of atoms cannot contain electrons. Consider the core radius to be 10~18 cm.

Given: R i \u003d 10 -15 m, \u003d 6.62 10 -34 J s.

Solution. The Heisenberg uncertainty relation is expressed by the formula

where is the uncertainty of the coordinate; - momentum uncertainty; is Planck's constant. If the uncertainty of the coordinate is taken equal to the radius of the nucleus, i.e., then the uncertainty of the electron momentum is expressed as follows: . Since then and . Let us calculate the uncertainty of the electron velocity:

Comparing the obtained value with the speed of light in vacuum c = 3·10 8 m/s, we see that , and this is impossible, therefore, nuclei cannot contain electrons.

3. The electron is in an infinitely deep one-dimensional potential well 1 nm wide in an excited state. Determine the minimum value of the electron energy and the probability of finding an electron in the interval of the second energy level.

Given: .

Find: , .

In quantum mechanics, information about the motion of particles is obtained from the wave function (T-function), which reflects the distribution of particles or systems over quantum states. These particles are characterized by discrete values ​​of energy, momentum, angular momentum; i.e. - function is a function of the state of particles in the microworld. Solving the Schrödinger equation, we obtain that for the considered case the eigenfunction has the form

, (1)

where = 1, 2, 3, ...; - particle coordinate; - hole width. Graphs of eigenfunctions are shown in fig. 17. According to the de Broglie relation, two projections of momentum that differ in sign correspond to two plane monochromatic de Broglie waves propagating in opposite directions along the axis. As a result of their interference, standing de Broglie waves arise, which are characterized by a stationary distribution along the axis of the oscillation amplitude. This amplitude is the wave function (x), the square of which determines the probability density of the electron being at the point with coordinate . As can be seen from fig. 17, for the value = 1, half the length of the standing de Broglie wave fits on the width of the well, for = 2 - the entire length of the standing de Broglie wave, etc., i.e., in the potential well there can only be de Broglie waves, the length of which satisfies condition

Thus, an integer number of half-waves must fit on the width of the well: . (2)

The total energy of a particle in a potential well depends on its width and is determined by the formula , (3) where is the particle mass; - 1, 2, 3... . The electron will have the minimum energy value at the minimum value , i.e. at =1. Consequently,

Substituting numerical values, we get

The probability that an electron will be found in the interval from to is equal to . The desired probability is found by integration in the range from 0 to:

Using the relation , we calculate the integral under the condition that the electron is in the second energy level:

4. The limiting wavelength K α - series of characteristic X-ray radiation for some element is 0.0205 nm. Define this element.

Given: .

Find Z.

Solution. From Moseley's formula

,

where λ is the wavelength of the characteristic radiation, equal to (c is the speed of light, v is the frequency corresponding to the wavelength λ); R is the Rydberg constant; Z is the serial number of the element from which the electrode is made; - shielding constant; - the number of the energy level to which the electron passes; - the number of the energy "level from which the electron passes (for K α - series \u003d 1, \u003d 2, \u003d 1), we find Z:

Ordinal number 78 has platinum.

Answer: Z = 78 (platinum).

5. A narrow monochromatic beam of γ-rays with a wavelength of 0.775 pm falls on the surface of the water. At what depth will the intensity of γ-rays decrease by 100 times!

Given: λ \u003d 0.775 pm \u003d 7.75 10 -13 m, \u003d 100.

Find

Solution. The weakening of the intensity of γ-rays is determined from the formula , (1) whence , where is the intensity of the incident beam of γ-rays; - their intensity at depth ; - coefficient of linear attenuation. Solving equation (1) with respect to , we find

To determine , we calculate the energy of γ-quanta , where is Planck's constant; c is the speed of light in vacuum. Substituting numerical values, we get

According to the graph of the dependence of the linear attenuation coefficient of γ-rays on their energy (Fig. 18), we find = 0.06 cm -1. Substituting this value of q into formula (2), we find

.

6. Determine how many nuclei in 1 g of radioactive decay within one year.

Given:

Find

Solution. To determine the number of atoms contained in 1 g, we use the relation

where is the Avogadro constant; - the number of moles contained in the mass of a given element; M is the molar mass of the isotope. There is a relation between the molar mass of an isotope and its relative atomic mass: M = 10 -3 A kg/mol. (2) For any isotope, the relative atomic mass is very close to its mass number A, i.e. for this case M = 10 -3 ·90 kg/mol = 9·10 -2 kg/mol.

Using the law of radioactive decay

where is the initial number of undecayed nuclei at the moment; N is the number of undecayed nuclei at the moment; λ is the radioactive decay constant, let's determine the number of decayed nuclei within 1 year:

Considering that the radioactive decay constant is related to the half-life by the relation λ = 1n 2/T, we obtain

Substituting (1), taking into account (2), into expression (5), we have

After performing calculations using formula (6), we find

Answer:

7. Calculate in megaelectron-volts the energy of a nuclear reaction:

Is energy released or absorbed in this reaction?

Solution. Nuclear reaction energy , (1), where is reaction mass defect; c is the speed of light in vacuum. If expressed in amu, then formula (1) will take the form . The mass defect is

Since the number of electrons before and after the reaction is the same, instead of the values ​​of the masses of the nuclei, we will use the values ​​of the masses of neutral atoms, which are given in the reference tables:

; ; ;

The reaction proceeds with the release of energy, since > 0:

Answer: \u003d 7.66 MeV.

8. Copper has a face-centered cubic lattice. The distance between the nearest copper atoms is 0.255 nm. Determine the density of copper and the lattice parameter.

Given: d \u003d 0.255 nm \u003d 2.55 10 -10 m, \u003d 4, M \u003d b3.54 10 -3 kg / mol.

Find: r, a.

Solution. We find the density of a copper crystal by the formula , (1) where M is the molar mass of copper; - molar volume. It is equal to the volume of one unit cell multiplied by the number of unit cells contained in one mole of the crystal: . (2)

The number of elementary cells contained in one mole of a crystal consisting of identical atoms can be found by dividing the Avogadro constant by the number of atoms per one elementary cell: . (3) For a cubic face-centered lattice = 4. Substituting (3) into (2), we obtain

Substituting (4) into (1), we finally have

.

The distance between the nearest neighboring atoms is related to the lattice parameter a by a simple geometric relation (Fig. 19):

Substituting the numerical values ​​into the calculation formulas, we find

Answer: ; .

9. Crystalline aluminum weighing 10 g is heated from 10 to 20 K. Using the Debye theory, determine the amount of heat required for heating. The characteristic Debye temperature for aluminum is 418 K. Assume that condition T is satisfied.

Given: = 0.01 kg, = 10 K, = 20 K, = 418 K, = 27 10 -3 kg / mol.

Solution. The amount of heat required to heat aluminum from temperature to , we will calculate by the formula

where is the mass of aluminum; c is its specific heat capacity, which is related to the molar heat capacity by the relation . Taking this into account, formula (1) can be written as

(2)

According to Debye's theory, if the condition T is satisfied, the molar heat capacity is determined by the limiting law

,

where R \u003d 8.31 J / (mol K) is the molar gas constant; is the characteristic Debye temperature; T - thermodynamic temperature. Substituting (3) into (2) and performing integration, we obtain

Substituting numerical values, we find

Answer: \u003d 0.36 J.

CONTROL WORK No. 6 (5)

1. Determine the kinetic energy of the proton and electron, for which the de Broglie wavelengths are equal to 0.06 nm.

2. The kinetic energy of a proton is equal to its rest energy. Calculate the de Broglie wavelength for such a proton.

3. Determine the de Broglie wavelengths of an electron and a proton that have passed the same accelerating potential difference of 400 V.

4. A proton has a kinetic energy equal to the rest energy. How many times will the de Broglie wavelength of a proton change if its kinetic energy is doubled?

5. The kinetic energy of an electron is equal to its rest energy. Calculate the de Broglie wavelength for such an electron.

6. The mass of a moving electron is 2 times the rest mass. Determine the de Broglie wavelength for such an electron.

7. Using Bohr's postulate, find the relationship between the de Broglie wavelength and the length of a circular electron orbit.

8. What kinetic energy must an electron have in order for the de Broglie wavelength of an electron to be equal to its Compton wavelength.

9. Compare the de Broglie wavelengths of an electron passing through a potential difference of 1000 V, a hydrogen atom moving at a speed equal to the mean square speed at a temperature of 27 ° C, and a 1 g ball moving at a speed of 0.1 m / s.

10. What kinetic energy must a proton have in order for the de Broglie wavelength of the proton to be equal to its Compton wavelength.

11. The average lifetime of a π° meson is 1.9·10 -16 s. What should be the energy resolution of the device with which it is possible to register the π° meson?

12. In a photograph taken with a cloud chamber, the width of the electron track is 0.8·10 -3 m. Find the uncertainty in finding its speed.

13. The average kinetic energy of an electron in an unexcited hydrogen atom is 13.6 eV. Using the uncertainty relation, find the smallest error with which you can calculate the coordinate of an electron in an atom.

14. An electron moving at a speed of 8·10 6 m/s is registered in a bubble chamber. Using the uncertainty relation, find the error in measuring the electron velocity if the diameter of the formed bubble in the chamber is 1 µm.

15. Show that for a particle whose position uncertainty (λ is the de Broglie wavelength), the uncertainty of its velocity is equal in order of magnitude to the velocity of the particle itself.

16. The average lifetime of a π+ meson is 2.5·10 -8 s. What should be the energy resolution of an instrument that can detect the π+ meson?

17. Based on the uncertainty relation, estimate the size of the atomic nucleus, assuming that the minimum energy of a nucleon in the nucleus is 8 MeV.

18. Using the uncertainty relation, estimate the energy of an electron in the first thieves orbit in a hydrogen atom.

19. Using the uncertainty relation, show that electrons cannot be in the nucleus. Take the linear dimensions of the nucleus equal to 5.8·10 -15 m. Take into account that the specific binding energy is on average 8 MeV/nucleon.

20. An atom emitted a photon with a wavelength of 0.550 microns. Duration of radiation 10 not. Determine the maximum error with which the wavelength of radiation can be measured.

21. A particle in a potential well wide is in an excited state. Determine the probability of finding a particle in the interval 0< < на третьем энергетическом уровне.

22. Calculate the ratio of the probabilities of finding an electron at the first and second energy levels of a one-dimensional potential well, the width of which is , in the interval 0< < .

23. Determine at what width of a one-dimensional potential well the discreteness of the electron energy becomes comparable with the energy of thermal motion at a temperature of 300 K.

24. An electron is in the ground state in a one-dimensional potential well with infinitely high walls, the width of which is 0.1 nm. Determine the momentum of an electron.

25. An electron is in the ground state in a one-dimensional potential well with infinitely high walls, the width of which is 0.1 nm. Determine the average pressure force exerted by the electron on the walls of the well.

26. An electron is in a one-dimensional potential well with infinitely high walls, the width of which is 1.4 10 -9 m. Determine the energy emitted during the transition of an electron from the third energy level to the second.

27. An electron is in a one-dimensional potential well with infinitely high walls, the width of which is 1 nm. Determine the smallest difference in the energy levels of an electron.

28. Determine at what temperature the discreteness of the energy of an electron located in a one-dimensional potential well, the width of which is 2·10 -9 m, becomes comparable with the energy of thermal motion.

29. A particle in a potential well wide is in an excited state. Determine the probability of finding a particle in the interval 0< < на втором энергетическом уровне

30. Determine the width of a one-dimensional potential well with infinitely high walls, if an energy of 1 eV is emitted during the transition of an electron from the third energy level to the second?

31. The boundary value of the wavelength of the K-series of the characteristic X-ray radiation of a certain element is 0.174 nm. Define this element.

32. Find the limiting wavelength of the K-series of X-rays from a platinum anticathode.

33. At what minimum voltage do lines of the K α -series appear on an X-ray tube with an iron anticathode?

34. What is the smallest potential difference that must be applied to an X-ray tube with a tungsten anticathode so that all K-series lines are in the tungsten emission spectrum?

35. The limiting wavelength of the K-series of the characteristic X-ray radiation of a certain element is 0.1284 nm. Define this element.

36. Determine the minimum wavelength of bremsstrahlung X-rays if voltages of 30 kV are applied to the X-ray tube; 75 kV,

37. The smallest wavelength of bremsstrahlung radiation obtained from a tube operating under a voltage of 15 kV is 0.0825 nm. Calculate Planck's constant from this data.

38. During the transition of an electron in a copper atom from the M-layer to the L-layer, rays with a wavelength of 12 10 -10 m are emitted. Calculate the screening constant in the Moseley formula.

39. The largest wavelength of the K-series of characteristic X-ray radiation is 1.94 10 -10 m. What material is the anti-cathode made of?

40. A voltage of 45,000 V is applied to an x-ray tube used in medicine for diagnostics. Find the boundary of the continuous x-ray spectrum.

41. The half-life of radioactive argon is 110 minutes. Determine the time during which 25% of the initial number of atoms decays.

42. Calculate the thickness of the half-absorption layer of lead through which a narrow monochromatic beam of γ-rays with an energy of 1.2 MeV passes.

43. The half-life of an isotope is approximately 5.3 years. Determine the decay constant and the average life of the atoms of this isotope.

44. A narrow monochromatic beam of γ-rays falls on an iron screen, the wavelength of which is 0.124 10 -2 nm. Find the thickness of the layer of half absorption of iron.

45. What is the energy of γ-rays if, when passing through a layer of aluminum 5 cm thick, the radiation intensity is weakened by 3 times?

46. ​​The half-life is 5.3 years. Determine what fraction of the initial number of nuclei of this isotope decays after 5 years,

48. In a year, 60% of some original radioactive element decayed. Determine the half-life of this element.

49. A narrow beam of γ-rays with an energy of 3 MeV passes through a screen consisting of two plates: lead 2 cm thick and iron 5 cm thick. Determine how many times the intensity of γ-rays will change when passing through this screen.

50. Determine the decay constant and the number of radon atoms that decayed during the day, if the initial mass of radon is 10 g.

51. Calculate the mass defect, the binding energy of the nucleus and the specific binding energy for the element.

52. Calculate the energy of a thermonuclear reaction

53. What element does it turn into after three α-decays and two β-transformations?

54. Determine the maximum energy of β-particles in the β-decay of tritium. Write the decay equation.

55. Determine the maximum kinetic energy of an electron emitted during the β-decay of a neutron. Write the decay equation.

56. Calculate the mass defect, binding energy and specific binding energy for the element.

57. A nucleus consisting of 92 protons and 143 neutrons ejected an α-particle. What nucleus was formed as a result of α-decay? Determine the mass defect and the binding energy of the formed nucleus.

58. In the thermonuclear interaction of two deuterons, two types of formations are possible: 1) and 2). Determine the thermal effects of these reactions.

59. How much energy is released when one proton and two neutrons combine to form an atomic nucleus?

60. Calculate the energy of a nuclear reaction

61. Molybdenum has a body-centered cubic crystal lattice. The distance between the nearest neighboring atoms is 0.272 nm. Determine the density of molybdenum.

62. Using the Debye theory, calculate the specific heat of iron at a temperature of 12 K. Take the characteristic Debye temperature for iron 467 K. Assume that condition T is satisfied.

63. Gold has a face-centered cubic crystal lattice. Find the density of gold and the distance between the nearest atoms if the lattice parameter is 0.407 nm.

64. Determine the impurity electrical conductivity of germanium, which contains indium with a concentration of 5 10 22 m -3 and antimony with a concentration of 2 10 21 m -3. The electron and hole mobilities for germanium are 0.38 and 0.18 m2/(V-s), respectively.

65. At room temperature, the density of rubidium is 1.53 g/cm3. It has a body-centered cubic crystal lattice. Determine the distance between the nearest neighboring rubidium atoms.

66. An ingot of gold weighing 500 g is heated from 5 to 15 K. Determine, using the Debye theory, the amount of heat required for heating. The characteristic Debye temperature for gold is 165 K. Assume that condition T is satisfied.

67. Determine the impurity electrical conductivity of germanium, which contains boron with a concentration of 2 10 22 m -3 and arsenic with a concentration of 5 10 21 m -3. The electron and hole mobilities for germanium are respectively 0.38 and 0.18 m 2 /(V·s).

68. Find the lattice parameter and the distance between the nearest neighboring atoms of silver, which has a face-centered cubic crystal lattice. The density of silver at room temperature is 10.49 g/cm3.

69. Using the Debye theory, find the molar heat capacity of zinc at a temperature of 14 K. The characteristic Debye temperature for zinc is 308 K. Assume that condition T is satisfied.

70. Determine the impurity electrical conductivity of silicon, which contains boron with a concentration of 5 10 22 m -3 and antimony with a concentration of 5 10 21 m -3. The electron and hole mobilities for silicon are 0.16 and 0.04 m 2 /(V·s), respectively.