# Irodov Solutions → Atomic and Nuclear Physics → Scattering of Particles. Rutherford-Bohr Atom

 6.2. An alpha particle with kinetic energy 0.27 MeV is deflected through an angle of 60° by a golden foil. Find the corresponding value of the aiming parameter. Free solution >> 6.3. To what minimum distance will an alpha particle with kinetic energy T = 0.40 MeV approach in the case of a head-on collision to (a) a stationary Pb nucleus; (b) a stationary free Li7 nucleus? Free solution >> 6.4. An alpha particle with kinetic energy T = 0.50 MeV is deflected through an angle of θ = 90° by the Coulomb field of a stationary Hg nucleus. Find: (a) the least curvature radius of its trajectory; (b) the minimum approach distance between the particle and the nucleus. Free solution >> 6.5. A proton with kinetic energy T and aiming parameter b was deflected by the Coulomb field of a stationary Au nucleus. Find the momentum imparted to the given nucleus as a result of scattering. Free solution >> 6.6. A proton with kinetic energy T = 10 MeV flies past a stationary free electron at a distance b = 10 pm. Find the energy acquired by the electron, assuming the proton's trajectory to be rectilinear and the electron to be practically motionless as the proton flies by. Free solution >> 6.8. A stationary ball of radius R is irradiated by a parallel stream of particles whose radius is r. Assuming the collision of a particle and the ball to be elastic, find: (a) the deflection angle θ of a particle as a function of its aiming parameter b; (b) the fraction of particles which after a collision with the ball are scattered into the angular interval between θ and θ + dθ; (c) the probability of a particle to be deflected, after a collision with the ball, into the front hemisphere (θ < π/2). Free solution >> 6.9. A narrow beam of alpha particles with kinetic energy 1.0 MeV falls normally on a platinum foil 1.0 μm thick. The scattered particles are observed at an angle of 60° to the incident beam direction by means of a counter with a circular inlet area 1.0 cm2 located at the distance 10 cm from the scattering section of the foil. What fraction of scattered alpha particles reaches the counter inlet? Free solution >> 6.12. A narrow beam of alpha particles with kinetic energy T = 0.50 MeV falls normally on a golden foil whose mass thickness is ρd = 1.5 mg/cm2. The beam intensity is I0 = 5.0*105 particles per second. Find the number of alpha particles scattered by the foil during a time interval τ = 30 min into the angular interval: (a) 59-61°; (b) over θ0 = 60°. Free solution >> 6.14. A narrow beam of alpha particles with kinetic energy T = 600 keV falls normally on a golden foil incorporating n = 1.1*1019 nuclei/cm2. Find the fraction of alpha particles scattered through the angles θ < θ0 = 20°. Free solution >> 6.15. A narrow beam of protons with kinetic energy T = 1.4 MeV falls normally on a brass foil whose mass thickness ρd = 1.5 mg/cm2. The weight ratio of copper and zinc in the foil is equal to 7:3 respectively. Find the fraction of the protons scattered through the angles exceeding θ0 = 30°. Free solution >> 6.16. Find the effective cross section of a uranium nucleus corresponding to the scattering of alpha particles with kinetic energy T = 1.5 MeV through the angles exceeding θ0 = 60°. Free solution >> 6.18. In accordance with classical electrodynamics an electron moving with acceleration w loses its energy due to radiation as dE/dt = -2e2w2/(3c3), where e is the electron charge, c is the velocity of light. Estimate the time during which the energy of an electron performing almost harmonic oscillations with frequency ω = 5*1015 s-1 will decrease η = 10 times. Free solution >> 6.19. Making use of the formula of the foregoing problem, estimate the time during which an electron moving in a hydrogen atom along a circular orbit of radius r = 50 pm would have fallen onto the nucleus. For the sake of simplicity assume the vector w to be permanently directed toward the centre of the atom. Free solution >> 6.25. Calculate the magnetic field induction at the centre of a hydrogen atom caused by an electron moving along the first Bohr orbit. Free solution >> 6.27. To what series does the spectral line of atomic hydrogen belong if its wave number is equal to the difference between the wave numbers of the following two lines of the Balmer series: 486.1 and 410.2 nm? What is the wavelength of that line? Free solution >> 6.30. What element has a hydrogen-like spectrum whose lines have wavelengths four times shorter than those of atomic hydrogen? Free solution >> 6.31. How many spectral lines are emitted by atomic hydrogen excited to the n-th energy level? Free solution >> 6.32. What lines of atomic hydrogen absorption spectrum fall within the wavelength range from 94.5 to 130.0 nm? Free solution >> 6.33. Find the quantum number n corresponding to the excited state of He+ ion if on transition to the ground state that ion emits two photons in succession with wavelengths 108.5 and 30.4 nm. Free solution >> 6.34. Calculate the Rydberg constant R if He+ ions are known to have the wavelength difference between the first (of the longest wavelength) lines of the Balmer and Lyman series equal to Δλ = 133.7 nm. Free solution >> 6.37. Find the binding energy of an electron in the ground state of hydrogen-like ions in whose spectrum the third line of the Balmer series is equal to 108.5 nm. Free solution >> 6.38. The binding energy of an electron in the ground state of He atom is equal to E0 = 24.6 eV. Find the energy required to remove both electrons from the atom. Free solution >> 6.39. Find the velocity of photoelectrons liberated by electromagnetic radiation of wavelength λ = 18.0 nm from stationary He+ ions in the ground state. Free solution >> 6.41. A stationary hydrogen atom emits a photon corresponding to the first line of the Lyman series. What velocity does the atom acquire? Free solution >> 6.45. According to the Bohr-Sommerfeld postulate the periodic motion of a particle in a potential field must satisfy the following quantization rule: ∫ p dr = 2πħn, where q and p are generalized coordinate and momentum of the particle, n are integers. Making use of this rule, find the permitted values of energy for a particle of mass m moving (a) in a unidimensional rectangular potential well of width l with infinitely high walls; (b) along a circle of radius r; (c) in a unidimensional potential field U = αx2/2, where α is a positive constant; (d) along a round orbit in a central field, where the potential energy of the particle is equal to U = -α/r (α is a positive constant). Free solution >> 6.47. For atoms of light and heavy hydrogen (H and D) find the difference (a) between the binding energies of their electrons in the ground state; (b) between the wavelengths of first lines of the Lyman series. Free solution >> 6.48. Calculate the separation between the particles of a system in the ground state, the corresponding binding energy, and the wavelength of the first line of the Lyman series, if such a system is (a) a mesonic hydrogen atom whose nucleus is a proton (in a mesonic atom an electron is replaced by a meson whose charge is the same and mass is 207 that of an electron); (b) a positronium consisting of an electron and a positron revolving around their common centre of masses. Free solution >>