Angle ot which a charged particle is defiected by the Coulomb field of a stationary atomic nueleus is detined by the formula:
\[
\tan \frac{\theta}{2}=\frac{9.4}{217} \text {, }
\]
where $b_{1}$ and $y$, are the charges of the particle and the nucleus, $b$ is the aiming parsmetier, $T^{\prime}$ ls the kisetic etergy of a strik: fare particle.
– Avetherford lormula. The relative sunber of particles sattered inte as elementary olid angle do at an angle 6 to their initial pro. pagation direction:
\[
\frac{d N}{N}=n\left(\frac{99_{1}}{4 r}\right)^{2} \frac{\pi}{\sin ^{2}(\theta / 2)} \text {, }
\]
where $n$ is the number of nuclei of the foll per unit area of its surface, do $=$ sin $\theta$ de dy.
– Geseralised Balmer lormula (Fig. 6.1):
where ois the transition frequency (in $s^{-1}$, between energy levels with quanbydreges-like los.
6.1. Employing Thomson’s model, calculate the radius of a hydrogen atom and the wavelength of emitted light if the ioaization energy of the atom is known to be equal to $E=13.6 \mathrm{eV}$.
6.2. An alpha particle with kinetic energy $0.27 \mathrm{MeV}$ is deflected through an angle of $60^{\circ}$ by a golden foil. Find the corresponding value of the aiming parameter.
6.3. To what minimum distance will an alpha particle with kinetic energy $r=0.40 \mathrm{MeV}$ approach in the case of a head-on collision to
(a) a stationary $\mathrm{Pb}$ nucleus;
(b) a stationary free $\mathrm{Li}^{+}$nucleus?
6.4. An alpha particle with kinetic energy $T=0.50 \mathrm{MeV}$ is deflected through an angle of $\theta-90^{\circ}$ by the Coulomb field of a stationary Hg aucleus, Find:
– All the formulas in this Part are given in the Gausian system of anits.
(a) the least curvature radius of its trajectory;
(b) the minimum approach distance between the particle and the aucteus.
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.
6.6. A proten with kinetic energy $T=10 \mathrm{MeV}$ nies past a stationary free electron at a distance $b=10 \mathrm{pm}$. Find the energy scquired by the electron, assuming the proton’s trajectory to be rectilinear and the electron to be practically motionless as the proton aies by.
6.7. A particle with kinetic energy $T$ is deflected by a spherical potential well of radius $R$ and depth $U_{\text {, }}$. i.e. by the field in which the potential energy of the particle takes the form
\[
\boldsymbol{U}=\left\{\begin{array}{r}
0 \text { for } r>\boldsymbol{R}, \\
-\boldsymbol{U}, \text { for } r<\boldsymbol{R},
\end{array}\right.
\]
where $r$ is the distance from the centre of the well. Find the relationship between the aiming parameter $b$ of the particle and the angle $\theta$ through which it deflects from the initial motion direction.
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 te be elastie, find:
(a) the deflection angle $\theta$ 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 inte the angular interval between $\theta$ and $\theta+\theta$;
(c) the probability of a particle to be deflected, after a collision with the ball, into the front hemisphere $\left(\theta<\frac{\pi}{2}\right)$.
6.9. A narrow beam of alpha particles with kinetic energy $1.0 \mathrm{MeV}$ falls normally on a platinum foil $1.0 \mu \mathrm{m}$ thick. The scattered particles are ebserved at an angle of $60^{\circ}$ to the incident beam direction by means of a counter with a circular inlet area $1.0 \mathrm{~cm}^{2}$ located at the distance $10 \mathrm{~cm}$ from the scattering section of the foil. What fraction of scattered alpha particles reaches the counter inlet?
6.10. A narrow beam of alpha particles with kinetic energy $T=$ $=0.50 \mathrm{MeV}$ and intensity $\mathrm{I}=5.0 \cdot 10^{\circ}$ particles per second falls normally on a golden foil. Find the thickness of the foil if at a distance $r=15 \mathrm{~cm}$ from a scattering section of that foil the flux density of scattered particles at the angle $\theta=60^{\circ}$ to the incident beas is equal to $J=40$ particles/( $\left.\mathrm{cm}^{2}, \mathrm{~s}\right)$.
6.11. A narrow beam of alpha particles falls nermally on a silver foil behind which a counter is set to register the scattered particles. On substitution of platinum foil of the same mass thickness for the silver foil, the number of alpha particles registered per unit time increased $\eta=1.52$ times. Find the atomic number of platinum, assuming the atomic number of silver and the atomic masses of both platinum and silver to be known.
6.12. A narrow beam of alpha particles with kinetic energy $T=$ – $0.50 \mathrm{MeV}$ falls nermally on a golden foil whese mass thickness is $\rho d=1.5 \mathrm{mg} / \mathrm{cm}^{2}$. The beam intensity is $I_{4}=5.0 \cdot 10^{5}$ particles per second. Find the number of alpha particles scattered by the foil during a time interval $\mathrm{T}-30 \mathrm{~min}$ inte the angular interval:
(a) $59-61^{\circ}$; (b) over $\theta_{0}=60^{\circ}$.
6.13. A narrow beam of protons with velecity $v=6-10^{4} \mathrm{~m} / \mathrm{s}$ falls normally on a silver foil of thickness $d=1.0 \mu \mathrm{m}$. Find the probability of the protons to be scattered into the rear hemisphere $\left(\theta>90^{\circ}\right)$.
6.14. A narrow beam of alpha particles with kinetie energy $T=$ $=600 \mathrm{keV}$ falls sormally on a golden foil incorporating $n=$ $=1.1 \cdot 10^{10}$ nuclei $/ \mathrm{cm}^{2}$. Find the fraction of alpha particles scattered through the angles $\theta<\theta_{0}=20^{\circ}$.
6.15. A narrow beam of protens with kinetic energy $T-1.4 \mathrm{MeV}$ falls normally on a brass foil whose mass thickness $p d=1.5 \mathrm{mg} \mathrm{cm}^{2}$. The weight ratio of copper and zine in the feil is equal to $7: 3$ respectively. Find the fraction of the protons scattered through the angles exceeding $\theta_{0}=30^{\circ}$.
6.16. Find the effective cross section of a urasium nuelens corresponding to the scattering of alpha particles with kinetic energy $T=1.5 \mathrm{MeV}$ through the angles exceeding $\theta_{6}=60^{\circ}$,
6.17. The effective cross section of a gold nucleus corresponding to the scattering of monoenergetic alpha particles within the angular interval from $90^{\circ}$ to $180^{\circ}$ is equal to $\Delta \sigma^{\circ}-0.50 \mathrm{~kb}$. Find:
(a) the energy of alpha particles;
(b) the differential cross section of scattering do/dQ ( $\mathrm{kb} / \mathrm{sr})$ corresponding to the angle $\theta=60^{\circ}$.
6.18. In accordance with classical electrodynamics an electron moving with acceleration w loses its energy due to radiation as
\[
\frac{d E}{d i}=-\frac{2 \pi}{2 \pi} w^{2} \text {, }
\]
where $e$ is the electron charge, $c$ is the velocity of light. Estimate the time during which the energy of an electron performing almost harmenie escillations with frequeney $\oplus-5 \cdot 10^{\text {als }} s^{-1}$ will decrease $\eta=10$ times.
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 $r=50 \mathrm{pm}$ would have fallen onte the necleus. For the sake of simplicity assume the vector $w$ to be permaaently directed toward the centre of the atom.
6.20. Demonstrate that the frequency of a photon emerging when an electron jumps between neighbouring circular orbits of a hydrogen-like ion satisfies the inequality $\omega_{n}>\omega>\omega_{n+1}$. where $a_{n}$ and $\omega_{n+1}$ are the frequencies of revolution of that electron around
268
the nucleus along the circular orbits. Make sure that as $n \rightarrow \infty$ the frequency of the photon $\omega \rightarrow \omega_{\mathrm{s}}$.
6.21. A particle of mass $m$ moves along a circular orbit in a centrosymmetrical potential field $U(r)=k r^{2} / 2$. Using the Behr quantisa: tion condition, find the permissible orbital radii and energy levels of that particle.
6.22. Caleulate for a bydregen atom and a He* ion:
(a) the radius of the first Bohr erbit and the velocity of an electron moving along it:
(b) the kinetic energy and the binding energy of an electron in the ground state:
(c) the ionization potential, the first excitation potential and the wavelength of the resonance line $\left(n^{\prime}-2 \rightarrow n=1\right.$ ).
6.23. Calculate the angular frequency of an electron occupying the second Bohr orbit of He* ion.
6.24. For hydrogen-like systems find the magnetic moment $\mu_{n}$ corresponding to the motion of an electron along the $n$-th orbit and the ratio of the magnetic and mechanical moments $\mu_{n} / M_{n}$. Calculate the magnetic mement of an electron occupying the first Bohr orbit.
6.25. Calculate the magnetic field induction at the centre of a hydrogen atom caused by an electron moving along the first Bohr orbit.
6.26. Calculate and draw on the wavelength scale the spectral intervals in which the Lyman. Balmer, and Paschen series for atomic hydrogen are confined. Show the visible portion of the spectrum.
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 $410.2 \mathrm{~nm}$ ? What is the wavelength of that line?
6.28. For the case of atemic hydrogen find:
(a) the wavelengths of the first three lines of the Balmer series;
(b) the minimum resolving power $2 / 8$ of a spectral instrument capable of resolving the first 20 lines of the Balmer series.
6.29. Radiation of atemic hydrogen falls normally on a diffraction grating of width $l=6.6 \mathrm{~mm}$. The $50 \mathrm{th}$ line of the Balmer series in the observed spectrum is elose to resolution at a diffraction angle $\theta$ (in accordance with Rayleigh’s criterion). Find that angle.
6.30. What element has a hydrogen-like spectrum whose lines have wavelengths four times shorter than those of atomic hydrogen?
6.31. How many spectral lines are emitted by atomic hydrogen excited to the $n$-th energy level?
6.32. What lines of atemic hydrogen absorption spectrum fall within the wavelength range from 9.5 to $130.0 \mathrm{~nm}$ ?
6.33. Find the quantum number $n$ corresponding to the excited state of $\mathrm{He}^{+}$ion if on transition to the ground state that ion emits twe photons in succession with wavelengths 108.5 and $30.4 \mathrm{~nm}$.
6.34. Calculate the Rydberg constant $n$ if He* ions are known to have the wavelength diflerence between the first (of the longest wavelength) lines of the Balmer and Lyman series equal to $\Delta \lambda$ $=133.7 \mathrm{~nm}$.
6.35. What hydrogen-like ion has the wavelength difference between the first lines of the Balmer and Lyman series equal to $59.3 \mathrm{~nm}$ ?
6.36. Find the wavelength of the first line of the He* ion spectral series whose interval between the extreme lines is $\Delta \omega=$ $-5.18 \cdot 10^{4 \mathrm{4}} \mathrm{s}-1$.
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 $108.5 \mathrm{~nm}$.
6.38. The binding energy of an electron in the ground state of He atom is equal to $E_{0}=24.6 \mathrm{eV}$. Find the energy required to remeve both electrons from the atom.
6.39. Find the velocity of photoelectrons liberated by electromagnetic radiation of wavelength $\lambda=18.0$ ns from stationary $H_{\text {e* }}$ ions in the ground state.
6.40. At what minimum kinetic energy must a hydrogen atom move for its inelastic head-on collision with anether, stationary, hydrogen atom to make ene of them eapable of emitting a photon? Both atoms are supposed to be in the ground state prior to the collision.
6.41. A stationary hydrogen atom emits a photon corresponding to the first line of the Lyman series. What velocity does the atom acquire?
6.42. From the conditions of the foregoing problem find how much (in per cent) the energy of the emitted photon differs from the energy of the corresponding transition in a hydregen atom.
6.43. A stationary He+ ion emitted a photon corresponding to the first line of the Lyman series. That photon liberated a photoelectron from a stationary hydrogen atom in the ground state. Find the velocity of the photoelectron.
6.44. Find the velocity of the exeited hydrogen atoms if the first line of the Lyman series is displaced by $\Delta \lambda=0.20 \mathrm{~nm}$ when their radiation is observed at an angle $\theta=45^{\circ}$ to their motion direction.
6.45. According to the Bohr-Sommerfeld postulate the periodic motion of a particle in a potential field must satisfy the following quantization rule:
\[
\oint_{P d q}=2 \pi A n,
\]
where $q$ and $p$ are generalized coordinate and momeatum of the particle, $n$ are integers. Making use of this rule, find the permitted values of energy for a partiele of mass $m$ meving
(a) in a unidimensional rectangular potential well of width $t$ with infinitely high walls:
(b) along a circle of radius $r$.
(c) in a unidimensional potential feld $U=\alpha x^{2} / 2$, where $\alpha$ is a positive constant;
(d) along a round orbit in a central field, where the potential energy of the particle is equal to $U=-a / r$ ( $a$ is a positive constant).
6.46. Taking into account the motion of the nucleus of a hydrogen atem, find the expressions for the electron’s binding energy in the ground state and for the Bydberg constant. How much (in per cent) do the binding energy and the Bydberg constant, obtained without taking inte account the motion of the nucleus, differ from the more accurnte cerresponding values of these quantities?
6.47. For atoms of light and heavy hydrogen (H and D) find the diflerence
(a) between the binding energies of their electrons in the ground state;
(b) between the wavelengths of first lines of the Lyman series.
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 aroend their common centre of wasses.