— Spectral labelling of terms: ${}^x(L{)}_J$, where $x=2S+1$ is the multiplicity, $L,S,J$ are quantum numbers,
$$\begin{array}{l}L=0,1,2,3,4,5,6,\dots \\ (L):S,P,D,F,G,H,I,\dots \end{array}$$
— Terms of alkali metal atoms:
$$T=\frac{R}{(n+\alpha {)}^2},$$
where $R$ is the Rydberg constant, $\alpha$ is the Rydberg correction.
Fig. 6.7 illustrates the diagram of a lithium atom terms.
Angular momenta of an atom:
$$M_L=\hslash \sqrt{L(L+1)},$$
with similar expressions for $M_S$ and $M_J$.
— Hund rules:
(1) For a certain electronic configuration, the terms of the largest $S$ value are the lowest in energy, and among the terms of $S_{max}$ that of the largest $L$ usually lies lowest;
Fig. 6.7.
Fig. 6.8.
(2) for the basic (normal) term $J=|L-S|$ if the subshell is less than halfflled, and $J=L+S$ in the remaining cases.
— Boltzmann’s formula:
$$\frac{N_2}{N_1}=\frac{g_2}{g_1}{\mathrm{e}}^{-(E_2-E_1)/kT},$$
where $g_1$ and $g_2$ are the statistical weights (degeneracies) of the corresponding levels.
— Probabilities of atomic transitions per unit time between level 1 and a higher level 2 for the cases of spontaneous radiation, induced radiation, and absorption:
$$P_{21}^{sp}=A_{21},P_{21}^{ind}=B_{21}{u}_{\omega },P_{12}^{abs}=B_{12}{u}_{\omega },$$
where $A_{21},B_{21},B_{12}$ are Einstein coefficients, $u_{\omega }$ is the spectral density of radiation corresponding to frequency $\omega$ of transition between the given levels.
— Relation between Einstein coefficients:
$$g_1{B}_{12}=g_2{B}_{21},B_{21}=\frac{{\pi }^2{c}^3}{\hslash {\omega }^3}{A}_{21}.$$
— Diagram showing formation of X-ray spectra (Fig. 6.8).
— Moseley’s law for $K_{\alpha }$ lines:
$${\omega }_{K_{\alpha }}=\frac{3}{4}R(Z-\sigma {)}^2,$$
where $\sigma$ is the correction constant which is equal to unity for light elements.
— Magnetic moment of an atom and Landé $g$ factor:
$$\mu =g\sqrt{J(J+1)}{\mu }_B,g=1+\frac{J(J+1)+S(S+1)-L(L+1)}{2J(J+1)}.$$
— Zeeman splitting of spectral lines in a weak magnetic field:
$$\mathrm{\Delta }\omega =(m_1{g}_1-m_2{g}_2){\mu }_{B}B/\hslash \text{. }$$
— With radiation directed along the magnetic field, the Zeeman components caused by the transition $m_1=m_2$ are absent.
6.97. The binding energy of a valence electron in a $\mathrm{Li}$ atom in the states $2S$ and $2P$ is equal to 5.39 and $3.54\mathrm{eV}$ respectively. Find the Rydberg corrections for $S$ and $P$ terms of the atom.
6.98. Find the Rydberg correction for the $3P$ term of a $\mathrm{Na}$ atom whose first excitation potential is $2.10\mathrm{V}$ and whose valence electron in the normal $3S$ state has the binding energy $5.14\mathrm{eV}$.
6.99. Find the binding energy of a valence electron in the ground state of a Li atom if the wavelength of the first line of the sharp series is known to be equal to ${\lambda }_1=813\mathrm{nm}$ and the short-wave cut-off wavelength of that series to ${\lambda }_2=350\mathrm{nm}$.
6.100. Determine the wavelengths of spectral lines appearing on transition of excited $\mathrm{Li}$ atoms from the state $3S$ down to the ground state $2S$. The Rydberg corrections for the $S$ and $P$ terms are -0.41 and -0.04 .
6.101. The wavelengths of the yellow doublet components of the resonance $\mathrm{Na}$ line caused by the transition $3P\to 3S$ are equal to 589.00 and $589.56\mathrm{nm}$. Find the splitting of the $3P$ term in $\mathrm{eV}$ units.
6.102. The first line of the sharp series of atomic cesium is a doublet with wavelengths 1358.8 and $1469.5\mathrm{nm}$. Find the frequency intervals (in $\mathrm{rad}/\mathrm{s}$ units) between the components of the sequent lines of that series.
6.103. Write the spectral designations of the terms of the hydrogen atom whose electron is in the state with principal quantum number $n=3$.
6.104. How many and which values of the quantum number $J$ can an atom possess in the state with quantum numbers $S$ and $L$ equal respectively to
(a) 2 and 3 ; (b) 3 and 3 ; (c) $5/2$ and 2 ?
6.105. Find the possible values of total angular momenta of atoms in the states ${}^4\mathrm{P}$ and ${}^5\mathrm{D}$.
6.106. Find the greatest possible total angular momentum and the corresponding spectral designation of the term
(a) of a $\mathrm{Na}$ atom whose valence electron possesses the principal quantum number $n=4$;
(b) of an atom with electronic configuration $1s^{2}2p3d$.
6.107. It is known that in $F$ and $D$ states the number of possible values of the quantum number $J$ is the same and equal to five. Find the spin angular momentum in these states.
6.108. An atem is in the state whose multiplicity is three and the total angular momentum is $n\sqrt{20}$. What can the corresponding quantum number $L$ be equal to?
6.109. Find the possible multiplicities $x$ of the terms of the types
(a) $\approx D_3;$ (b) $\approx P_{3/s};$ (c) $\approx F_1$.
6.110. A certain atom has three electrons $(p,p$, and $d)$, in addition to filled shells, and is in a state with the greatest possible total mechanical moment for a given configuration. In the corresponding vector model of the atom find the angle between the spin momentum and the total angular momentum of the given atom.
6.111. An atom possessing the total angular momestum a $\sqrt{6}$ is in the state with spin quantum number $S=1$. In the corresponding vector model the angle between the spis momentum and the total angular momentum is $\theta =73.2$. Write the spectral symbol for the term of that state.
6.112. Write the spectral symbols for the terms of a two-electron system consisting of one $P$ electron and one $d$ electron.
6.113. A system comprises an atom in ${}^2{P}_{\mathrm{z}}$, state and a d electron. Find the possible spectral terms of that system.
6.114. Find out which of the following transitions are forbidden by the selection rules: ${}^2{D}_{2/}\to {}^2{P}_{1/n},{}^3{P}_1\to {}^2{S}_{1/2},{}^3{F}_3\to {}^3{P}_1$,
6.115. Determine the overall degeneracy of a $3D$ state of a $\mathrm{Li}$ atom. What is the physical meaning of that value?
6.116. Find the degeneracy of the states ${}^{2}P,{}^{2}D$, and $4P$ possessing the greatest possible values of the total angular momeatum.
6.i17. Write the spectral designation of the term whose degeneracy is equal to seven and the quantum numbers $L$ and $S$ are interrelated as $L=3s$.
6.118. What element has the atom whose $K,L$, and $M$ shells and $4s$ subshell are filled completely and $4p$ subshell is half-filled?
6.119. Using the Hund rules, find the basic term of the atom whose partially filled subshell contains
(a) three $P$ electrons; (b) four $p$ electrons.
6.120. Using the Hund rules, find the total angular momentum of the atom in the ground state whose partially filled subshell contains
(a) three $d$ electrons; (b) seven $d$ electrons.
6.121. Making use of the Hund rules, find the number of electrons in the enly partially filled subahell of the atom whose basic term is (a) ${}^2{F}_{\mathrm{si}}$ (b) ${}^2{P}_{\mathrm{y}/\mathrm{i}}$ (c) ${}^{\ast }{S}_{\mathrm{M}/\mathrm{r}}$,
6.122. Using the Hund rules, write the spectral symbol of the basic term of the atom whose only partially filled subshell
(a) is filled by $1/3$, and $S=1$ :
(b) is filled by $70\mathrm{\%}$, and $s=3/2$.
6.123. The only partially filled subshell of a certain aton contains three electrons, the basic term of the atom having $L=3$. Using
the Hund rules, write the spectral symbol of the ground state of the given atom.
6.124. Using the Hund rules, find the magnetie moment of the ground state of the atom whose epen subshell is half-filled with five electrons.
6.125. What fraction of hydrogen atoms is in the state with the prineipal quantum number $n=2$ at a temperature $T=3000\mathrm{K}$ ?
6.126. Find the ratio of the number of atoms of gaseous sodium in the state $3P$ to that in the ground state $3S$ at a temperature $T=$ $=2400\mathrm{K}$. The spectral line corresponding to the transition $3P\to$ $\to 3S$ is known to have the wavelength $\lambda =589\mathrm{nm}$.
6.127. Calculate the mean lifetime of excited atoms if it is known that the intensity of the spectral line appearing due to transition to the ground state diminishes by a factor $\eta =25$ ever a distance $l=2.5\mathrm{mm}$ along the stream of atoms whose velocity is $v=$ $-600\mathrm{m}/\mathrm{s}$.
6.128. Rarefied Hg gas whose atems are practically all in the ground state was lighted by a mercury lamp emitting a resonance line of wavelength $\lambda =253.65\mathrm{nm}$. As a result, the radiation power of $\mathrm{Hg}$ gas at that wavelength turned out to be $P=35\mathrm{mW}$. Find the number of atoms in the state of resonance excitation whose mean lifetime is $\tau =0.55\mathrm{s}$.
6.129. Atomic lithium of concentration $n=3.6\cdot {10}^{-4}{\mathrm{cm}}^{-3}$ is at — temperature $T=1500\mathrm{K}$. In this case the power emitted at the resonant line’s wavelength $\lambda =671$ nn $(2P\to 2S)$ per unit volume of gas is equal to $P=0.30\mathrm{W}/{\mathrm{cm}}^2$. Find the mean lifetime of $\mathrm{Li}$ atoms in the resonasce exeitation state.
6.130. Atomic hydrogen is in thermodynamic equilibrium with its radiation. Find:
(a) the ratio of probabilities of induced and spontaneous radiations of the atoms from the level $2P$ at a temperature $T=3000\mathrm{K}$;
(b) the temperature at which these probabilities become equal.
6.131. A beam of light of frequency a, equal to the resonant frequency of transition of atoms of gas, passes through that gas heated to temperature $T$. In this case $h_{\omega }>kT$. Taking into account induced radiation, demonstrate that the absorption coefficient of the gas $x$ varies as $x=x_0(1-e^{-t=\ast })$, where $x_0$ is the absorption coefineient for $T\to 0$.
6.132. The wavelength of a resonant mercury line is $\lambda =$ $=253.65\mathrm{mm}$. The mean lifetime of mercury atoms in the state of resonance excitation is $\tau =0.15\mu \mathrm{s}$. Evaluate the ratio of the Doppler line broadening to the natural linewidth at a gas temperature $T=300\mathrm{K}$.
6.133. Find the wavelength of the $K_a$ line in copper $(Z-29)$ if the wavelength of the $K_{\text{a line }}$ in iron $(Z^{-}=26)$ is known to be equal to 193 pm.
6.134. Proceeding from Moseley’s law find:
(a) the wavelength of the $K_a$ line in aluminium and cobalt:
(b) the difference in binding energies of $K$ and $L$ electrons in vanadium.
6.135. How many elements are there in a row between those whose wavelengths of ${\mathit{K}}_{\text{, lines are equal to }}250$ and $179\mathrm{pm}$ ?
6.136. Find the voltage applied to an $X$-ray tube with sickel anticathode if the wavelength diflerence between the $K_{\mathrm{a}}$ line and the short-wave cut-off of the continuous X-ray spectram is equal to $84\mathrm{pm}$.
6.137. At a certain voltage applied to an $X$-ray tube with aluminium anticathode the short-wave cut-of wavelength of the continuous $X$-ray spectrum is equal to $0.50\mathrm{mm}$. Will the $K$ series of the characteristic spectrum whose exeitation potential is equal to $1.56\mathrm{kV}$ be also observed in this case?
6.138. When the voltage applied to an $X$-ray tube increased from $V_1=10\mathrm{kV}$ to $V_1=20\mathrm{kV}$, the wavelength interval between the $K_a$ line and the short-wave cut-of of the continuous $\mathrm{X}$-ray spectrum increases by a factor $n=3.0$. Find the atomic number of the element of which the tube’s anticathode is made.
6.139. What metal has in its absorption spectrum the difference between the frequencies of $X$-ray $K$ and $L$ absorption edges equal to $\mathrm{\Delta }\omega =6.85\cdot {10}^{14}{\mathrm{s}}^{-1}$ ?
6.140. Calculate the binding energy of a $K$ electron in vanadium whose $L$ absorption edge has the wovelength ${\lambda }_L=2.4\mathrm{nm}$.
6.141. Find the binding energy of an $L$ electron in titanium if the wavelength difference between the first line of the $K$ series and its short-wave cut-ol is $\mathrm{\Delta }\lambda =26\mathrm{pm}$.
6.142. Find the kinetic energy and the velocity of the photeelectrons liberated by $K$, radiation of zine from the $K$, shell of iron whose $K$ band absorption edge wavelength is ${\lambda }_K=174\mathrm{pm}$.
6.143. Calculate the Lande $g$ factor for atoms
(a) in $S$ states; (b) in singlet states.
6.144. Calculate the Lande $g$ factor for the following terms:
6.145. Calculate the magrotic moment of an atom (in Bohr magnetons)
(a) in if state;
(b) in ‘ $D_N$, state;
(c) in the state in which $S=1,L=2$, and Landé factor $g=4/3$.
6.146. Determine the spin angular momentum of an atom in the state $D_1$ if the maximum value of the magnetic moment projection in that state is equal to four Bohr maznetons.
6.147. An atom in the state with quantum numbers $L=2$, $S=1$ is located in a weak magnetic field. Find its magnetic moment if the least possible angle between the angular momentum and the field direction is known to be equal to ${30}^{\circ }$.
6.148. A valence electron in a sodium atom is in the state with principal quantum number $n=3$, with the total angular momentum being the greatest possible. What is its magnetic moment in that state?
6.149. An excited atom has the electronic configuration $1s^{2}2s^{2}2p3d$ being in the state with the greatest possible total angular momentum. Find the magnetic moment of the atom in that state.
6.150. Find the total angular momentum of an atom in the state with $S=3/2$ and $L=2$ if its magnetic moment is known to be equal to tero.
6.151. A certais atom is in the state in which $s=2$, the total angular momentum $M=V2\hat{n}$, and the magnetic moment is equal to zero. Write the spectral symbol of the corresponding term.
6.152. An atom in the state ${}^2{P}_{y_2}$ is located in the external magnetie field of induction $B=1.0\mathrm{kG}$. In terms of the vector model find the angular precession velocity of the total angular momentum of that atom.
6.153. An atom in the state ${}^1{P}_{1/1}$ is located on the axis of a loop of radius $r=5\mathrm{cm}$ carrying a current $I=10\mathrm{A}$. The distance between the atom and the centre of the loop is equal to the radius of the latter. How great may be the maximum force that the magnetic field of that current exerts on the atom?
6.154. A hydrogen atom in the normal state is located at a distance $r=2.5\mathrm{cm}$ from a long straight conductor carrying a current $I=10\mathrm{A}$. Find the force acting on the atom.
6.155. A narrow stream of vasadium atoms in the ground state * $F_{\mathrm{sa}}$ is passed through a transverse strongly inhomogeneous magnetie field of length $h_1-5.0\mathrm{cm}$ as in the Stem-Gerlach experiment. The beam splitting is observed on a screen located at a distance $l_2=15\mathrm{cm}$ from the magnet. The kinetic energy of the atoms is $t=22\mathrm{MeV}$. At what value of the gradient of the magnetic field induction $B$ is the distance between the extreme components of the split beam on the screen equal to $\delta =2.0\mathrm{mm}$ ?
6.156. Inte what number of sublevels are the following terms split in a weak magnetic field:
6.157. An atom is located in a magnetic field of induction $B=$ $=2.50\mathrm{kG}$. Find the value of the total splitting of the following terms (expressed in ${\mathrm{eV}}^{\mathrm{V}}$ units):
(a) ${}^{1}D;$ (b) ${}^2{F}_{\mathrm{c}}$ :
6.158. What kind of Zeeman effect, normal or anomalous, is observed in a weak magnetic field in the case of spectral lines caused by the following transitions:
6.159. Determine the spectral symbol of an atomic singlet term If the total splitting of that term in a weak magnetic feld of induction $B=3.0\mathrm{kG}$ amounts to $\mathrm{\Delta }E=104\mu \mathrm{V}$.
6.160. It is knows that a spectral line $\lambda =612\mathrm{nm}$ of an atom is caused by a transition between singlet terms. Calculate the interval $\mathrm{\Delta }\lambda$ between the extreme components of that line in the magnetic feld with induction $B=10.0\mathrm{kG}$.
6.161. Find the minimum magnitude of the magnetic field induction $B$ at which a spectral instrument with resolving power $N\mathrm{\Delta }\lambda =$ $=1.0\cdot {10}^n$ is capable of resolving the components of the spectral line $\lambda =536\mathrm{nm}$ caused by a transition between singlet terms. The observation line is at right angles to the magnetic field direction.
6.162. A spectral line caused by the transition ${}^D{D}_1\to {}^{\prime }P$, experiences the Zeeman splitting in a weak magnetic field. When observed at right angles to the magnetic field direction, the interval between the neighbouring components of the split line is $\mathrm{\Delta }\omega =1.32\cdot {10}^{10}{\mathrm{s}}^{-1}$. Find the magnetic field induction $B$ at the point where the source is located.
6.163. The wavelengths of the $\mathrm{Na}$ yellow doublet $({}^{2}P\to {}^{1S})$ are equal to 589.59 and $589.00\mathrm{nm}$. Find:
(a) the ratio of the intervals between neighbouring sublevels of the Zeeman splitting of the terms ‘ $P_{\mathrm{y}}$ and ${}^{\prime }{P}_{\mathrm{y}}$, in a weak magnetic field:
(b) the magnetic feld induction $B$ at which the interval between neighbouring sublevels of the Zeeman splitting of the term ‘ $P_{\mathrm{s}/}$, is $\eta =50$ times smaller than the natural splitting of the term ‘ $P$.
6.164. Draw a diagram of permitted transitions between the terms $\text{»}{P}_{y/2}$ and $1S_{y/2}$ is a weak magnetic field. Find the displacements (in rad/sunits) of Zeeman components of that line in a magnetic field $B=4.5\mathrm{kG}$.
6.165. The same spectral line undergoing
Fie. 6.2. anomalous Zeeman splitting is observed in direction $I$ and, after reflection from the $m$ irror $M$ (Fig. 6.9), in direction 2. How many Zeeman components are observed in both directions if the spectral line is caused by the transition
6.166. Calculate the total splitting $\mathrm{\Delta }\omega$ of the spectral line ${}^{3}D,\to$ $\to P_1$ in a weak magnetic field with induction $B=3.4\mathrm{kG}$.