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become equal. 5.44. ~A. beam of light of frequency 00, equal to the resonant fre- quency of transition of atoms of gas Uu» » kT), passes through that gas heated to temperature T. Taking into account induced radiation, demonstrate that the absorption coefficient of the gas varies as x (T) =- x 0 (1 - e-li U) / IiT), where %0 is the absorption coefficient at T == 0 I\.. 5.45. Under what conditions can light passing through matter be amplified ? Find the ratio of the populations of levels ID 2 and Ip}.(ED> E p ) in atoms of gas at which a beam of monochromatic light with a frequency equal to the frequency of transition between these levels passes through the gas without attenuation. 5.46. Suppose that a quantum system (Fig. 20) is excited to level 2 and the reverse transition occurs only via level 1. Demonstrate that in this case light with frequency 0)21 can be amplified, if the condition g1A10> g2A21 is satisfied, where gl and g 2 are the statistical weights of levels 1 and 2 and A l o and A 21 are the Einstein coeffi- cients for the corresponding transitions. 5.47. Let q be the number of atoms excited to level 2 per unit t ime (Fig. 20). Find the number of atoms in level 1 after the time interval t following the beginning of excitation. The Einstein coeffi- cients A 20 , .11 21 , and A I O are supposed to be known, The induced transitions are to be ignored. 0 5.48. A spectral line 'A == 5320 A appears due to transition in an a torn between t\VO excited states whose mean lifetimes are 1.2· -10-s and 2.0."10-8 s. Evaluate the natural width of that line, ~'A. 47 5.49. The distribution of radiation intensity within a spectral line with natural broadening takes the form (1'/2)2 J (I) -.-.=:: J 0 ( )2 -1 ( /2)"> •(U-(r)o - Y - where J 0 is the spectral intensity at the line's centre (at ro === (Oo); l' is the constant which is characteristic for every line (e.g. when an excited state relaxes directly down to the ground state, l' === 1/1', T being the mean lifetime of the excited state). Using this formula, find: (a) the naturallinewidth <Sw, if the value of l' is known; (b) the mean lifetime of mercury atoms in the 61P state, if the transition to the ground state is known to result in emission of a line A ~ 1850 A with natural width <SA ~ 1.5.10-4 1\. Note. The linewidth is the width of the line's contour measured at half its height. 5.50. Making use of the formula of the foregoing problem: (a) dem- onstrate that half the total intensity of a line is confined within its Iinewid th , that is, within the width of line's contour at the half of its height; (b) find the total intensity of a line whose natural width is ou) and spectral intensity at the centre J o. 5.51. The distribution of radiation intensity in a spectral line with Doppler's broadening takes the form: J w == Joe- a (W-Wo)2/ (,)g; a === m.c2/2kT, where J 0 is the spectral intensity at the line's centre (at (D ~ (Do); m ·is the mass of the atom; T is the temperature of gas, K. (a) Derive this formula, using l\-1axwell's distribution. (b) Demonstrate that the Doppler width of line An, i.e. the width of line's contour at the half of its height, is equal to 6ADop:== 2Aov(In 2)/0. 5.52. The wavelength of the I-Ig resonance line is A === 2536.5 A. The mean lifetime of the resonance level is 'T ~ 1.5.10-7 s. Estimate the ratio of the Doppler broadening of that line to its natural width at a temperature of 11 === 300 K. 5.53. At what temperature is the Doppler broadening of each component of the spectral doublet 22P - 128 of atomic hydrogen equal to the interval between these components? 5.54. To obtain spectral lines without Doppler's broadening, a narrow slightly divergent beam of excited atoms is used, the obser- vation heing performed at right angles to the beam. Estimate the beam apex angle in the case of sodium atoms, if the Doppler broad- eninz of the resonance line A === 5896 A. is ten times one tenth of its nat 11ra] w id t h, the vel oc it Y 0 f a to ms is 1000 m Is, and the III ean l i fe- t ime of resonance excitation stale is 1.G·10-~ s. CHARACTERISTIC X-RAY SPECTRA Jl Fig. 21 ..... ~ ~ ~ ~ ~ ~----~-----4 ..o 5.55. Proceeding from Moseley's law, calculate the wavelengths and energies of photons corresponding to the K a line in aluminium and cobalt. 5.56. Determine the wavelength of the K a line of the element of the Periodic Table, beginning from which the appearance of the L series of characteristic X-ray radiation is to be expected. 5.57. Assuming the correction (J in Moseley's law to be equal to unity, find: (a) to what elements belong ~he Ka. lines with the wavelengths of 1.935,1.787,1.656, and 1.434 A; what is the wavelength of the Ka- line of the element omitted in this sequence; (b) how many elements there are in the sequence between the elements whose K a line wavelengths are equal to 2.50 and 1.79 A. 5.58. The correction in Moseley's law differs considerably from unity for heavy elements. Make sure that this is true in the case of tin, cesium, and tungsten, whose Ka, line wavelengths are equal to 0.492, 0.402, and 0.210 A respectively. 5.59. Find the voltage applied to an X-ray tube with nickel anticath- ode, if the wavelength difference between the K a line and the short- wave cut-off of the continuuus X-ray spectrum is equal to 0.84 A. 5.60. When the voltage applied to an X-ray tube is increased from 10 to 20 kV, the wavelength interval between the K a line and the short-wave cut-off of the continuous X-ray spectrum increases three- fold. What element is used as the tube anticathode? 5.61. How will the X-ray radiation spectrum vary, if the voltage applied to an X-ray tube increases gradually? Using the tables of the Appendix, calculate the lowest voltage to be applied to X-ray tubes with vanadium and tungsten anticathodes, at which the Ka, lines of these ele- ments start to appear. 5.62. What series of the character- istic spectrum are excited in molyb- denum and silver by Ag Ka, radiation? 5.63. Figure 21 shows the K absorption edge of X-ray radiation and the K a and K (1 emission lines. (a) Explain the nature of the abrupt discontinuity in absorption. (b) Calculate and plot to scale the diagram of K, L, and M levels of the a!om for which AKa = 2.75 A, AKB===2.51 J\, and AK = = 2.49 A. Of what element is this atom? What is the wavelength of its La emission line? 5.64. Knowing the wavelengths of K and L absorption edges in vanadium, calculate (neglecting the fine structure): (a) the binding energies of K and L electrons; (b) the wavelength of the K a line in vanadium. 4-0339 49 5.65. Find the binding energy of an L electron in titanium, if the wavelength difference between the first line of the K series and its absorption edge is I1Iv == 0.26 A. 5.66. In the first approximation, the X-ray radiation terms can be described in the form T == R (Z - a)2/n2, where R is the Rydberg constant, Z is the atomic number, a is the screening correction, n is the principal quantum number of a distant electron. Calculate the correction a for the K and L terms of titanium whose K absorption edge has the wavelength Iv K == 2.49 A. 5.67. Find the kinetic energy of electrons ejected from the K shell of molybdenum atoms by Ag K a radiation. 5.68. Carbon subjected to Al K a radiation emits photoelectrons whose spectrum comprises several monoenergetic groups. Find the binding energy of the electrons ejected from carbon atoms with the kinetic energy of 1.21 keY. 5.69. On irradiation of krypton atoms with monochromatic X-rays of wavelength lv, it was found that in some cases the atoms emit two electrons, namely, a photoelectron removed from the K shell and an electron ejected from the L shell due to the Auger effect. The binding energies of the K and L electrons are equal to 14.4 and 2.0 keY respectively.