irodov problems in atomic and nuclear physics

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.