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irodov problems in atomic and nuclear physics

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Breit-Wigner formula, identify the conditions
under which the cross-section of radiative capture of neutrons obeys
the llv law.
14.31. On irradiation of a magnesium target with neutrons
(2.5 MeV), it was observed that in addition to elastically scattered
neutrons there is a group of inelastically scattered neutrons whose
energy corresponds to a certain excited level of transitional nuclei
(1.3 MeV). Determine the relative width of the given level for inelas-
tic scattering, if it is known that the total cross-section of the process
Otot = 2.2 h and the elastic scattering cross-section <leI = 1.6 b,
44 % of which represents potential scattering.
8-0339 113
PROPAGATION OF NEUTRONS THROUGH MATTER
14.32. What must the thickness of cadmium plate be to reduce
the flux of thermal neutrons 100-fold after passing through it?
14.33. How many times will a narrow beam of thermal neutrons
be attenuated after passing through a layer of heavy water 1.0 em
in thickness?
14.34. Evaluate in what proportion a narrow beam of fast neu-
trons with an energy of 10 l\IeV attenuates on passing through a lead
plate 4.0 em in thickness. The effective nuclear cross-section is
assumed to be (J = 2n (R + t. )2; R is the radius of the nucleus;
~ is the neutron wavelength.
14.35. In the centre of a spherical graphite layer whose inside
and outside radii are r1 = 1.0 em and r 2 == 10.0 cm a point source
of monochromatic neutrons is located, emitting 1 0 = 2.0.104 neu-
trons/s with an energy of 2.0 Mev , The interaction of neutrons of
such an energy with carbon nuclei is characterized by a cross-section
(J = 1.6 b. Determine the neutron flux density at the outside surface
of the layer, counting only neutrons that penetrated the layer with-
out collisions.
14.36. The intensity of a narrow beam of slow monochromatic
neutrons diminishes 20-fold on passing through a plate of natural
boron with a mass thickness of 1.0 g/cm'', Determine the energy of
neutrons, taking into account that the 1/u law is valid in this case.
14.37. A narrow beam of neutrons with an energy of 10.0 eV passes
a distance l = 15.0 em along the axis of a counter filled up with
BF 3 at NTP (natural boron is used). Determine the counter efficiency
provided that the cross-section of the reaction (n, ex) is known to
obey the llv law.
14.38. In a neutron counter with LiI crystal sensor the reaction
(n, ex) in 6Li nuclei is used. Determine the efficiency of the counter
for a thermal neutron beam, if the thickness of the crystal is known
to be 2.0 em and density 4.0 g/cm3 (natural lithium is used). The
scattering of neutrons is to be neglected.
14.39. Find the decrease in efficiency (%) of a neutron detector,
a thin lOB layer, that was irradiated for a week by a plane flux of
thermal neutrons with a density J == 1.00-1013 neutrons/Icm--s).
14.40. A non-monochromatic beam of slow neutrons falls on a thin
target activating a nuclear reaction whose cross-section is (J ex: 11v.
Demonstrate that in this case the mean cross-section of the reaction
(averaged over all neutron velocities) (0" (v) = (J ( (v»).
14.41. A beam of neutrons with energies falling within the inter-
val, in which the cross-section of the reaction (n, ex) is proportional
to llv, passes through a thin 6Li foil 10 mg/cm'' in thickness. What
is the mean velocity of the neutrons, if the yield of the reaction
(n, a) is known to be 0.40 in this case?
14.42. A neutron counter with a volume of 100 cm'' filled up with
BF 3 gas at NTP is placed in the uniform field of slow neutrons (boron
114
:#
of natural isotopic composition is used). Assuming that the reaction
cross-section ana oc 1lv, determine: (a) the volume density of neu-
trons if 1.0.1012 reactions occur in the counter per one second;
(b) the number of reactions occurring in the counter per one second
if <D === 1.1.1010 neutrons/Icmv- s) and the neutron temperature is
300 K.
14.43. Demonstrate that in a thin target exposed to an isotropic
field of neutrons the reaction rate is twice that in the case when a par
allel flux of neutrons with the same energy spectrum falls normally
on the target's surface. The number of neutrons hitting the target
is the same in both cases.
14.44. How long does it take to irradiate a thin layer of lOB
nuclide in a field of thermal neutrons with a volume density n =
=== 4.0.108 neutrons/em" to decrease the number of lOB nuclei by
50 percent? I t is known that the reaction cross-section ana ex: 1lu.
14.45. A thin sample of metallic sod ium of 0.40 g mass was
placed in an isotropic field of thermal neutrons with <D === 1.0 X
X 1010 neutrons/(cm2·s). Assuming the 24Na radionuclide production
rate constant, determine: (a) the activity of the saturated sample
and the fraction of 24Na nuclei accumulated in such a sample; (b) the
irradiation time required to raise the sample's activity up to 75%
of its saturation activity.
14.46. The specific activity of a neutron-activated golden foil
equals A == 1.1.108 dis/Is-g) == 3.0 mCi/g. For how long has this
foil to be additionally exposed to the field of thermal neutrons with
$ === 1.0.1010 neutrons/Icmv-s) to increase its activity by a factor of
YI == 10?
14.47. A thin copper plate is exposed to the isotropic field of
thermal neutrons with <1> == 0.9.1012 neutrons/fcm--s). Determine
the specific activity of the plate t == 2.0 h after the beginning of the
exposure.
14.48. A thin 1151n foil of mass 0.20 g was exposed to an isotropic
thermal neutron flux for 't == 2.0 h. In t == 0.50 h after the exposure-
was discontinued, the foil activity turned out to be A === 0.07 meL
Determine the neutron flux density <D.
14.49. A 51V sample of mass 0.50 g is activated up to saturation
in a thermal neutron field. During 't = 5.0 min immediately after'
completion of irradiation, N == 0.8-109 pulses were registered, the ..
count efficiency being ~ == 0.010. Determine the volume density'
of neutrons, assuming the activation cross-section to obey the 1/w
law in this case.
14.50. An 1151n foil whose both sides are covered with thin layers
of cadmium was exposed to an isotropic neutron field. Taking into
account that the cross-section of indium activation obeys the 1llJ
law in the case of thermal neutrons, determine the specific saturation
activity of the foil, if the volume density of thermal neutrons n ='
== 3.1 .10 4 cm -3 and a cadmium ratio of Red = 20. Cadmium is
s* 115
supposed to absorb all thermal neutrons and let through above-
thermal ones. N ate. Red is the ratio of saturation activities of the
naked foil and cadmium-coated one.
MODERATION AND DIFFUSION OF NEUTRONS
14.51. What fraction of its kinetic energy does a neutron lose in:
(a) an elastic head-on collision with initially stationary nuclei 2H,
12C, and 238U; (b) an elastic scattering through the angle 'ft by an
initially stationary deuteron, if the angle -fr is equal to 30, 90, and
150°?
14.52. Neutrons with the kinetic energy To are elastically scat-
tered by nuclei with the mass number~. Determine: (a) the energy of
neutrons scattered through the angle {} in the C frame; (b) the frac-
tion of neutrons that after single scattering possess a kinetic energy
whose value falls within the interval (T, T + dT) provided the
scattering in the C frame is isotropic. Plot the distribution of scat-
tered neutrons in terms of energy.
14.53. Neutrons with a kinetic energy of To = 1.00 MeV are
elastically scattered by initially stationary 4He nuclei. Determine
the mean energy value of singly scattered neutrons, assuming the
scattering in the C frame to be isotropic.
14.54. Determine the probability that after a single elastic scat-
tering of a neutron by a deuteron the neutron energy becomes less .
than half the initial value; the scattering in the C frame is isotropic.
14.55. Neutrons are scattered by initially stationary protons.
Assuming this scattering