irodov problems in atomic and nuclear physics

result of the nuclear reaction (p, n) whose
yield w = 1.2.10-3 a 56CO radionuclide is produced with a half-life
of 77.2 days. Determine the activity of the target 't' = 2.5 h after the
beginning of irrad iation , if the proton's current J = 21 ~A.
13.64. A target of metallic sodium was irradiated with a beam of
deuterons with an energy of 14 MeV and a current of 10 f.tA for
a long period of time. Ftndthe yield of the reaction (d, p) producing
a 24Na radionuclide, if the activity of the target 10 h after the end
of irradiation is 1.6 Ci.
13.65. A thin phosphorus plate of thickness 1.0 g/cm 2 was irra-
diated for 't' = 4.0 h with a neutron flux of 2.0.1010 neutrons/s with
a kinetic energy of 2 l\IeV. One hour after the end of irradiation, the
activity of the plate turned out to be 105 f.tCi. The activity is known
to result from 31Si nuclide produced by the reaction (n, p). Determine
the cross-section of the given reaction.
13.66. A thick* aluminium target irradiated with a beam of
a-particles with an energy of 7.0 l\leV emits 1.60.109 neutrons/s
resulting from the reaction (a, n). Find the yield and mean cross-
section of the given reaction, if the current of a-particles is equal
to 50 ~"A.
13.67. A thick* beryllium target is bombarded with a-particles
with an energy of 7.0 ~IeV. Determine the mean cross-section of the
reaction (a, n), if its yield amounts to 2.50.10-4 •
13.68. A thick* target made of 7Li nuclide is bombarded with
a-particles with an energy of 7.0 ~1eV. Find the mean cross-section
of the reaction 7Li(a, n)lOB - 4.4 l\IeV, if its yield w = 2.8.10-5•
* A target is referred to as "thick" when its thickness exceeds the range of
a striking particle in the target's material.
107
13.69. A beam of a-particles with an energy of 7.8 MeV enters
a chamber filled with air at NTP. The length of the chamber along
the beam exceeds the range of a-particles of the given energy. Find
the mean cross-section of the reaction 14N(a, p) 170 - 1.20 ~leV, if
the yield of the reaction is 2.0.10-6. The nitrogen content in air
is 78 % by volume.
13.70. A beam of neutrons with an energy of 14 MeV falls nor-
mally on the surface of a beryllium plate. Evaluate the thickness
of the plate sufficient for the 10 % reproduction of neutrons by means
of the reaction (n, 2n) whose cross-section (1 = 0.50 b for the given
energy of neutrons. Other processes are assumed non-existent, the
secondary neutrons are not to be absorbed in the plate.
13.71. A thick target containing no nuclei/ems is irradiated with
heavy charged particles. Find how the cross-section of a nuclear
reaction depends on the kinetic energy T of striking particles, if the
reaction yield as a function of the particles' energy, w (T), and the
expression for ionization loss of energy of these particles, dT/dx =
= f (T), are known.
13.72. When a deuterium target is irradiated with deuterons, the
following reaction occurs: d + d ~ 3He + n, Q = +3.26 Me'V.
Making use of the detailed balancing principle, find the spin of a sHe
if the cross-section of this process equals 01 for energy of deuterons
T == 10.0 MeV, while the cross-section of the reverse process for the
corresponding energy of striking neutrons (12 = 1.80't. The spins
of a neutron and a deuteron are supposed to he known.
13.73. Using the detailed balancing principle, find the cross-
section <11 of the reaction a + 6Li -+ 9Be + p - 2.13 MeV, if
the energy of striking a-particles is T = 3.70 ~IeV and the cross-
section of the reverse reaction with the corresponding energy of pro-
tons is 0'2 == 0.050 mho
13.74. Using the detailed balancing principle, demonstrate that
the cross-section of an endoergic reaction A (P, n) B activated due
to irradiation of a target with protons of energy T p is proportional
to V Tp - Tp th in the vicinity of the threshold, if in the case of slow
neutrons the cross-section of the reverse reaction is proportional
to 1/vn , Vn being the velocity of the neutrons.
13.75. The cross-section of the deuteron photodisintegration
reaction 'V + d --+ n + p, Q = -2.22 l\1eV is 0'1 = 0.150 mh for
an energy of 'V-quanta nw = 2.70 MeV. Using the detailed balancing
principle, find the cross-section (12 of the reverse process for the cor-
responding energy T'; of striking neutrons. Calculate this value of Tn.
14
NEUTRON PHYSICS
• Aiming parameter of a neutron
b= t. yrl (l+1),
where ~ = "A/2rr. is its wavelength, 'l is the orbital quantum number.
• Breit-Wigner formula for an individual level gives the cross-section
mation of compound nucleus by slow s-neutrons (I = 0):
rrn
oa = rtl 2g---~--­(T -TO) 2 + (r/2)2
. 21+.1
g == 2 (21+1) ,
(14.1)
of Ior-
(14.2)
---0
---[,
Fig. 39
T
where ~ and Tare the wavelength and kinetic energy of an incoming neutron,
To is the kinetic e:ifergy of a neutron corresponding to the given level of the
compound nucleus M* (Fig. 39), g is the statisti-
cal weight, I is the spin of the target nucleus, J
is the spin of the given level of the compound
nucleus, rand r n is the total and neutron width
of the level, r n depends on the wavelength of the
incoming neutron, ~rn == ~orno, ~o and rno are
the neutron's wavelength and neutron width of
the level at T = To.
• Ra te of nuclear reaction:
R == (~) <1> reaction/(em" -s), (14.3)
where (1:) = N (0) is the mean macroscopic cross-
section of reaction, N is the concentration of nu-
clei, cI> = n (v) is the flux density of neutrons, 12
is the concentration of neutrons, and (v) is their
mean velocity.
• Mean value of the cosine of the angle at which neutrons are scattered due to
elastic collisions with stationary nuclei of mass number A:
2(cos {t) == 3A • (14.4)
• Logarithmic loss of energy is In (T0/T), where To and T are the initial and
final kinetic energies of a neutron., .
• Mean logarithmic loss of energy of a neutron undergoing a single elastic col-
lision with a nucleus:
t = 1-1-_a._In «:
'"' I i-a ' (
A-1 )2a~ A+1 ' (14.5)
where A is the mass number of the nucleus.
109
(14.6)
• Age of neutrons moderated from energy To down to T:
Tor 1 dT <).
't= J 36!s~tr T cm - ,
T
~tT= ~s (1-(cos {t»),
where ~ is the mean logarithmic loss of energy, ~ sand L tr are the macroscopic
scattering and transport cross-sections.
• Moderation density q (E) is the number of neutrons in .. Cll1 3 crossing a given
energy level E per one second in the process of moderation. For a point source
of fast monoergic neutrons in an infinite homogeneous moderating medium
( ) n -r2/4 tqE r = 4Jt't e , (14.7)
(14.8)
where n is the source intensity, neutrons/a, t' is the neutron age, cm~, r is the
distance from the source, em.
• Neutron diffusion equation for a medium without multiplication:
dndt=DV2<I>-~Q<I>;
1 ..r-
D= 3~tr ; LdU= r D/Ia t
where n is the concentration of neutrons, D is the diffusion coefficient, V2 is
the Laplace operator, <1> is the flux density of neutrons, ~ a is the macroscopic
absorption cross-section, Ldtf is the diffusion leng-th.
• Neutron albedo ~ is the probability of neutrons being reflected after multiple
scattering in a medium.
NEUTRON SPECTROSCOPY
14.1. One of the first designs for mechanical selection of neutrons
consists of two discs fixed to an axle rotating at a speed of n rps.
The distance between discs is L. Each disc has a radial slit displaced
relative to each other by the angle Ct. Find the energy of neutrons
filtered through such a selector, if n = 100 rps, L = 54 em, and
ex = 8°.
14.2. In a mechanical neutron selector, constructed as a stack
of alternating aluminium plates of a thickness of 0.75 mm and thin
cadmium layers, the total length of the stack is equal to 50 mm.
What must be the speed of rotation of the stack to arrest neutrons
with energies below 0.015 eV? What is the neutron pulse duration
in this case?
14.3. A mechanical time-of-flight neutron selector