Q.231. A radionuclide A_{1} goes through the transformation chain A_{1 }→ A_{2} → A_{3 } (stable) with respective decay constants λ_{1} and λ_{2}. Assuming that at the initial moment the preparation contained only the radionuclide A_{1} equal in quantity to N_{10} nuclei, find the equation describing accumulation of the stable isotope A_{3}.
Ans. Here we have the equations
From problem 229
Then
or
since N_{3} = 0 initially
So
Q.232. A Bi21° radionuclide decays via the chain
where the decay constants are 5.80.10^{8} s^{1}. Calculate alpha and betaactivities of the Bi^{210} preparation of mass 1.00 mg a month after its manufacture.
Ans We have the chain
of the previous problem initially
A month after preparation
N_{1 =} 4.54 x 10^{16 }
N_{2} = 2.52 x 10^{18}
using the results of the previous problem.
Then
Aβ = λ_{1}N_{1} = 0.725 x 10^{11 }dis/sec
Aα = λ_{2}N_{2} = 1.46 x 10^{11} dis/sec
Q.233. (a) What isotope is produced from the alpharadioactive Ra^{228} as a result of five alphadisintegrations and four βdisintegrations?
(b) How many alpha and βdecays does U^{ 238 } experience before turning finally into the stable Pb^{206 } isotope?
Ans. (a) Ra has Z  88, A  226 After 3 a emission and 4 p (electron) emission
A  206
Z  88 + 45x2  82
The product is ^{82}pb^{206 }
(b) We require
 ΔZ = 10 = 2 n  m
 ΔA = 32 = ft x 4
Here n = no. of α emissions
m = no. of β emissions
Thus n = 8, m = 6
Q.234. A stationary Pb^{200} nucleus emits an alphaparticle with kinetic energy T_{α} = 5.77 MeV. Find the recoil velocity of a daughter nucleus. What fraction of the total energy liberated in this decay is accounted for by the recoil energy of the daughter nucleus?
Ans. The momentum of the αparticle is
This is also the recoil m om entum o f the daughter nuclear in opposite direction. The recoil velocity of the daughter ntideus is
= 3.39x10^{5} m/S
The eneigy of the daughter nucleus is and this represents a fraction
of total eneigy. Here Md is the mass of the daughter nudeus.
Q.235. Find the amount of heat generated by 1.00 mg of a Po^{210} preparation during the mean lifetime period of these nuclei if the emitted alphaparticles are known to possess the kinetic energy 5.3 MeV and practically all daughter nuclei are formed directly in the ground state.
Ans. The number of nuclei initially present is
In the mean life time of these nuclei the number decaying is the fraction Thus the eneigy released is 2.87 x 10^{18} x 0.632 x 5.3 x 1.602 x 10^{13} J = 1.54MJ
Q.236. The alphadecay of Po^{210} nuclei (in the ground state) is accompanied by emission of two groups of alphaparticles with kinetic energies 5.30 and 4.50 MeV. Following the emission of these particles the daughter nuclei are found in the ground and excited states. Find the energy of gammaquanta emitted by the excited nuclei.
Ans. We neglect all recoil effects. Then the following diagram gives the eneigy of the gamma ray
Q.237. The mean path length of alphaparticles in air under standard conditions is defined by the formula R = 0.98.10^{27} v^{3}_{o} cm, where v_{0} (cm/s) is the initial velocity of an alphaparticle. Using this formula, find for an alphaparticle with initial kinetic energy 7.0 MeV:
(a) its mean path length;
(b) the average number of ion pairs formed by the given alphaparticle over the whole path R as well as over its first half, assuming the ion pair formation energy to be equal to 34 eV.
Ans. (a) For an alpha particle with initial K.E. 7.0 MeV, the initial velocity is
= 1.83 x 10^{9}an/sec
Thus R = 6.02an
(b) Over the whole path the number of ion pairs is
Over the first half of the path We write the formula for the mean path as RαE^{1/2 }where E is the initial energy. Thus if the energy of the aparticle after traversing the first half of the path is E_{1 }then
Hence number of ion pairs formed in the first half of the path length is
Q.238. Find the energy Q liberated in β^{} and β^{+}decays and in Kcapture if the masses of the parent atom M_{P}, the daughter atom M_{d} and an electron m are known.
Ans. In β^{} decay
since M_{p}, M_{d} are the masses of the atoms. The binding energy of the electrons in ignored. In K capture
In β^{+} decay
Then
Q.239. Taking the values of atomic masses from the tables, find the maximum kinetic energy of betaparticles emitted by Be^{10} nuclei and the corresponding kinetic energy of recoiling daughter nuclei formed directly in the ground state.
Ans. The reaction is
For maximum K.E. of electrons we can put the energy of to be zero. The atomic masses are
Be^{10 =} 10.016711 amu
B^{10} = 10.016114 amu
So the K.E. of electrons is (see previous problem)
597 x 10^{6} amu x c^{2} = 0.56 MeV
The momentum of electrons with this K.E. is 0.941
and the recoil energy of the daughter is
Q.240. Evaluate the amount of heat produced during a day by a β^{}active Na^{24} preparation of mass m = 1.0 mg. The betaparticles are assumed to possess an average kinetic energy equal to 1/3 of the highest possible energy of the given decay. The halflife of Na^{24} is T = 15 hours.
Ans. The masses are
Na^{24 }= 24  0.00903 amu and Mg^{24} = 24  0.01496 amu
The reaction is
The maximum K.E. of electrons is
0.00593 x 93 MeV = 5.52 MeV
Average K.E. according to the problem is then = 1.84 MeV
The initial number of Na^{24} is
The fraction decaying in a day is
1  (2) ^{ 24/15} = 0.67
Hence the heat produced in a day is
0.67 x 2.51 x 10^{19 }x 1.84 x 1.602 x 10^{13} Joul = 4.95MJ
Q.241. Taking the values of atomic masses from the tables, calculate the kinetic energies of a positron and a neutrino emitted by C^{11} nucleus for the case when the daughter nucleus does not recoil.
Ans. We assume that the parent nucleus is at rest Then since the daughter nucleus does not recoil, we have
i.e. positron & v mometum are equal and opposite. On the other hand
total energy released. (Here we have used the fact that energy
of the neutrino is
Now
Then
Thus c p = 0.646 M eV = energy of neutrino
Also K.E. of electron = 1.47  0.646  0.511 = 0.313 MeV
Q.242. Find the kinetic energy of the recoil nucleus in the positronic decay of a Nn nucleus for the case when the energy of positrons is maximum.
Ans. The K.E. of the positron is maximum when the energy of neutrino is zero. Since the recoil energy of the nucleus is quite small, it can be calculated by successive approximation.
The reaction is
The maximum energy available to the positron (including its rest energy) is
c^{2} (Mass of N^{13 }nucleus  Mass of C^{13} nucleus)
= c^{2} (Mass of N^{13} atom  Mass of C^{13} atom  m_{e})
= 0.00239 c^{2}  m_{e}c^{2}
= (0.00239 x 931  0.511) MeV
= 1.71 MeV
The momentum corresponding to this energy is 1.636 MeV/c .
The recoil energy of the nucleus is then
= 111 eV = 0.111 keV
on using Mc^{2}  13 x 931 MeV
Q.243. From the tables of atomic masses determine the velocity of a nucleus appearing as a result of Kcapture in a Bel atom provided the daughter nucleus turns out to be in the ground state.
Ans. The process is
The energy available in the process is
Q = c^{2} (Mass of Be^{7} atom  Mass of Li^{7} atom)
= 0.00092 x 931 MeV = 0.86 MeV
The momentum of a K electron is negligible. So in the rest frame of the Be^{7} atom, most of the energy is taken by neutrino whose momentum is very nearly 0.86MeV/c The momentum of the recoiling nucleus is equal and opposite. The velocity of recoil is
= 3.96 x10^{6} cm/s
Q.244. Passing down to the ground state, excited Ag^{109} in nuclei emit either gamma quanta with energy 87 keV or K conversion electrons whose binding energy is 26 keV. Find the velocity of these electrons.
Ans. In internal conversion, the total energy is used to knock out K electrons. The KE. of these electrons is energy availableB.E. of K electrons
= (87  26) = 61 keV
The total energy including rest mass of electrons is 0.511 + 0.061 = 0.572 MeV
The momentum corresponding to this total energy is
0257 MeV/c.
The velocity is then =
Q.245. A free stationary Ir^{191} nucleus with excitation energy E = 129 keV passes to the ground state, emitting a gamma quantum. Calculate the fractional change of gamma quanta energy due to recoil of the nucleus.
Ans. With recoil neglected, the yquantram will have 129 keV eneigy. To a first approximation, its momentrum will be 129 keV/c and the energy of recoil will be
= 4.18 x 10^{ 8} MeV
In the next approximation we therefore write
Therefore
Q.246. What must be the relative velocity of a source and an absorber consisting of free Ir^{191} nuclei to observe the maximum absorption of gamma quanta with energy ε = 129 keV?
Ans. For maximum (resonant) absorption, the absorbing nucleus must be moving with enough speed to cancel the momentum of the oncoming photon and have just right eneigy (ε = 129 keV) available for transition to the excited state.
Since and momentum of the photon is these condition can be satisfied if the velocity of the nucleus is
218m/s = 0.218km/s
Q.247. A source of gamma quanta is placed at a height h = 20m above an absorber. With what velocity should the source be displaced upward to counterbalance completely the gravitational variation of gamma quanta energy due to the Earth's gravity at the point where the absorber is located?
Ans. Because of the gravitational shift the frequency of the gamma ray at the location of the absorber is increased by
For this to be compensated by the Doppler shift (assuming that resonant absorption is possible in the absence of gravitational field) we must have
Q.248. What is the minimum height to which a gamma quanta source containing excited Zn^{67 } nuclei has to be raised for the gravitational displacement of the Mossbauer line to exceed the line width itself, when registered on the Earth's surface? The registered gamma quanta are known to have an energy ε = 93 keV and appear on transition of Zn^{67} nuclei to the ground state, and the mean lifetime of the excited state is ζ = 14μS.
Ans. The natural life time is
Thus the condition implies
= 4.64 metre
(h here is height of the place, not planck’s constant.)
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