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6.200. Up to what temperature has one to heat classical electronic gas to make the mean energy of its electrons equal to that of free
electrons in copper at T = 0? Only one free electron is supposed to
correspond to each copper atom. 6.201. Calculate the interval (in eV units) between neighbouring levels of free electrons in a metal at T = 0 near the Fermi level, if the concentration of free electrons is n = 2.0.1022 cm-3and the
volume of the metal is V = 1.0 cm3.
6.202. Making use of Eq. (6.4g), find at 7' = 0: (a) the velocity distribution of free electrons; (b) the ratio of the mean velocity of free electrons to their maxi- mum velocity. 6.203. On the basis of Eq. (6.4g) find the number of free electrons in a metal at 7' = 0 as a function of de Broglie wavelengths. 6.204. Calculate the electronic gas pressure in metallic sodium,
at T = 0, in which the concentration of free electrons is n =
= 2.5.1022 cm-3. Use the equation for the pressure of ideal gas. 6.205. The increase in temperature of a cathode in electronic tube
by OT = 1.0 K from the value 7' '= 2000 K results in the increase
of saturation current by ----- 1.4%. Find the work function of electron for the material of the cathode. 6.206. Find the refractive index of metallic sodium for electrons
with kinetic energy T = 135 eV. Only one free electron is assumed
to correspond to each sodium atom. 6.207. Find the minimum energy of electron-hole pair formation in an impurity-free semiconductor whose electric conductance increases = 5.0 times when the temperature increases from T1 = = 300 K to T2 = 400 K. 6.208. At very low temperatures the photoelectric threshold short wavelength in an impurity-free germanium is equal to (^) 1 th = 1.7 p,m. Find the temperature coefficient of resistance of this germanium sample at room temperature. 6.209. Fig. 6.11 illustrates logarithmic electric conductance as a function of reciprocal I/ 8
temperature (7' in kK (^) units) for some
7 2 J^4 j-/ Fig. 6.11.
n-type semiconductor. Using this plot, find (^) the width of the forbid- den band of the semiconductor and the activation energy of donor levels. 6.210. The resistivity of an impurity-free semiconductor at room temperature is p = 50 Q • cm. It becomes equal to pi= 40 Q• cm
when the semiconductor is illuminated with light, and t = 8 ms
after switching off the light source the resistivity becomes equal to p, = 45 I2-cm. Find the mean lifetime of conduction electrons and holes.
6.211. In Hall effect measurements a plate of width h = 10 mm
and length 1 = 50 mm made of p-type semiconductor was placed
in a magnetic field with induction B = 5.0 kG. A potential differ-
ence V = 10 V was applied across the edges of the plate. In this
case the Hall field is (^) VH = 50 mV and resistivity p = 2.5 52•cm. Find the concentration of holes and hole mobility. 6.212. In Hall effect measurements in a magnetic field with
induction B = 5.0 kG the transverse electric field strength in an
impurity-free germanium turned out to be rl = 10 times less than the longitudinal electric field strength. Find the difference in the mobilities of conduction electrons and holes in the given semicon- ductor. 6.213. The Hall effect turned out to be not observable in a semi- conductor whose conduction electron mobility was 7.1 = 2.0 times that of the hole mobility. Find the ratio of hole and conduction electron concentrations in that semiconductor.
6.5. RADIOACTIVITY
- Fundamental law of radioactive decay:
N=Noe —xt. (6.5a)
- Relation between the decay constant X., the mean lifetime T, and the half-life T: 1 ln 2 T •
- Specific activity is the activity of a unit mass df a radioisotope.
6.214. Knowing the decay constant X of a nucleus, find: (a) the probability of decay of the nucleus during the time from 0 to t; (b) the mean lifetime ti of the nucleus. 6.215. What fraction of the radioactive cobalt nuclei whose half- life is 71.3 days decays during a month? 6.216. How many beta-particles are emitted during one hour by 1.0 p,g of Na24radionuclide whose half-life is 15 hours? 6.217. To investigate the beta-decay of Mg23radionuclide, a coun-
ter was activated at the moment t = 0. It registered Nibeta-parti-
cles by a moment t, = 2.0 s, and by a moment t2 = 3t1the number
(6.5b)
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initial moment the preparation contained only the radionuclide A1, find: (a) the equation describing accumulation of the radionuclide A 2 With time; (b) the time interval after which the activity of radionuclide A 2 reaches the maximum value. 6.230. Solve the foregoing problem if Xi = X2 = X. 6.231. A radionuclide A lgoes through the transformation chain A l -.-A2 —.A3(stable) with respective decay constants Xiand X2. Assuming that at the initial moment the preparation contained only the radionuclide A1equal in quantity to N10 nuclei, find the equation describing accumulation of the stable isotope A3. 6.232. A Bi21° radionuclide decays via the chain Bpi° .-). P0210 (^) Pbao6(stable), a.,
where the decay constants are X, = 1.60.10-8 s-1, X2 = = 5.80.10-8 8 -1. Calculate alpha- and beta-activities of the Bi21° preparation of mass 1.00 mg a month after its manufacture. 6.233. (a) What isotope is produced from the alpha-radioactive Ra228as a result of five alpha-disintegrations and four (3'-disintegra- tions? (b) How many alpha- and P --decays does U238experience before turning finally into the stable Pb206isotope? 6.234. A stationary Pb2" nucleus emits an alpha-particle with kinetic energy 7'„ = 5.77 MeV. Find the recoil velocity of a daught- er nucleus. What fraction of the total energy liberated in this decay is accounted for by the recoil energy of the daughter nucleus? 6.235. Find the amount of heat generated by 1.00 mg of a Po21° preparation during the mean lifetime period of these nuclei if the emitted alpha-particles are known to possess the kinetic energy 5.3 MeV and practically all daughter nuclei are formed directly in the ground state. 6.236. The alpha-decay of Po21° nuclei (in the ground state) is accompanied by emission of two groups of alpha-particles 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 gamma-quanta emitted by the excited nuclei. 6.237. The mean path length of alpha-particles in air under standard conditions is defined by the formula R = 0.98.10-27 v^3 ocm, where v0(cm/s) is the initial velocity of an alpha-particle. Using this formula, find for an alpha-particle with initial kinetic energy 7.0 MeV: (a) its mean path length; (b) the average number of ion pairs formed by the given alpha- particle over the whole path R as well as over its first half, assuming the ion pair formation energy to be equal to 34 eV.
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6.238. Find the energy Q liberated in (3 -- and 13+-decays and in K-capture if the masses of the parent atom MP, the daughter atom M d and an electron m are known. 6.239. Taking the values of atomic masses from the tables, find the maximum kinetic energy of beta-particles emitted by Be" nuclei and the corresponding kinetic energy of recoiling daughter nuclei formed directly in the ground state. 6.240. Evaluate the amount of heat produced during a day by a P --active Na24preparation of mass m = 1.0 mg. The beta-particles are assumed to possess an average kinetic energy equal to 1/3 of the highest possible energy of the given decay. The half-life of Na24 is T = 15 hours. 6.241. Taking the values of atomic masses from the tables, calcu- late the kinetic energies of a positron and a neutrino emitted by Cu nucleus for the case when the daughter nucleus does not recoil. 6.242. Find the kinetic energy of the recoil nucleus in the positron- ic decay of a Nn nucleus for the case when the energy of positrons is maximum. 6.243. From the tables of atomic masses determine the velocity of a nucleus appearing as a result of K-capture in a Bel atom provided the daughter nucleus turns out to be in the ground state. 6.244. Passing down to the ground state, excited Agin 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. 6.245. A free stationary Irm nucleus with excitation energy E = 129 keV passes to the ground state, emitting a gamma quan- tum. Calculate the fractional change of gamma quanta energy due to recoil of the nucleus. 6.246. What must be the relative velocity of a source and an absorber consisting of free Iris' nuclei to observe the maximum absorp- tion of gamma quanta with energy g = 129 keV? 6.247. A source of gamma quanta is placed at a height h = 20 m 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? 6.248. What is the minimum height to which a gamma quanta source containing excited Zn°7nuclei has to be raised for the gravi- tational 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 c = 93 keV and appear on transition of Zn67nuclei to the ground state, and the mean lifetime
of the excited state is i = 14 fiS.
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