Discrete and Combinatorial Mathematics An Applied Introduction 5th edition Ralph P. Grimaldi - Solutions

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A proton with kinetic energy T = 1.5 MeV is captured by a deuteron H a. Find the excitation energy of the formed nucleus.
The yield of the nuclear reaction C13 (d, n) N14 has maximum magnitudes at the following values of kinetic energy Ti of bombarding deuterons: 0.60, 0.90, 1.55, and 1.80 MeV. Making use of the table of atomic masses, find the corresponding energy levels of the transitional nucleus through which this
A narrow beam of thermal neutrons is attenuated η = 360 times after passing through a cadmium plate of thickness d = 0.50 ram. Determine the effective cross-section of interaction of these neutrons with cadmium nuclei.
Determine how many times the intensity of a narrow beam of thermal neutrons will decrease after passing through the heavy water layer of thickness d = 5.0 cm. The effective cross-sections of interaction of deuterium and oxygen nuclei with thermal neutrons are equal to σ1 = 7.0 b and σ2 =
A narrow beam of thermal neutrons passes through a plate of iron whose absorption and scattering effective cross-sections are equal to σa, = 2.5 b and σ8 =11 b respectively. Find the fraction of neutrons quitting the beam due to scattering if the thickness of the plate is d = 0.50 cm.
The yield of a nuclear reaction producing radio nuclides may be described in two ways: either by the ratio w of the number of nuclear reactions to the number of bombarding particles, or by the quantity k, the ratio of the activity of the formed radio nuclide to the number of bombarding particles,
Thermal neutrons fall normally on the surface of a thin gold foil consisting of stable Au197 nuclide. The neutron flux density is J = 1.0 ∙ 1010 part./(s ∙ cm2). The mass of the foil is m = l0 mg. The neutron capture produces beta-active Au198 nuclei with half-life T = 2.7 days. The
A thin foil of certain stable isotope is irradiated by thermal neutrons falling normally on its surface. Due to the capture of neutrons a radionuclide with decay constant λ appears. Find the law describing accumulation of that radionuclide N (t) per unit area of the foil's surface. The neutron
A gold foil of mass m = 0.20 g was irradiated during t = 6.0 hours by a thermal neutron flux falling normally on its surface. Following τ = 12 hours after the completion of irradiation the activity of the foil became equal to A = 1.9.107 dis/s. Find the neutron flax density if the effective
How many neutrons are there in the hundredth generation if the fission process starts with No = 1000 neutrons and takes place in a medium with multiplication constant k = 1.05?
Find the number of neutrons generated per unit time in a uranium reactor whose thermal power is P = 100 MW if the average number of neutrons liberated in each nuclear splitting is v = 2.5. Each splitting is assumed to release an energy E = 200 MeV.
In a thermal reactor the mean lifetime of one generation of thermal neutrons is τ = 0.10 s. Assuming the multiplication constant to be equal to k = 1.010, find: (a) How many times the number of neutrons in the reactor, and consequently its power, will increase over t = 1.0 min; (b) The
Calculate the de Broglie wavelengths of an electron, proton, and uranium atom, all having the same kinetic energy 100 eV.
What amount of energy should be added to an electron to reduce its de Broglie wavelength from 100 to 50 pm?
A neutron with kinetic energy T = 25eV strikes a stationary deuteron (heavy hydrogen nucleus). Find the de Broglie wavelengths of both particles in the frame of their centre of inertia.
Two identical non-relativistic particles move at right angles to each other, possessing de Broglie wavelengths λ1 and λ2. Find the de Broglie wavelength of each particle in the frame of their centre of inertia.
Find the de Broglie wavelength of hydrogen molecules, which corresponds to their most probable velocity at room temperature.
Calculate the most probable de Broglie wavelength of hydrogen molecules being in thermodynamic equilibrium at room temperature.
Derive the expression for a de Broglie wavelength λ of a relativistic particle moving with kinetic energy T. At what values of T does the error in determining λ using the non-relativistic formula not exceed 1 % for an electron and a proton?
At what value of kinetic energy is the de Broglie wavelength of an electron equal to its Compton wavelength?
Find the de Broglie wavelength of relativistic electrons reaching the anticathode of an X-ray tube if the short wavelength limit of the continuous X-ray spectrum is equal to λsh = 10.0pm?
A parallel stream of mono energetic electrons falls normally on a diaphragm with narrow square slit of width b = 1.0μm. Find the velocity of the electrons if the width of the central diffraction maximum formed on a screen located at a distance l = 50 cm from the slit is equal to ∆x =
A parallel stream of electrons accelerated by a potential difference V = 25 V falls normally on a diaphragm with two narrow slits separated by a distance d = 50μm. Calculate the distance between neighbouring maxima of the diffraction pattern on a screen located at a distance l = 100 cm from
A narrow stream of mono energetic electrons falls at an angle of incidence θ = 30° on the natural facet of an aluminium single crystal. The distance between the neighbouring crystal planes parallel to that facet is equal to d = 0.20 nm. The maximum mirror reflection is observed at a certain
A narrow beam of mono energetic electrons falls normally on the surface of a Ni single crystal. The reflection maximum of fourth order is observed in the direction forming an angle θ = 55° with the normal to the surface at the energy of the electrons equal to T = 180 eV. Calculate the
A narrow stream of electrons with kinetic energy T = 10 keV passes through a polycrystalline aluminium foil, forming a system of diffraction fringes on a screen. Calculate the interplanar distance corresponding to the reflection of third order from a certain system of crystal planes if it is
A stream of electrons accelerated by a potential difference V falls on the surface of a metal whose inner potential is Vi = 15 V. Find: (a) The refractive index of the metal for the electrons accelerated by a potential difference V = 150 V; (b) The values of the ratio V/Vi at which the
A particle of mass m is located in a unidimensional square potential well with infinitely high walls. The width of the well is equal to l. Find the permitted values of energy of the particle taking into account that only those states of the particle's motion are realized for which the whole number
Describe the Bohr quantum conditions in terms of the wave theory: demonstrate that an electron in a hydrogen atom can move only along those round orbits which accommodate a whole number of de Broglie waves.
Estimate the minimum errors in determining the velocity of an electron, a proton, and a ball of mass of t mg if the coordinates of the particles and of the centre of the ball are known with uncertainly 1μm.
Employing the uncertainty principle, evaluate the indeterminancy of the velocity of an electron in a hydrogen atom if the size of the atom is assumed to be l = 0.10 nm. Compare the obtained magnitude with the velocity of an electron in the first Bohr orbit of the given atom.
Show that for the particle whose coordinate uncertainty is ∆x = λ /2π, where λ is its de Broglie wavelength, the velocity uncertainty is of the same order of magnitude as the particle's velocity itself.
A free electron was initially confined within a region with linear dimensions l = 0.10 nm. Using the uncertainty principle, evaluate the time over which the width of the corresponding train of waves becomes η = 10 times as large.
Employing the uncertainty principle, estimate the minimum kinetic energy of an electron confined within a region whose size is l = 0.20 nm.
An electron with kinetic energy T ≈ 4 eV is confined within a region whose linear dimension is l = 1μm. Using the uncertainty principle, evaluate the relative uncertainty of its velocity.
An electron is located in a unidimensional square potential well with infinitely high walls. The width of the well is l. From the uncertainty principle estimate the force with which the electron possessing the minimum permitted energy acts on the walls of the well.
A particle of mass rn moves in a unidimensional potential field U = kx2/2 (harmonic oscillator). Using the uncertainty principle, evaluate the minimum permitted energy of the particle in that field.
Making use of the uncertainty principle, evaluate the minimum permitted energy of an electron in a hydrogen atom and its corresponding apparent distance from the nucleus.
A parallel stream of hydrogen atoms with velocity v = 600 m/s falls normally on a diaphragm with a narrow slit behind which a screen is placed at a distance l = 1.0 m. Using the uncertainty principle, evaluate the width of the slit 5 at which the width of its image on the screen is minimum.
Find a particular solution of t-he time-dependent Schrödinger equation for a freely moving particle of mass m.
A particle in the ground state is located in a unidimensional square potential well of length l with absolutely impenetrable walls (0 < x < l). Find the probability of the particle staying within a region 1/3 l < x < 2/3 l.
A particle is located in a unidimensional square potential well with infinitely high walls. The width of the well is l. Find the normalized wave functions of the stationary states of the particle, taking the midpoint of the well for the origin of the x coordinate.
Demonstrate that the wave functions of the stationary states of a particle confined in a unidimensional potential well with infinitely high walls are orthogonal, i.e. they satisfy the conditionHere l is the width of the well, n are integers.
An electron is located in a unidimensional square potential well with infinitely high walls. The width of the well equal to l is such that the energy levels are very dense. Find the density of energy levels dN/dE, i.e. their number per unit energy interval, as a function of E. Calculate dN/dE for E
A particle of mass m is located in a two-dimensional square potential well with absolutely impenetrable walls. Find:(a) The particle's permitted energy values if the sides of the well are l1 and l2; (b) The energy values of the particle at the first four levels if the well has the shape of a square
A particle is located in a two-dimensional square potential well with absolutely impenetrable walls (0 < x < a, 0 < y < b). Find the probability of the particle with the lowest energy to be located within a region 0 < x < a/3.
A particle of mass m is located in a three-dimensional cubic potential well with absolutely impenetrable walls. The side of the cube is equal to a. Find: (a) The proper values of energy of the particle; (b) The energy difference between the third and fourth levels; (c) The energy of the sixth level
Using the Schrödinger equation, demonstrate that at the point where the potential energy U (x) of a particle has a finite discontinuity, the wave function remains smooth, i.e. its first derivative with respect to the coordinate is continuous.
A particle of mass m is located in a unidimensional potential field U (x) whose shape is shown in Fig. 6.2, where U (0) = ∞. Find: (a) The equation defining the possible values of energy of the particle in the region E < Uo; reduce that equation to the form sin kl = ± kl √h2/2ml2Uo,
Making use of the solution of the foregoing problem, determine the probability of the particle with energy E = Uo/2 to be located in the region x > l, if l2Uo = (3/4π) 2 h2/m.
Find the possible values of energy of a particle of mass m located in a spherically symmetrical potential well U (r) = 0 for r < ro and U(r) = ∞ for r = r0, in the case when the motion of the particle is described by a wave function ψ (r) depending only on r. When solving the Schr6dinger
From the conditions of the foregoing problem find: (a) Normalized Eigen functions of the particle in the states for which ψ (r) depends only on r; (b) The most probable value rpr for the ground state of the particle and the probability of the particle to be in the region r < rpr.
A particle of mass m is located in a spherically symmetrical potential well U(r) = 0 for r r0. (a) By means of the substitution ψ (r) = χ (r)/r find the equation defining the proper values of energy E of the particle for E(b) Calculate the value of the quantity r20Uo at which the first
The wave function of a particle of mass m in a unidimensional potential field U (x) = kx2/2 has in the ground state the form ψ (x) = Ae –ax2, where A is a normalization factor and a is a positive constant. Making use of the Schrödinger equation, find the constant a and the energy E of the
Find the energy of an electron of a hydrogen atom in a stationary state for which the wave function takes the form ψ (r) = A (1 + ar) e –ar, where A, a, and a are constants.
The wave function of an electron of a hydrogen atom in the ground state takes the form ψ (r) = Ae-r/r1, where A is a certain constant, r1 is the first Bohr radius. Find: (a) The most probable distance between the electron and the nucleus; (b) The mean value of modulus of the Coulomb force
Find the mean electrostatic potential produced by an electron in the centre of a hydrogen atom if the electron is in the ground state for which the wave function is ψ (r) = Ae -r/r1, where A is a certain constant, r1 is the first Bohr radius.
Particles of mass m and energy E move from the left to the potential barrier shown in Fig. 6.3 Find:(a) The reflection coefficient R of the barrier for E = Uo;(b) Te effective penetration depth of the particles into the region x > 0 for E
Employing Eq. (6.2e), find the probability D of an electron with energy E tunneling through a potential barrier of width l and height U0 provided the barrier is shaped as shown:(a) In Fig. 6.4;(b) In Fig. 6.5.
Using Eq. (6.2e), find the probability D of a particle of mass m and energy E tunnelling through the potential barrier shown in Fig. 6.6, where U (x) ---- Uo (1 €“x2/l2).
Knowing the decay constant λ of a nucleus, find: (a) The probability of decay of the nucleus during the time from 0 to t; (b) The mean lifetime τ of the nucleus.
What fraction of the radioactive cobalt nuclei whose half-life is 71.3 days decays during a month?
How many beta-particles are emitted during one hour by 1.0μg of Na24 radionuclide whose half-life is 15 hours?
To investigate the beta-decay of Mg23 radionuclide, a counter was activated at the moment t = 0. It registered N1 beta-particles by a moment t1 = 2.0s, and by a moment t2 = 3t1 the number of registered beta-particles was 2.66 times greater. Find the mean lifetime of the given nuclei.
The activity of a certain preparation decreases 2.5 times after 7.0 days. Find its half-life.
At the initial moment the activity of a certain radionuclide totaled 650 particles per minute. What will be the activity of the preparation after half its half-life period?
Find the decay constant and the mean lifetime of Co55 radionuclide if its activity is known to decrease 4.0% per hour. The decay product is non-radioactive.
A U238 preparation of mass 1.0 g emits 1.24 ∙ 104 alpha-particles per second. Find the half-life of this nuclide and the activity of the preparation.
Determine the age of ancient wooden items if it is known that the specific activity of C14 nuclide in them amounts to 3/5 of that in lately felled trees. The half-life of C14 nuclei is 5570 years.
In a uranium ore the ratio of U238 nuclei to Pb206 nuclei is η = 2.8. Evaluate the age of the ore, assuming all the lead Pb206 to be a final decay product of the uranium series. The half-life of U238 nuclei is 4.5 ∙ 109 years.
Calculate the specific activities of Na24 and U235 nuclides whose half-life are 15 hours and 7.1.10 s years respectively.
A small amount of solution containing Na24 radionuclide with activity A = 2.0 ∙ 103 disintegrations per second was injected in the bloodstream of a man. The activity of 1 cm 3 of blood sample taken t = 5.0 hours later turned out to be A = 16 disintegrations per minute per cm 3. The half-life
The specific activity of a preparation consisting of radioactive Co58 as and non-radioactive Co59 ao is equal to 2.2.1012 dis /(s ∙ g). The half-life of Co58 as is 71.3 days. Find the ratio of the mass of radio active cobalt in that preparation to the total mass of the preparation (in per
A certain preparation includes two beta-active components with different half-life. The measurements resulted in the following dependence of the natural logarithm of preparation activity on time t expressed in hours:Find the half-life of both components and the ratio of radioactive nuclei of these
A p32 radionuclide with half-life T = 14.3 days is produced in a reactor at a constant rate q = 2.7.109 nuclei per second. How soon after the beginning of production of that radionuclide will its activity be equal to A = 1.0 ∙ 109 dis/s?
A radionuclide A1 with decay constant λ1 transforms into a radionuclide A2 with decay constant λ2. Assuming that at the initial moment the preparation contained only the radionuclide A1, find: (a) The equation describing accumulation of the radionuclide A2 with time; (b) The time
Solve the foregoing problem if λ1 = λ2 = λ.
A radionuclide A1 goes through the transformation chain A1+ A2 → A3 (stable) with respective decay constants λ1 = and λ2. Assuming that at the initial moment the preparation contained only the radionuclide Ax equal in quantity to N10 nuclei; find the equation describing accumulation
A Bi210 radionuclide decays via the chain where the decay constants are λ1 = 1.60.10 -6 s -1, λ2 = 5.80 ∙ 10 -8 s -1.Calculate alpha- and beta-activities of the Bi210 preparation of mass i.00 mg a month after its manufacture.
(a) What isotope is produced from the alpha-radioactive Ra226 as a result of five alpha-disintegrations and four β-disintegrations? (b) How many alpha- and β-decays does U238 experience before turning finally into the stable Pb200 isotope?
A stationary Pb200 nucleus emits an alpha-particle with kinetic energy Ta = 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?
Find the amount of heat generated by 1.00 mg of a Po210 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.
The alpha-decay of Po210 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
The mean path length of alpha-particles in air under standard conditions is defined by the formula R = 0.98.10-27 v3o cm, where vo (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)
Find the energy Q liberated in β-- and β+- decays and in K-capture if the masses of the parent atom Mp, the daughter atom Md, and an electron m are known.
Evaluate the amount of heat produced during a day by a β--active Na24 preparation of mass m = 1.0 rag. 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.
Taking the values of atomic masses from the tables, calculate the kinetic energies of a positron and a neutrino emitted by C11 nucleus for the case when the daughter nucleus does not recoil.
Find the kinetic energy of the recoil nucleus in the positron-ic decay of a N13 nucleus for the case when the energy of positrons is maximum?
From the tables of atomic masses determine the velocity of a nucleus appearing as a result of K-capture in a Be7 atom provided the daughter nucleus turns out to be in the ground state.
Passing down to the ground state, excited Ag 109 nuclei emit either gamma quanta with energy 87keV or K conversion electrons whose binding energy is 26 keV. Find the velocity of these electrons.
A free stationary Ir191 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.
What must be the relative velocity of a source and an absorber consisting of free Ir TM nuclei to observe the maximum absorption of gamma quanta with energy e = 129 keV?
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?
What is the minimum height to which a gamma quanta source containing excited Zn67 nuclei has to be raised for the gravitational displacement of the Moss Bauer line to exceed the line width itself, when registered on the Earth's surface? The registered gamma quanta are known to have an energy e = 93
Taking the values of atomic masses from the tables, find the maximum kinetic energy of beta-particles emitted by Be10 nuclei and the corresponding kinetic energy of recoiling daughter nuclei formed directly in the ground state.
Determine the angular rotation velocity of an S2 molecule promoted to the first excited rotational level if the distance between its nuclei is d = 189 pm.
For an HC1 molecule find the rotational quantum numbers of two neighbouring levels whose energies differ by 7.86 meV. The nuclei of the molecule are separated by the distance of 127.5 pm.
Find the angular momentum of an oxygen molecule whose rotational energy is E = 2.16 meV and the distance between the nuclei is d = 121 pm.
Show that the frequency intervals between the neighbouring spectral lines of a true rotational spectrum of a diatomic molecule are equal. Find the moment of inertia and the distance between the nuclei of a CH molecule if the intervals between the neighbouring lines of the true rotational spectrum
For an HF molecule find the number of rotational levels located between the zeroth and first excited vibrational levels assuming rotational states to be independent of vibrational ones. The natural vibration frequency of this molecule is equal to 7.79.1014 rad/s, and the distance between the nuclei

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Discrete and Combinatorial Mathematics An Applied Introduction 5th edition Ralph P. Grimaldi - Solutions

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