A hypothetical diatomic molecule of oxygen (mass = 2.656 * 10-26 kg) and hydrogen (mass = 1.67 * 10-27 kg) emits a photon of wavelength 2.39 µm when it makes a transition from one vibrational state to the next lower state. If we model this molecule as two point masses at opposite ends of a massless spring, a. what is the force constant of this spring, and b. how many vibrations per second is the molecule making?
> The most common radium isotope found on earth, 226Ra, has a half-life of about 1600 years. If the earth was formed well over 109 years ago, why is there any radium left now?
> One problem in radiocarbon dating of biological samples, especially very old ones, is that they can easily be contaminated with modern biological material during the measurement process. What effect would such contamination have on the estimated age? Why
> In Example 43.9 (Section 43.4), the activity of atmospheric carbon before 1900 was given. Discuss why this activity may have changed since 1900. From Example 43.9 EXAMPLE 43.9 RADIOCARBON DATING Then, from Eq. (43.16), Before 1900 the activity per
> In the process of internal conversion, a nucleus decays from an excited state to a ground state by giving the excitation energy directly to an atomic electron rather than emitting a gamma-ray photon. Why can this process also produce x-ray photons?
> Why is the alpha, beta, or gamma decay of an unstable nucleus unaffected by the chemical situation of the atom, such as the nature of the molecule or solid in which it is bound? The chemical situation of the atom can, however, have an effect on the half-
> In a nuclear decay equation, why can we represent an electron as 1 0
> If (_Z^A)Eli represents the initial nuclide, what is the decay process or processes if the final nuclide is
> For a particle in a box, what would the probability distribution function Ψ 2 look like if the particle behaved like a classical (Newtonian) particle? Do the actual probability distributions approach this classical form when n is very large? Explain.
> Compared to α particles with the same energy, β particles can much more easily penetrate through matter. Why is this?
> In the 238U decay series shown in Fig. 43.7, some nuclides in the series are found much more abundantly in nature than others, even though every 238U nucleus goes through every step in the series before finally becoming 206Pb. Why don’t
> Neutrons have a magnetic dipole moment and can undergo spin flips by absorbing electromagnetic radiation. Why, then, are protons rather than neutrons used in MRI of body tissues? (See Fig. 43.1.) From Fig. 43.1 43.1 Magnetic resonance imaging (MRI)
> In what ways do atoms in a diatomic molecule behave as though they were held together by a spring? In what ways is this a poor description of the interaction between the atoms?
> The air you are breathing contains primarily nitrogen (N2) and oxygen (O2). Many of these molecules are in excited rotational energy levels (l = 1, 2, 3, ……), but almost all of them are in the vibrational ground level (n = 0). Explain this difference bet
> Various organic molecules have been discovered in interstellar space. Why were these discoveries made with radio telescopes rather than optical telescopes?
> Discuss the differences between the rotational and vibrational energy levels of the deuterium (“heavy hydrogen”) molecule D2 and those of the ordinary hydrogen molecule H2. A deuterium atom has twice the mass of an ordinary hydrogen atom.
> Many radioactive decays occur within a sequence of decays—for example, 92 234
> Radioisotopes are used in a variety of manufacturing and testing techniques. Wear measurements can be made using the following method. An automobile engine is produced using piston rings with a total mass of 100 g, which includes 9.4 µCi of 59Fe whose ha
> The H2+ molecule consists of two hydrogen nuclei and a single electron. What kind of molecular bond do you think holds this molecule together? Explain.
> If Ψ is normalized, what is the physical significance of the area under a graph of Ψ 2 versus x between x1 and x2? What is the total area under the graph of Ψ 2 when all x are included? Explain.
> Consider a simple model of the helium atom in which two electrons, each with mass m, move around the nucleus (charge +2e) in the same circular orbit. Each electron has orbital angular momentum ħ (that is, the orbit is the smallest-radius Bohr
> Each of 2N electrons (mass m) is free to move along the x-axis. The potential-energy function for each electron is U(x)= 1/2 k′x2, where k′ is a positive constant. The electric and magnetic interactions between electrons can be ignored. Use the exclusion
> How could you make compensated silicon that has twice as many acceptors as donors?
> Protons, neutrons, and many other particles are made of more fundamental particles called quarks and antiquarks (the antimatter equivalent of quarks). A quark and an antiquark can form a bound state with a variety of different energy levels, each of whic
> The WKB approximation (see Challenge Problem 40.64) can be used to calculate the energy levels for a harmonic oscillator. In this approximation, the energy levels are the solutions to the equation. Here E is the energy, U(x) is the potential-energy fun
> It can be a challenge to solve the Schrödinger equation for the bound-state energy levels of an arbitrary potential well. An alternative approach that can yield good approximate results for the energy levels is the WKB approximation (named f
> There are several methods for removing electrons from the surface of a semiconductor. Can holes be removed from the surface? Explain.
> What is the essential characteristic for an element to serve as a donor impurity in a semiconductor such as Si or Ge? For it to serve as an acceptor impurity? Explain.
> Why are materials that are good thermal conductors also good electrical conductors? What kinds of problems does this pose for the design of appliances such as clothes irons and electric heaters? Are there materials that do not follow this general rule?
> The assumptions of the free-electron model of metals may seem contrary to reason, since electrons exert powerful electric forces on each other. Give some reasons why these assumptions actually make physical sense.
> For the particle in a box, we chose k = nπ/L with n = 1, 2, 3, …………. to fit the boundary condition that Ψ = 0 at x = L. However, n = 0, -1, -2, -3, …… also satisfy that boundary condition. Why didn’t we also choose those values of n?
> An isolated zinc atom has a ground-state electron configuration of filled 1s, 2s, 2p, 3s, 3p, and 4s subshells. How can zinc be a conductor if its valence subshell is full?
> Use the band model to explain how it is possible for some materials to undergo a semiconductor-to-metal transition as the temperature or pressure varies.
> Speeds of molecules in a gas vary with temperature, whereas speeds of electrons in the conduction band of a metal are nearly independent of temperature. Why are these behaviors so different?
> Consider a collision in which a stationary particle with mass M is bombarded by a particle with mass m, speed v0, and total energy (including rest energy) Em. a. Use the Lorentz transformation to write the velocities vm and vM of particles m and M in te
> What factors determine whether a material is a conductor of electricity or an insulator? Explain.
> When the pressure p on a material increases by an amount ∆p, the volume of the material will change from V to V + ∆V, where ∆V is negative. The bulk modulus B of the material is defined to be the rati
> Consider a system of N free electrons within a volume V. Even at absolute zero, such a system exerts a pressure p on its surroundings due to the motion of the electrons. To calculate this pressure, imagine that the volume increases by a small amount dV.
> An electron in a hydrogen atom is in an s level, and the atom is in a magnetic field B = B
> In the ground state of the helium atom one electron must have “spin down” and the other “spin up.” Why?
> In the Stern–Gerlach experiment, why is it essential for the magnetic field to be inhomogeneous (that is, nonuniform)?
> If a particle is in a stationary state, does that mean that the particle is not moving? If a particle moves in empty space with constant momentum p and hence constant energy E = p2/2m, is it in a stationary state? Explain your answers.
> a. If two electrons in hydrogen atoms have the same principal quantum number, can they have different orbital angular momenta? How? b. If two electrons in hydrogen atoms have the same orbital quantum number, can they have different principal quantum num
> The Stern–Gerlach experiment is always performed with beams of neutral atoms. Wouldn’t it be easier to form beams using ionized atoms? Why won’t this work?
> Why is the analysis of the helium atom much more complex than that of the hydrogen atom, either in a Bohr type of model or using the Schrödinger equation?
> For a body orbiting the sun, such as a planet, comet, or asteroid, is there any restriction on the z-component of its orbital angular momentum such as there is with the z-component of the electron’s orbital angular momentum in hydrogen? Explain.
> Repeat Discussion Question Q41.24 for the wave function Ψ(r1, r2) = Ψα(r1)Ψα(r2). From Q41.24: A system of two electrons has the wave function Ψ(r1, r2)=(1/ 2 )[Ψa(r1) Ψb(r2)- Ψb(r1)Ψa(r2)], where ca is a normalized wave function for a state with Sz =
> The binding energy of a potassium chloride molecule (KCl) is 4.43 eV. The ionization energy of a potassium atom is 4.3 eV, and the electron affinity of chlorine is 3.6 eV. Use these data to estimate the equilibrium separation between the two atoms in the
> a. The equilibrium separation of the two nuclei in an NaCl molecule is 0.24 nm. If the molecule is modeled as charges +e and -e separated by 0.24 nm, what is the electric dipole moment of the molecule (see Section 21.7)? b. The measured electric dipole
> When a diatomic molecule undergoes a transition from the l = 2 to the l = 1 rotational state, a photon with wavelength 54.3 µm is emitted. What is the moment of inertia of the molecule for an axis through its center of mass and perpendicular to the line
> You have entered a graduate program in particle physics and are learning about the use of symmetry. You begin by repeating the analysis that led to the prediction of the Ω- particle. Nine of the spin-3/2 baryons are four ∆ particles, each with mass 1232
> The decay products from the decay of shortlived unstable particles can provide evidence that these particles have been produced in a collision experiment. As an initial step in designing an experiment to detect short-lived hadrons, you make a literature
> While tuning up a medical cyclotron for use in isotope production, you obtain the data given in the table. B is the uniform magnetic field in the cyclotron, and Kmax is the maximum kinetic energy of the particle being accelerated, which is a proton. Th
> A system of two electrons has the wave function Ψ(r1, r2)=(1/ 2 )[Ψa(r1) Ψb(r2)- Ψb(r1)Ψa(r2)], where ca is a normalized wave function for a state with Sz = + 1/2 ħ and Ψb is a normalized wave function for a state with Sz = - 1/2 ħ. a. If Sz for electr
> The K0 meson has rest energy 497.7 MeV. A K0 meson moving in the +x-direction with kinetic energy 225 MeV decays into a π+ and a π-, which move off at equal angles above and below the +x-axis. Calculate the kinetic energy of the π+ and the angle it makes
> A Σ- particle moving in the +x-direction with kinetic energy 180 MeV decays into a π- and a neutron. The π- moves in the +y-direction. What is the kinetic energy of the neutron, and what is the direction of its velocity? Use relativistic expressions for
> A Ξ- particle at rest decays to a Λ0 and a π-. a. Find the total kinetic energy of the decay products. b. What fraction of the energy is carried off by each particle? (Use relativistic expressions for momentum and energy.)
> The densities of ordinary matter and dark matter have decreased as the universe has expanded, since the same amount of mass occupies an ever-increasing volume. Yet observations suggest that the density of dark energy has remained constant over the entire
> A ϕ meson (see Problem 44.45) at rest decays via ϕ → K+ + K-. It has strangeness 0. a. Find the kinetic energy of the K+ meson. (Assume that the two decay products share kinetic energy equally, since their masses are equal.) b. Suggest a reason the dec
> One proposed proton decay is p+ → e+ + π0, which violates both baryon and lepton number conservation, so the proton lifetime is expected to be very long. Suppose the proton half-life were 1.0 * 1018 y. a. Calculate the energy deposited per kilogram of b
> Estimate the energy width (energy uncertainty) of the ψ if its mean lifetime is 7.6 * 10-21 s. What fraction is this of its rest energy?
> The ϕ meson has mass 1019.4 MeV/c2 and a measured energy width of 4.4 MeV/c2. Using the uncertainty principle, estimate the lifetime of the f meson.
> An η0 meson at rest decays into three p mesons. a. What are the allowed combinations of π0, π+, and π- as decay products? b. Find the total kinetic energy of the π mesons.
> Each of the following reactions is missing a single particle. Calculate the baryon number, charge, strangeness, and the three lepton numbers (where appropriate) of the missing particle, and from this identify the particle. a. p + p → p + Λ0 + ? ; b. K
> Can a hydrogen atom emit x rays? If so, how? If not, why not?
> Calculate the threshold kinetic energy for the reaction π- + p → Σ0 + K0 if a π- beam is incident on a stationary proton target. The K0 has a mass of 497.7 MeV/c2.
> Calculate the threshold kinetic energy for the reaction p + p → p + p + K+ + K- if a proton beam is incident on a stationary proton target.
> A proton and an antiproton collide head-on with equal kinetic energies. Two γ rays with wavelengths of 0.720 fm are produced. Calculate the kinetic energy of the incident proton.
> Beams of π- mesons are used in radiation therapy for certain cancers. The energy comes from the complete decay of the π- to stable particles. a. Write out the complete decay of a π- meson to stable particles. What are th
> In your job as a health physicist, you measure the activity of a mixed sample of radioactive elements. Your results are given in the table. Time (h)………….Decays,s 0………………7500 0.5……………4120 1.0…………….2570 1.5……………1790 2.0……………1350 2.5……………1070 3.0……………..872
> Your company develops radioactive isotopes for medical applications. In your work there, you measure the activity of a radioactive sample. Your results are given in the table. Time (h)…..Decays,s 0……………20,000 0.5………….14,800 1.0………….11,000 1.5…………….8130 2
> As a scientist in a nuclear physics research lab, you are conducting a photodisintegration experiment to verify the binding energy of a deuteron. A photon with wavelength l in air is absorbed by a deuteron, which breaks apart into a neutron and a proton.
> Consider the fusion reaction 1 2
> In the 1986 disaster at the Chernobyl reactor in eastern Europe, about 1/8 of the 137Cs present in the reactor was released. The isotope 137Cs has a half-life of 30.07 y for b decay, with the emission of a total of 1.17 MeV of energy per decay. Of this,
> A bone fragment found in a cave believed to have been inhabited by early humans contains 0.29 times as much 14C as an equal amount of carbon in the atmosphere when the organism containing the bone died. (See Example 43.9 in Section 43.4.) Find the approx
> An atom in its ground level absorbs a photon with energy equal to the K absorption edge. Does absorbing this photon ionize this atom? Explain.
> Nuclear weapons tests in the 1950s and 1960s released significant amounts of radioactive tritium ( 1 3
> A 60Co source with activity 2.6 * 10-4 Ci is embedded in a tumor that has mass 0.200 kg. The source emits g photons with average energy 1.25 MeV. Half the photons are absorbed in the tumor, and half escape. a. What energy is delivered to the tumor per s
> The nucleus 8 15
> Measurements indicate that 27.83% of all rubidium atoms currently on the earth are the radioactive 87Rb isotope. The rest are the stable 85Rb isotope. The half-life of 87Rb is 4.75 * 1010 y. Assuming that no rubidium atoms have been formed since, what pe
> A 70.0-kg person experiences a whole-body exposure to α radiation with energy 4.77 MeV. A total of 7.75 * 1012 α particles are absorbed. a. What is the absorbed dose in rad? b. What is the equivalent dose in rem? c. If the source is 0.0320 g of 226Ra
> In 1952 spectral lines of the element technetium-99 (99Tc) were discovered in a red giant star. Red giants are very old stars, often around 10 billion years old, and near the end of their lives. Technetium has no stable isotopes, and the half-life of 99T
> A person ingests an amount of a radioactive source that has a very long lifetime and activity 0.52 µCi. The radioactive material lodges in her lungs, where all of the emitted 4.0-MeV α particles are absorbed within a 0.50-kg mass of tissue. Calculate the
> Calculate the mass defect for the β+ decay of 6 11
> A neutral pion (π0) has a mass of 264 times the electron mass and decays with a lifetime of 8.4 * 10-17 s to
> The radiocarbon in our bodies is one of the naturally occurring sources of radiation. Let’s see how large a dose we receive. 14C decays via β- emission, and 18% of our body’s mass is carbon. a. Write out t
> a. Can you show that the orbital angular momentum of an electron in any given direction (e.g., along the z-axis) is always less than or equal to its total orbital angular momentum? In which cases would the two be equal to each other? b. Is the result in
> The polonium isotope 84 210
> The atomic mass of 12 25
> Thorium 90 230
> One of the problems of in-air testing of nuclear weapons (or, even worse, the use of such weapons!) is the danger of radioactive fallout. One of the most problematic nuclides in such fallout is strontium-90 (90Sr), which breaks down by β- decay with a ha
> The isotope 47 110
> a. Calculate the minimum energy required to remove one neutron from the nucleus 8 17
> a. Calculate the minimum energy required to remove one proton from the nucleus 6 12
> a. When gasoline is burned, it releases 1.3 * 108 J of energy per gallon (3.788 L). Given that the density of gasoline is 737 kg/m3, express the quantity of energy released in J/g of fuel. b. During fission, when a neutron is absorbed by a 235U nucleus,
> A p­n junction is part of the control mechanism for a wind turbine that is used to generate electricity. The turbine has been malfunctioning, so you are running diagnostics. You can remotely change the bias voltage V applied to the junction an
> The table gives the occupation probabilities f(E) as a function of the energy E for a solid conductor at a fixed temperature T To determine the Fermi energy of the solid material, you are asked to analyze this information in terms of the Fermiâ&
> The nucleus of a gold atom contains 79 protons. How does the energy required to remove a 1s electron completely from a gold atom compare with the energy required to remove the electron from the ground level in a hydrogen atom? In what region of the elect
> Why must the wave function of a particle be normalized?
> To determine the equilibrium separation of the atoms in the HCl molecule, you measure the rotational spectrum of HCl. You find that the spectrum contains these wavelengths (among others): 60.4 µm, 69.0 µm, 80.4 µm, 96.4 µm, and 120.4 µm. a. Use your mea
> Metallic lithium has a bcc crystal structure. Each unit cell is a cube of side length a = 0.35 nm. a. For a bcc lattice, what is the number of atoms per unit volume? Give your answer in terms of a. (Hint: How many atoms are there per unit cell?) b. Use
