- Compute the normalization constant C210 in Equation 7-34. 210 = C210e-Zr/2an cos 0 ao 7-34
- Consider muonic atoms (see Problem 4-19). (a) Draw a correctly scaled and labeled partial energy level diagram including levels with n = 1, 2, 3, 4, 5, and ∞ for muonic hydrogen.(b) Compute
- (a) What is the ratio of the number of particles per unit area on the screen scattered at 10º to those at 1º? (b) What is the ratio of those scattered at 30º to those at 1º?
- The series of hydrogen spectral lines with m = 4 is called Brackett’s series. Compute the wavelengths of the first four lines of Brackett’s series.
- Compute the wavelength and frequency of the series limit for the Lyman, Balmer, and Paschen spectral series of hydrogen.
- Compute (x) and (x2) for (a) The ground state, (b) The first excited state, and (c) The second excited state of the harmonic oscillator.
- Write down the wave function for the hydrogen atom when the electron’s quantum numbers are n = 3, ℓ = 4, and mℓ = -1. Check to be sure that the wave function is normalized.
- Verify that the wave function found in Problem 7-28 is a solution of the time-independent Schrödinger equation, Equation 7-9.Problem 7-28Write down the wave function for the hydrogen atom when the
- The force on a magnetic moment with z component μz moving in an inhomogeneous magnetic field is given by Equation 7-51. If the silver atoms in the Stern- Gerlach experiment traveled horizontally 1 m
- Show that the expectation value of r for the electron in the ground state of a one-electron atom is (r) = (3/2)a0/Z.
- Since the P states and the D states of sodium are all doublets, there are four possible energies for transitions between these states. Indicate which three transitions are allowed and which one is
- Find the number density N/V for electrons such that (a) e-a = 1 and(b) e-a = 10-6.
- Use the Maxwell distribution of molecular speeds to calculate (v2) for the molecules of a gas.
- Compute the maximum fractional contribution to the heat capacity of solid iron that can be made by the electrons.
- The separation of nearest-neighbor ions in the KCl crystal (an FCC structure) is 0.315 nm. Use this information to determine the density of KCL.
- Show that the H+ - H- system cannot be ionically bonded.
- Compute the dissociation energy of molecular NaBr in kilocalories per mole.
- Calculate the Fermi energy for magnesium in a long, very thin wire.
- Show that for T = 300 K, about 0.1 percent of the free electrons in metallic silver have an energy greater than EF.
- At what temperature is the heat capacity due to the electron gas in copper equal to 10 percent of that due to lattice vibrations?
- Show that the magnetic susceptibility Χ is a dimensionless quantity.
- Arsenic has five valence electrons. If arsenic is used as a dopant in silicon, compute(a) The ionization energy and(b) The orbit radius of the fifth arsenic electron. The effective mass for electrons
- Gallium has three valence electrons. If gallium is used to dope germanium, compute(a) The ionization energy of the hole and (b) The orbit radius of the hole. The effective mass of holes in
- Approximating atoms in an FCC crystal as hard spheres of radius r with a being the length of each side of the unit cube, what fraction of the volume of the cube (and hence the crystal) is occupied by
- Crystallographers and materials scientists use the density of a metallic sample to infer its likely crystal structure. The density of copper (Cu) is 8.96 g/cm3 and its atomic radius is 0.128 nm. Is
- In the anomalous Zeeman effect, the external magnetic field is much weaker than the internal field seen by the electron as a result of its orbital motion. In the vector model the vectors L and S
- Recalling that the light-year c · y is the distance light travels in one year, compute in meters the distance equivalent to 1 light-second, 1 light-minute, 1 light-hour, and 1 light-day.
- A gold foil of thickness 2.0 μm is used in a Rutherford experiment to scatter α particles with energy 7.0 MeV. (a) What fraction of the particles will be scattered at angles greater than
- The wavelength of a particular line in the Balmer series is measured to be 379.1 nm. What transition does it correspond to?
- An astronomer finds a new absorption line with λ = 164.1 nm in the ultraviolet region of the Sun’s continuous spectrum. He attributes the line to hydrogen’s Lyman series. Is he right? Justify
- In a sample that contains hydrogen, among other things, four spectral lines are found in the infrared with wavelengths 7460 nm, 4654 nm, 4103 nm, and 3741 nm. Which one does not belong to a hydrogen
- Figure 4-26 shows an energy loss spectrum for He measured in an apparatus such as that shown in Figure 4-24a. Use the spectrum to construct and draw carefully to scale an energy-level diagram for
- A sample of hydrogen atoms are all in the n = 5 state. If all the atoms return to the ground state, how many different photon energies will be emitted, assuming all possible transitions occur? If
- A finite square well 1.0 fm wide contains one neutron. How deep must the well be if there are only two allowed energy levels for the neutron?
- An electron is confined to a finite square well whose “walls” are 8.0 eV high. If the ground-state energy is 0.5 eV, estimate the width of the well.
- Use conservation of energy to obtain an expression connecting x2 and p2 for a harmonic oscillator, then use it along with the result from Problem 6-34 to compute (p2) for the harmonic oscillator
- (a) Using A0 from Problem 6-34, write down the total wave function Ψ0(x, t) for the ground state of a harmonic oscillator.(b) Use the operator for px from Table 6-1 to compute (p2).Problem
- At what values of r/a0 is the radial function R30 equal to zero? (See Table 7-2.) Table 7-2 n=1 n=2 n=3 Table 7-2 Radial functions for hydrogen l=0 e = 0 l=1 l = 0 l = 1 l =
- Use dimensional analysis to show that the expression for the energy levels of hydrogenlike atoms given by Equation 7-25 has the units of energy. 22 kZe² E. - (46) +- 2 ħ 2n² n² 7-25
- Write down all possible sets of quantum numbers for an electron in a(a) 4f, (b) 3d, and (c) 2p subshell.
- Show that the wave function of Equation 7-57 satisfies the Schrödinger equation (Equation 7-55) with V = 0 and find the energy of this state. ħ² 2m (X1, X₂) ax² ħ2² 2² (X₁, X₂) ax² 2m +
- The molar mass of oxygen gas (O2) is about 32 g/mol and that of hydrogen gas (H2) about 2 g/mol. Compute(a) The rms speed of O2 and (b) The rms speed of H2 when the temperature is 0°C.
- (a) Compute e-a from Equation 8-44 for O2 gas at standard conditions. (b) At what temperature is e-a = 1 for O2? e Nh³ 2 (2πm₂kT) ³/² V 3/2 or e 3/2 2(2πm₂kT) ³/² V Nh³ 8-44
- Like 4He, the most common form of neon, 20Ne, is a rare gas and the 20Ne atoms have zero spin and so are bosons. But unlike helium, neon does not become superfluid at low temperatures. Show that this
- Find the average energy of an oscillator at (a) T = 10hf/k,(b) T = hf/k, and(c) T = 0.1hf/k, and compare your results with those from the equipartition theorem.
- (a) Show that the rule of Dulong-Petit follows directly from Einstein’s specific heat formula (Equation 8-62) as T → ∞.(b) Show that CV → 0 as T → 0. Cy= du dT hf 2 ehf/kT = 3 (17) AK
- Using Figure 8-13, compute the (approximate) frequency of atomic oscillations in silicon and in aluminum at 200 K.Figure 8-13 Cy, kcal/kmol K 7 6 5 4 3 2 1 0 T 200 Lead Aluminum Silicon Carbon
- Use Equation 8-62 to calculate the value of CV for a solid at the Einstein temperature TE = hf/k. du Cy= = dT 2 3NAK 3 №₁k (17) ² - ehf/kT (ehf/kT - 1)² 8-62
- Use Equation 8-69 to plot an accurate graph of nFD(E) /V for electrons whose Fermi energy is 4.8 eV from E = 4.5 eV to E = 5.1 eV at T = 300 K. Determine from the graph the number of electrons per
- The molar heat capacity data given in Table 8-2 are taken from AIP Handbook, 2d ed. (McGraw-Hill, New York, 1963). Plot the data for these solids all on one graph and sketch in the curves CV versus
- Consider a gas of electrons (fermions) and a gas of photons (bosons). Which has more states available at T = 1 K? Explain why.
- Electrons emitted in b decay have energies of the order of 1 MeV or smaller. Use this fact and the uncertainty principle to show that electrons cannot exist inside the nucleus.
- The spin of the ground state of Li, which constitutes 7.5 percent of natural lithium, is zero. Show that this value is not compatible with a model of the nucleus that consists of protons and
- If the assumptions leading to the Bose-Einstein distribution are modified so that the number of particles is not assumed constant, the resulting distribution has ea = 1. This distribution can be
- Consider a small silicon crystal measuring 100 nm on each side.(a) Compute the total number N of silicon atoms in the crystal. (The density of silicon is 2.33 g/cm3.)(b) If the conduction band in
- When arsenic is used to dope silicon, the fifth arsenic electron and the As+ ion act like a hydrogen atom system, except that the potential function V(r) and the electron mass must be modified as
- What are the number of protons and the number of neutrons in each of the following isotopes? 18F, 25Na, 51V, 84Kr, 120Te, 148Dy, 175W, and 222Rn.
- Suppose that the deuteron really did consist of two protons and one electron. (It doesn’t!) Compute the spin and magnetic moment of such a deuteron’s ground state and compare the results with the
- The magnetic moment of 14N is 0.4035 μN. Show that this value is not compatible with a model of the nucleus that consists of protons and electrons.
- Give the symbols for at least two isotopes and two isotones of each of the following nuclides:(a) 18F,(b) 208Pb, and (c) 120Sn.
- Give the symbols for at least two isobars and one isotope of each of the following nuclides: (a) 14O, (b) 63Ni, and (c) 236Np.
- Approximating the mass of a nucleus with mass number A as A x u and using Equation 11-3, compute the nuclear density in SI units. R = R₁A¹/³ with Ro= 1.2 0.2 fm 11-3
- Use the masses in the table in Appendix A to compute the total binding energy and the binding energy per nucleon of the following nuclides:(a) 9Be, (b) 13C, and(c) 57Fe.
- Compute the “charge distribution radius” from Equation 11-5 and the “nuclear force radius” from Equation 11-7 for the following nuclides:(a) 16O, (b) 63Cu, and(c) 208Pb. R= (1.07 0.02)
- 39Ca and 39K are a mirror pair, 39Ca decaying into 39K. Use Equations 11-1 and 11-2 to compute the radius of 40Ca. U = 319² 3 1 q 5 4περ R 11-1
- 62Cu is produced at a constant rate [e.g., by the (ϒ, n) reaction on 63Cu placed in a high-energy x-ray beam] and decays by β+ decay with a half-life of about 10 min. How long does it take to
- The decay constant of 235U is 9.8 x 10-10 y-1.(a) Compute the half-life. (b) How many decays occur each second in a 1.0 μg sample of 235U? (c) How many 235U atoms will remain in the 1.0 mg
- The decay constant of 22Na is 0.266 y-1.(a) Compute the half-life. (b) What is the activity of a sample containing 1.0 g of 22Na? (c) What is the activity of the sample after 3.5 years have
- Using Figure 11-16, find the parameters A and B in Equation 11-30.Figure 11-16 a decay half-life, s 10:20 1015 1010 105 10⁰ 10-5 10-10 4 5 1 7 6 8 a kinetic energy, MeV 9 10% y 10º y 10³ y 1y 1
- Make a diagram like Figure 11-18 for the (4n + 1) decay chain that begins with 237Np, a nuclide that is no longer present in nature. N 140 135 130 125 Thorium series (4n) 212pb 208
- Show that the a particle emitted in the decay of 232Th carries away 4.01 MeV, or 98 percent, of the total decay energy.
- 7Be decays exclusively by electron capture to 7Li with a half-life of 53.3 d. Would the characteristics of the decay be altered and, if so, how if (a) A sample of 7Be were placed under very high
- Compute the energy carried by the neutrino in the electron capture decay of 67Ga to the ground state of 67Zn.
- Compute the maximum energy of the β- particle emitted in the decay of 72Zn.
- With the aid of Figures 11-19 and 11-20, list the energies of all of the possible g rays that may be emitted by 223Ra following the a decay of 227Th.Figures 11-19 Number of
- In Example 11-13 we saw that 233Np could decay by emitting an a particle. Show that decay by emission of a nucleon of either type is forbidden for this nuclide.
- 8Be is very unusual among low-Z nuclides: it decays by emitting two a particles. Show why 8Be is unstable toward a decay.
- 80Br can undergo all three types of b decay.(a) Write down the decay equation in each case. (b) Compute the decay energy for each case.
- Assuming that the average separation between two protons in 12C is equal to the nuclear diameter, compute the Coulomb force of repulsion and the gravitational force of attraction between the protons.
- Suppose the range of the nuclear force was 5 fm. Compute the mass (in MeV/c2) of an exchange particle that might mediate such a force.
- The repulsive force that results in the “hard core” of the nucleus might be due to the exchange of a particle, just as the strong attractive force is. Compute the mass of such an exchange
- The nuclei listed below have filled j shells plus or minus one nucleon. (For example, 2914Si has the 1d5/2 shell filled for both neutrons and protons plus one neutron in the 2s1/2 shell.) Use the
- Use the shell model to predict the nuclear magnetic moments of the isotopes listed in Problem 11-38.Problem 11-38The nuclei listed below have filled j shells plus or minus one nucleon. (For example,
- The atomic spectral lines of 14N exhibit a hyperfine structure indicating that the ground state is split into three closely spaced levels. What must be the spin of the 14N ground state?
- Sketch diagrams like Figure 11-9 for the ground states of 3H, 3He, 14N, 14C, 15N, 15O, and 16O.Figure 11-9
- Which of the following nuclei have closed neutron shells? 36S, 50V, 50Ca, 53Mn, 61Ni, 82Ge, 88Sr, 93Ru, 94Ru, 131In, and 145Eu?
- Which of the following nuclei have closed proton shells: 3He, 19F, 12C, 40Ca, 50Ti, 56Fe, 60Ni, 60Cu, 90Zr, 124Sn, 166Yb, and 204Pb?
- (a) Use Figure 11-35 to draw a diagram like Figure 11-9 for 13N. (b) What value would you predict for the value of j? (c) What value would you predict for j for the first excited
- Use Figure 11-35 to predict the values of j for the ground states of 30Si, 37Cl, 55Co, 90Zr, and 107In.Figure 11-35
- (a) Find the Q value for the reaction 3H + 1H → 3He + n + Q. (b) Find the threshold for this reaction if stationary 1H nuclei are bombarded with 3H nuclei from an accelerator. (c) Find
- What is the compound nucleus for the reaction of deuterons on 14N? What are the possible product nuclei and particles for this reaction?
- Using data from Appendix A, compute the Q value for the reaction(a) 12C(a, p) 15N, and (b) 16O(p, d)17O.
- The cross section for the reaction 75As(n, ϒ)76As is 4.5 b for thermal neutrons. A sample of natural As in the form of a crystal 1 cm x 2 cm that is 30 mm thick is exposed to a thermal neutron flux
- Write three different reactions that could produce the products (a) n + 23Na, (b) p + 14C, and (c) d + 31P.
- Write down the correct symbol for the particle or nuclide represented by the x in the following reactions:(a) 14N(n, p)x, (b) 208Pb(n, x)208Pb, (c) x(a, p)61Cu,(d) 9Be(x, n)12C, (e)
- A few minutes after the Big Bang the first fusion reaction occurred in the early universe. It was n + p → d + ϒ. Compute the Q for this reaction.
- Write down the several reactions possible when 235U captures a thermal neutron and 1n, 2n, 3n, or 4n are produced.
- Assuming an average energy release of 17.6 MeV/fusion, calculate the rate at which 2H must be supplied to a 500 MW fusion reactor.
- Compute the total energy released in the following set of fusion reactions. This is the proton-proton cycle, the primary source of the Sun’s energy ¹H + ¹H-¹H + e + ve 2H + ¹H³He + y He He He
- A particular nuclear power reactor operates at 1000 MWe (megawatts electric) with an overall efficiency in converting fission energy to electrical energy of 30 percent. What mass of 235U must fission

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