35. Special Relativity

A sample of 1.0 x 10¹⁰ atoms that decay by alpha emission has a half-life of 100 min. How many alpha particles are emitted between t = 50 min and t = 200 min?

A Geiger counter is used to measure the decay of a radioactive isotope produced in a nuclear reactor. Initially, when the sample is first removed from the reactor, the Geiger counter registers 15,000 decays/s. 15 h later the count is down to 5500 decays/s. At what time after the sample's removal from the reactor is the count 1200 decays/s?

For those that are not stable, identify both the decay mode and the daughter nucleus.

It might seem strange that in beta decay the positive proton, which is repelled by the positive nucleus, remains in the nucleus while the negative electron, which is attracted to the nucleus, is ejected. To understand beta decay, let's analyze the decay of a free neutron that is at rest in the laboratory. We'll ignore the antineutrino and consider the decay n → p⁺ + e⁻. The analysis requires the use of relativistic energy and momentum, from Chapter 36. Write the equation that expresses the conservation of relativistic energy for this decay. Your equation will be in terms of the three masses m_n, m_p and m_e and the relativistic factors y_p and y_e.

It might seem strange that in beta decay the positive proton, which is repelled by the positive nucleus, remains in the nucleus while the negative electron, which is attracted to the nucleus, is ejected. To understand beta decay, let's analyze the decay of a free neutron that is at rest in the laboratory. We'll ignore the antineutrino and consider the decay n → p⁺ + e⁻. The analysis requires the use of relativistic energy and momentum, from Chapter 36. Write the equation that expresses the conservation of relativistic momentum for this decay. Let v represent speed, rather than velocity, then write any minus signs explicitly.

It might seem strange that in beta decay the positive proton, which is repelled by the positive nucleus, remains in the nucleus while the negative electron, which is attracted to the nucleus, is ejected. To understand beta decay, let's analyze the decay of a free neutron that is at rest in the laboratory. We'll ignore the antineutrino and consider the decay n → p⁺ + e⁻. The analysis requires the use of relativistic energy and momentum, from Chapter 36. What is the total kinetic energy, in MeV, of the proton and electron?

The starship Enterprise, of television and movie fame, is powered by combining matter and antimatter. If the entire $400$-kg antimatter fuel supply of the Enterprise combines with matter, how much energy is released? How does this compare to the U.S. yearly energy use, which is roughly $1.0\times {10}^{20}$ J?

FIGURE EX39.14 is a graph of |ψ(x)|² for an electron. What is the probability that the electron is located between x = 1.0 nm and x = 2.0 nm?

Consider the electron wave function where x is in cm. If 10⁴ electrons are detected, how many will be in the interval 0.00 cm ≤ x ≤ 0.50 cm?

A particle is described by the wave function where L = 2.0 mm. Calculate the probability of finding the particle within 1.0 mm of the origin.

For a particle in a finite potential well of width L and depth U₀, what is the ratio of the probability Prob(in δx at x=L+η) to the probability Prob(in δx at x = L)?

Potassium and gold cathodes are used in a photoelectric-effect experiment. For each cathode, find: The stopping potential if the wavelength is 220 nm.

A particle is described by the wave function where L = 2.0 mm. Interpret your answer to part b by shading the region representing this probability on the appropriate graph in part a.

What is the speed of a particle when its kinetic energy equals its rest energy? Does the mass of the particle affect the result?

INT Model an atom as an electron in a rigid box of length 0.100 nm, roughly twice the Bohr radius. What are the four lowest energy levels of the electron?