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Ch 39: Wave Functions and Uncertainty
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 39: Wave Functions and UncertaintyProblem 43b
Chapter 39, Problem 43b

Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted. The probability that a nucleus will undergo alpha decay is proportional to the frequency with which the alpha particle reflects from the walls of the nucleus. What is that frequency (reflections/s) for a maximum-speed alpha particle within a 238U nucleus?

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1
Step 1: Understand the problem. The question asks for the frequency of reflections of an alpha particle within a uranium-238 nucleus. This frequency is proportional to the speed of the alpha particle and the dimensions of the nucleus. We need to calculate the frequency using the formula for frequency in terms of speed and distance.
Step 2: Identify the relevant formula. The frequency of reflections can be calculated using the formula: \( f = \frac{v}{2d} \), where \( v \) is the speed of the alpha particle and \( d \) is the approximate diameter of the nucleus. The factor of 2 accounts for the round trip of the alpha particle across the nucleus.
Step 3: Estimate the diameter of the nucleus. The diameter of a uranium-238 nucleus can be approximated using the formula for nuclear radius: \( R = R_0 A^{1/3} \), where \( R_0 \) is the nuclear radius constant (approximately \( 1.2 \times 10^{-15} \, \text{m} \)) and \( A \) is the mass number of the nucleus (238 for uranium-238). The diameter \( d \) is then \( 2R \).
Step 4: Determine the maximum speed of the alpha particle. The maximum speed can be estimated using the kinetic energy of the alpha particle, which is typically on the order of a few MeV. Use the relationship \( KE = \frac{1}{2}mv^2 \) to solve for \( v \), where \( m \) is the mass of the alpha particle (approximately \( 6.64 \times 10^{-27} \, \text{kg} \)).
Step 5: Combine the values into the frequency formula. Substitute the calculated values of \( v \) and \( d \) into \( f = \frac{v}{2d} \) to find the frequency of reflections. Ensure units are consistent throughout the calculation (e.g., meters for distance, seconds for time).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable nucleus emits an alpha particle, which consists of two protons and two neutrons (essentially a helium nucleus). This process reduces the atomic number of the original nucleus by two and the mass number by four, leading to the formation of a new element. Understanding alpha decay is crucial for analyzing the stability of heavy nuclei and the mechanisms of nuclear reactions.
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Nuclear Potential Well

The nuclear potential well is a model that describes the forces acting within a nucleus, where nucleons (protons and neutrons) are bound together by the strong nuclear force. The potential well creates a region where particles can exist, and the energy levels within this well determine the behavior of particles, including alpha particles. The concept is essential for understanding how alpha particles can be emitted from the nucleus and the conditions that affect their stability.
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Reflection Frequency

Reflection frequency refers to the rate at which an alpha particle bounces off the boundaries of the nuclear potential well. This frequency is influenced by the particle's speed and the size of the nucleus. In the context of alpha decay, a higher reflection frequency increases the likelihood of the alpha particle escaping the nucleus, making it a key factor in determining the probability of decay events.
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Related Practice
Textbook Question

Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted. A 238U nucleus, which decays by alpha emission, is 15 fm in diameter. Model an alpha particle within a 238U nucleus as being in a one-dimensional box. What is the maximum speed an alpha particle is likely to have?

Textbook Question

A pulse of light is created by the superposition of many waves that span the frequency range f₀ − (1/2) Δf ≤ f ≤ f₀ + (1/2) Δf, where f₀ = c/λ is called the center frequency of the pulse. Laser technology can generate a pulse of light that has a wavelength of 600 nm and lasts a mere 6.0 fs (1 fs = 1 femtosecond =10−15 s). What is the spatial length of the laser pulse as it travels through space?

Textbook Question

Consider the electron wave function ψ(x)={c1−x2∣x∣≤1 cm0∣x∣≥1 cm\(\psi\) (x)=\(\begin{cases}\) c\(\sqrt{1-x^{2}\)} & \(\left\)|x\(\right\)|\(\leq\) 1\(\text{ cm}\) \\ 0 & \(\left\)|x\(\right\)|\(\geq\) 1\(\text{ cm}\) \(\end{cases}\) where x is in cm. If 104 electrons are detected, how many will be in the interval 0.00 cm ≤ x ≤ 0.50 cm?