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Ch 42: Nuclear Physics
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 42: Nuclear PhysicsProblem 66c
Chapter 42, Problem 66c

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.

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Step 1: Begin by understanding the principle of conservation of momentum. In any physical process, the total momentum before and after the event must remain the same. Here, the neutron is initially at rest, so its initial momentum is zero.
Step 2: Write the relativistic momentum expression for each particle involved in the decay. The relativistic momentum is given by the formula: p = mv1 - v2c2, where m is the rest mass, v is the speed, and c is the speed of light.
Step 3: Assign variables to the particles. Let the proton have momentum pp and the electron have momentum pe. Since the neutron is initially at rest, the total momentum after the decay must also sum to zero.
Step 4: Write the conservation of momentum equation. Since the neutron's initial momentum is zero, the momentum of the proton and the electron must cancel each other out: pp + pe = 0. Explicitly include the direction of momentum by assigning a negative sign to one of the terms: pp - pe = 0.
Step 5: Substitute the relativistic momentum expressions for the proton and electron into the equation. For the proton: pp = mpvp1 - vp2c2, and for the electron: pe = meve1 - ve2c2. Solve for the relationship between the proton's and electron's momenta.

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Key Concepts

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

Beta Decay

Beta decay is a type of radioactive decay in which a neutron is transformed into a proton, emitting an electron (beta particle) and an antineutrino. This process occurs in unstable nuclei to achieve a more stable configuration. Understanding beta decay is crucial for analyzing nuclear reactions and the behavior of subatomic particles.
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Conservation of Momentum

The conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it. In the context of beta decay, this principle allows us to relate the momenta of the neutron before decay and the proton and electron after decay. This is essential for deriving equations that describe the dynamics of the decay process.
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Relativistic Energy and Momentum

Relativistic energy and momentum take into account the effects of special relativity, particularly at high speeds close to the speed of light. The equations for energy and momentum are modified from classical mechanics to include factors such as rest mass and Lorentz factor. This framework is necessary for accurately describing particle interactions in processes like beta decay.
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Related Practice
Textbook Question

All the very heavy atoms found in the earth were created long ago by nuclear fusion reactions in a supernova, an exploding star. The debris spewed out by the supernova later coalesced into the gases from which the sun and the planets of our solar system were formed. Nuclear physics suggests that the uranium isotopes ²³⁵U and ²³⁸U should have been created in roughly equal numbers. Today, 99.28% of uranium is ²³⁸U and only 0.72% is ²³⁵U. How long ago did the supernova occur?

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Textbook Question

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 mn, mp and me and the relativistic factors yp and ye.

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Textbook Question

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?

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