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Chapter 7: Problem 16

At low energies, protons and neutrons behave roughly like hard spheres of equal mass and radius about \(1.3 \times 10^{-14} \mathrm{~m} .\) A parallel beam of neutrons, with a flux of \(3 \times 10^{10}\) neutrons \(\mathrm{m}^{-2} \mathrm{~s}^{-1}\), strikes a target containing \(4 \times 10^{22}\) protons. A circular detector of radius \(20 \mathrm{~mm}\) is placed \(0.7 \mathrm{~m}\) from the target, in a direction making an angle of \(30^{\circ}\) to the beam direction. Calculate the rate of detection of neutrons, and of protons.

Short Answer

Expert verified
Question: Calculate the rate of detection of neutrons and protons in the following scenario: Neutrons with a flux of \(3 \times 10^{10}\) neutrons/m²s interact with protons in a target. Neutrons and protons are approximated as hard spheres with a radius of \(1.3 \times 10^{-14}\) m. The target contains \(4 \times 10^{22}\) protons. A circular detector with a radius of 0.02 m is placed at a distance of 0.7 m from the target at an angle of \(30^{\circ}\). Compute the rate of detection of neutrons and protons. Answer: First, follow the steps in the solution to calculate the Interaction Area, Interaction Rate/Area, Interaction Rate, Detector Area, Solid Angle, and Effective Solid Angle. Step 1: Interaction Area = \(\pi (2(1.3 \times 10^{-14}))^2 = 2.13 \times 10^{-28}\) m² Step 2: Interaction Rate/Area = \(3 \times 10^{10}\) neutrons/m²s × \(2.13 \times 10^{-28}\) m² = \(6.39 \times 10^{-18}\) interactions/m²s Step 3: Interaction Rate = \(6.39 \times 10^{-18}\) interactions/m²s × \(4 \times 10^{22}\) protons = \(2.56 \times 10^{5}\) interactions/s Step 4: Detector Area = \(\pi(0.02)^2 = 1.26 \times 10^{-3}\) m² Solid Angle = \(\frac{1.26 \times 10^{-3} \, m²}{4 \pi (0.7)^2 \, m²} = 8.19 \times 10^{-5}\) Effective Solid Angle = \(8.19 \times 10^{-5} \cos{30^{\circ}} = 7.09 \times 10^{-5}\) Step 5: Rate of Detection = \(2.56 \times 10^{5}\) interactions/s × \(7.09 \times 10^{-5}\) = 18.15 interactions/s Hence, the rate of detection of neutrons and protons in this scenario is approximately 18.15 interactions per second.

Step by step solution

01

Calculate the interaction area

We will approximate protons and neutrons as hard spheres of radius 1.3 x 10^{-14}m. For them to interact with each other, their surfaces must be in contact. So, the interaction area, which is the cross-sectional area where the particles collide, can be calculated as: Interaction Area \(= \pi (2r)^{2}\) Where \(r = 1.3 \times 10^{-14}\) m
02

Calculate the total number of neutron-proton interactions per unit area per second

The neutron flux \(\phi\) is given as \(3 \times 10^{10}\) neutrons/m²s. To calculate the total number of neutron-proton interactions happening per unit area per second, we will multiply the neutron flux by the interaction area: Interaction Rate/Area = \(\phi\) x Interaction Area Where \(\phi = 3 \times 10^{10}\) neutrons/m²s
03

Calculate the total number of neutron-proton interactions in the target per second

To find the total number of interactions happening in the target per second, we will multiply the interaction rate per unit area by the number of protons in the target: Interaction Rate = Interaction Rate/Area x Number of Protons in the target Where the Number of Protons in the target = \(4 \times 10^{22}\)
04

Calculate the fraction of interactions detected by the detector

The detector is placed 0.7m away from the target at an angle of \(30^{\circ}\). As a result, not all neutron-proton interactions will be detected. To determine the fraction of interactions detected, we will use the geometry of the detector. The area of the detector can be calculated as: Detector Area = \(\pi R^2\) Where \(R = 0.02\) m Also, we will approximate the solid angle \(\Omega\) subtended by the detector at the target as: \(\Omega = \frac{Detector \, Area}{4 \pi d^2}\) Where \(d = 0.7\) m Finally, since the detector is at an angle of \(30^{\circ}\), the effective solid angle is: \(\Omega_{eff} = \Omega \cos{30^{\circ}}\)
05

Calculate the rates of neutron and proton detection

Now, using the interaction rate and the effective solid angle, we can find the rate of detection of neutrons and protons as: Rate of Detection = Interaction Rate x \(\Omega_{eff}\) Compute these values using the numbers obtained in previous steps to find the rate of detection of neutrons and protons.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Neutron Flux
Neutron flux is a fundamental concept in nuclear physics and reactor design that quantifies the flow of neutrons in a material or space. When working with nuclear reactions, such as the interactions between neutrons and protons, we need to understand this flow to predict how reactions occur over time.

The neutron flux, denoted by \( \phi \) and typically measured in neutrons per square meter per second (neutrons/m²s), indicates how many neutrons pass through a unit area in one second. A high value of neutron flux means that more neutrons are available to initiate reactions, and therefore, a higher likelihood of interactions. It's crucial for calculations such as determining the reaction rate in a nuclear reactor or, in this case, estimating the number of neutron-proton collisions.

By knowing the neutron flux and the cross-sectional area of the target particles, we can calculate the interaction rate. In the given problem, the neutron flux is \( 3 \times 10^{10} \) neutrons/m²s. This value allows us to estimate how many neutrons engage with protons within the target, ultimately affecting the rate of detection by a detector.
Cross-Sectional Area Calculation
The cross-sectional area plays a pivotal role when particles collide. In the context of neutron-proton interactions, we treat the particles as spheres, assuming that an interaction occurs when the surfaces of these spheres make contact. This basic approach simplifies the understanding of a rather complex quantum mechanical process.

To calculate the cross-sectional area where the neutron and proton collide, we use the formula, \( \text{{Area}} = \pi (2r)^{2} \), where \( r \) is the radius of each particle. Since they are hard spheres with a radius of about \(1.3 \times 10^{-14} \text{{~m}}\), we can determine the area of interaction necessary for estimating the interaction rate. For truly hard spheres, this area is equivalent to the 'target' that the incoming neutron needs to hit for a collision to occur.

Understanding the concept of interaction cross-sectional area helps in various fields, including particle physics, materials science, and even medicine, where understanding such collisions at a microscopic level is crucial for applications such as radiation therapy.
Solid Angle Estimation
In three-dimensional space, the concept of a solid angle is analogous to the two-dimensional angle we are familiar with. The solid angle, measured in steradians (sr), represents a portion of a sphere's surface area in relation to its radius squared and is crucial when estimating how much of a signal, such as particles or light, is captured by a detector.

To estimate the solid angle \( \Omega \) subtended by a detector at the target, we take the formula \( \Omega = \frac{{\text{{Detector Area}}}}{4 \pi d^2} \), where \( d \) is the distance from the detector to the source, and \( \text{{Detector Area}} \) is \( \pi R^2 \) with \( R \) being the radius of the detector.

In our problem, however, the detector is not directly facing the target but is angled. Hence, we need to adjust our solid angle calculation to reflect this orientation. We do this by multiplying the calculated solid angle by the cosine of the angle between the detector and the neutron beam. This 'effective solid angle' accounts for the actual area of the detector that is exposed to the particles or signals from the source.

Understanding solid angle estimations is vital for applications ranging from astrophysics, where it helps in the analysis of the luminosity of stars, to medical imaging, influencing the detection efficiency of radiological equipment.

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