In the lecture, Professor Murayama talked about how we can use cosmic ray muons to map...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
In the lecture, Professor Murayama talked about how we can use cosmic ray muons to map otherwise invisible things. A particularly novel example he discussed was Luis Alvarez looking for a hidden chamber in an Egyptian pyramid. In this problem we explore the basic idea a bit further. Muons, since they are charged, lose energy as they pass through material. Energetic muons (1000 GeV E 1 GeV) are in the so-called "minimum ionizing case" where they lose a constant amount of energy for most of their passage. Just before they stop, they rapidly lose their enery (just like the protons of last week's proton beam therapy question). If you want to see a plot of energy loss per distance of muons as a function of energy, see figure 30.1 (page 4) here: http://pdg.lbl.gov/2013/reviews/rpp2012-rev-passage-particles-matter.pdf In the minimum ionizing case, the "stopping power" of a material on muons is MeV cm² dE dr 2. g (don't worry about the derivative sign-calculus is not required for this problem unless you want!). What stopping power means is that for a given density of material, px dE is the energy lost per distance of the muon passing through the material (check the units to make sure this is the case!). We will do some warm up in parts (a) and (b) of this problem to make sure we understand how to work with the quantities given and what they mean. In part (c) we utilize the same concepts in a more realistic (yet challenging) scenario. (a) How far will a 100 GeV muon travel through rock? Take the density of rock to be 2.7 g/cm³ and give your answer to two significant figures. (b) Now imagine you have three different detectors as shown in figure 1. Each detector has an area of 1 m². We will be asking how many muons reach each detector. Imagine the flux of muons (here flux = number per area per time per energy, area time energy) to be constant # up to 100 GeV, and then zero for energies above that. 100 m 28-¹GeV-1 0 for E< 100 GeV for E 100 GeV (note that this constant flux in energy is not true in the real world see the optional part of this problem). How many muons reach detectors (B) and (C) after one hour? As is often the case, dimensional analysis provides the guidance for setting up the necessary equation. Give your answer to two significant figures. Hint: If you do this for detector A, you get Ndetect. A = 3.6 x 107 muons. If you need a further hint, online in the quiz feedback, I show the steps for calculating the answer for detector A. mim P-3 plom P₁-2 g/m² H 50 m с B 25 m P₁ P₂ 25 m Figure 1: Muons impingent on three different detectors. Each detector has an area of 1 m². In (A) there is no material above the detector. Detector (B) is beneath 50 m of rock with a density of p1 = 3 g/cm³. Detector (C) is beneath a tower of two different types of rock, p1 = 3 g/cm³ and p₂ = 2 g/cm³ (c) Note: below are two variations of the same problem. One requires calculus, the other doesn't. Either answer will be graded as correct. The actual flux is energy dependent, decreasing with energy as ox E-27. For concrete- ness, the flux is 7 (GEV) 27 for E> 1 GeV 100 m² s GeV where E is the energy measured in GeV (so that at E = 1 GeV the flux is the value used in part (b) of the problem). Variation of problem requiring calculus: Using the flux equation above, how many muons with initial energies between 1 and 100 GeV reach detectors (B) and (C) after one hour? Give your answer to two significant figures. Hints/tips/possible problems: If we were really doing this properly, we should specify a detector resolution, i.e. what range of muon energies can our detector measure. We won't worry about this (I've avoided this problem by not asking you about the flux at detector A where you might notice a problem if you tried to send the energy to zero, as well as by asking you only about the flux for given range of initial energies not their energies at the detector). As a check so you don't need to worry about if you are entering answers into the website correctly, if we let the maximum initial energy extend from 100 GeV to infinity, you would find 653 muons at detector B after one hour. Variation of problem without calculus: An approximate form of the muon flux is shown in figure 2. Using this graph, how many muons reach detectors (B) and (C) after one hour? Give your answer to two signficant figures. Hint: part (b) of this problem is like having the histogram at the height of my for 1 GeV < E < 100 GeV. GeV Flux (Gevms) 100 10 0.1 0.01 0.001 (1 GeV) (10 GeV) - 20 100 m² GeV 0.200 ² GeV 4(20 GeV)- 3.07x10 m² s GeV 40 60 80 Energy(GeV) Figure 2: Approximate flux of muons. The height of each block in the histogram is determined by o(E) at the points E= 1, 10, 20, 30, 40, 50, 60, 70, 80, and 90 GeV. The first few values at E = 1, 10, and 20 are shown in the figure. Use the equation for to get the values at the other points. To those that have done both ways of the problem, the histogram is an overestimation on the actual answer using calculus. For a better approximation, you could do the height of the histogram at the midway points, i.e. take the height as o(E) at E= 5, 15, 25, 35, GeV. In the lecture, Professor Murayama talked about how we can use cosmic ray muons to map otherwise invisible things. A particularly novel example he discussed was Luis Alvarez looking for a hidden chamber in an Egyptian pyramid. In this problem we explore the basic idea a bit further. Muons, since they are charged, lose energy as they pass through material. Energetic muons (1000 GeV E 1 GeV) are in the so-called "minimum ionizing case" where they lose a constant amount of energy for most of their passage. Just before they stop, they rapidly lose their enery (just like the protons of last week's proton beam therapy question). If you want to see a plot of energy loss per distance of muons as a function of energy, see figure 30.1 (page 4) here: http://pdg.lbl.gov/2013/reviews/rpp2012-rev-passage-particles-matter.pdf In the minimum ionizing case, the "stopping power" of a material on muons is MeV cm² dE dr 2. g (don't worry about the derivative sign-calculus is not required for this problem unless you want!). What stopping power means is that for a given density of material, px dE is the energy lost per distance of the muon passing through the material (check the units to make sure this is the case!). We will do some warm up in parts (a) and (b) of this problem to make sure we understand how to work with the quantities given and what they mean. In part (c) we utilize the same concepts in a more realistic (yet challenging) scenario. (a) How far will a 100 GeV muon travel through rock? Take the density of rock to be 2.7 g/cm³ and give your answer to two significant figures. (b) Now imagine you have three different detectors as shown in figure 1. Each detector has an area of 1 m². We will be asking how many muons reach each detector. Imagine the flux of muons (here flux = number per area per time per energy, area time energy) to be constant # up to 100 GeV, and then zero for energies above that. 100 m 28-¹GeV-1 0 for E< 100 GeV for E 100 GeV (note that this constant flux in energy is not true in the real world see the optional part of this problem). How many muons reach detectors (B) and (C) after one hour? As is often the case, dimensional analysis provides the guidance for setting up the necessary equation. Give your answer to two significant figures. Hint: If you do this for detector A, you get Ndetect. A = 3.6 x 107 muons. If you need a further hint, online in the quiz feedback, I show the steps for calculating the answer for detector A. mim P-3 plom P₁-2 g/m² H 50 m с B 25 m P₁ P₂ 25 m Figure 1: Muons impingent on three different detectors. Each detector has an area of 1 m². In (A) there is no material above the detector. Detector (B) is beneath 50 m of rock with a density of p1 = 3 g/cm³. Detector (C) is beneath a tower of two different types of rock, p1 = 3 g/cm³ and p₂ = 2 g/cm³ (c) Note: below are two variations of the same problem. One requires calculus, the other doesn't. Either answer will be graded as correct. The actual flux is energy dependent, decreasing with energy as ox E-27. For concrete- ness, the flux is 7 (GEV) 27 for E> 1 GeV 100 m² s GeV where E is the energy measured in GeV (so that at E = 1 GeV the flux is the value used in part (b) of the problem). Variation of problem requiring calculus: Using the flux equation above, how many muons with initial energies between 1 and 100 GeV reach detectors (B) and (C) after one hour? Give your answer to two significant figures. Hints/tips/possible problems: If we were really doing this properly, we should specify a detector resolution, i.e. what range of muon energies can our detector measure. We won't worry about this (I've avoided this problem by not asking you about the flux at detector A where you might notice a problem if you tried to send the energy to zero, as well as by asking you only about the flux for given range of initial energies not their energies at the detector). As a check so you don't need to worry about if you are entering answers into the website correctly, if we let the maximum initial energy extend from 100 GeV to infinity, you would find 653 muons at detector B after one hour. Variation of problem without calculus: An approximate form of the muon flux is shown in figure 2. Using this graph, how many muons reach detectors (B) and (C) after one hour? Give your answer to two signficant figures. Hint: part (b) of this problem is like having the histogram at the height of my for 1 GeV < E < 100 GeV. GeV Flux (Gevms) 100 10 0.1 0.01 0.001 (1 GeV) (10 GeV) - 20 100 m² GeV 0.200 ² GeV 4(20 GeV)- 3.07x10 m² s GeV 40 60 80 Energy(GeV) Figure 2: Approximate flux of muons. The height of each block in the histogram is determined by o(E) at the points E= 1, 10, 20, 30, 40, 50, 60, 70, 80, and 90 GeV. The first few values at E = 1, 10, and 20 are shown in the figure. Use the equation for to get the values at the other points. To those that have done both ways of the problem, the histogram is an overestimation on the actual answer using calculus. For a better approximation, you could do the height of the histogram at the midway points, i.e. take the height as o(E) at E= 5, 15, 25, 35, GeV.
Expert Answer:
Related Book For
Quantitative Analysis for Management
ISBN: 978-0132149112
11th Edition
Authors: Barry render, Ralph m. stair, Michael e. Hanna
Posted Date:
Students also viewed these physics questions
-
Consequentialist Analysis: We have covered how we can use thought experiments and consequentialist analysis in order to make sense of ethical dilemmas. In phase two, you'll apply those tools to your...
-
Answer the following question: How we can use correlation or regression analysis?
-
Explain how we can use the constant growth DDM to estimate the cost of firms' internal common equity, as well as the cost of new common share issues.
-
Which of the following is not necessary to do before you can run a Java program? a. Coding b. Compiling c. Debugging d. Saving
-
(a) What is the difference between Werner's concepts of primary valence and secondary valence? What terms do we now use for these concepts? (b) Why can the NH3 molecule serve as a ligand but the BH3...
-
Aquatech is a U.S.-based company that manufactures, sells, and installs water purification equipment. On April 11, the company sold a system to the City of Nagasaki, Japan, for installation in...
-
A gray gas with an absorption coefficient of \(\kappa\) is contained between two plates separated by a distance \(L\). Using the diffusion approximation for radiation, derive the relation...
-
Fred's Freight employs three drivers who are paid $20 per hour for regular time and $30 for overtime. A single pickup and delivery requires, on average, one hour of driver time. Drivers are paid for...
-
At the beginning of Year 1, Copeland Drugstore purchased a new computer system for $170,000. It is expected to have a five-year life and a $30,000 salvage value. Required a. Compute the depreciation...
-
The table below shows Lanark's production possibilities. Wheat Cars A 0 51 B 50 49 C 88 44 D 114 34 E 126 19 F 131 e a. If Lanark is producing 39 cars, it can produce approximately b. If Lanark is...
-
An engine causes a car to move 10 meters with a force of 100 N. The engine produces 10,000 J of energy. What is the efficiency of this engine?
-
If I needed to make an investment in equipment or find a third-party company, I would work with accounting to see what resources our company can spend, while also doing my own research on outside...
-
Saltz, Inc. maintains a perpetual inventory system and uses the First-in, First-out (FIFO) method of assigning costs. Purchases and sales of inventory for the month of July are as follows: Date...
-
What features of the Financial Reform Act address the causes of the financial crisis? What features of the Financial Reform Act address issues raised by the Abacus transaction? What features of the...
-
OopsieOil Corporation is a private company that follows ASPE. The corporation is currently facing a lawsuit due to illegal oil dumping. After consulting both their accounting and legal teams, it...
-
Assume for this part that Velocity Vehicles follows the warranty revenue approach at the start of 2020, to facilitate its transition when it goes public. Given repair costs are not evenly distributed...
-
The financial statements of Carla Vista Inc. are presented here: CARLA VISTA INC. Income Statement Year Ended December 31, 2021 Service revenue $2,185,500 Expenses Operating expenses $1,887,500...
-
Nate prepares slides for his microscope. In 1 day he prepared 12 different slides. Which equation best represents y, the total number of slides Nate prepares in x days if he continues at this rate? A...
-
Woofer Pet Foods produces a low-calorie dog food for overweight dogs. This product is made from beef products and grain. Each pound of beef costs $0.90, and each pound of grain costs $0.60. A pound...
-
Small boxes of NutraFlakes cereal are labeled "net weight 10 ounces." Each hour, random samples of size boxes are weighed to check process control. Five hours of observations yielded the following:...
-
Burger City is a large chain of fast-food restaurants specializing in gourmet hamburgers. A mathematical model is now used to predict the success of new restaurants based on location and demographic...
-
N = 230, n = 15, k = 200 Compute the mean and standard deviation of the hypergeometric random variable X.
-
N = 60, n = 8, k = 25 Compute the mean and standard deviation of the hypergeometric random variable X.
-
One study showed that in a certain year, airline fatalities occur at the rate of 0.011 deaths per 100 million miles. Find the probability that, during the next 100 million miles of flight, there will...
Study smarter with the SolutionInn App