Съдържание

Условие Отговори Решение

USAPhO 2024, Полуфинал — Задача 4. Question B1 · The Muon Shot

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от USAPhO 2024

AAPT AIP UNITED STATES PHYSICS TEAM 2024 — USA Physics Olympiad Exam · 25 т.

Important Instructions for the Exam Supervisor

DO NOT DISTRIBUTE THIS PAGE

This examination has two parts. Each part has three questions and lasts for 90 minutes.
For each student, print out one copy of the exam and one copy of the answer sheets. Print everything single-sided, and do not staple anything. Divide the exam into the instructions (pages 2–3), Part A questions (pages 4–7), and Part B questions (pages 8–10). Page 11 is a graph page, in case students wish to use it as an additional page for problem B3.
Begin by giving students the instructions and all of the answer sheets. Let the students read the instructions and fill out their information on the answer sheets. They can keep the instructions for both parts of the exam. Also give students blank sheets of paper to use as scratch paper throughout the exam.
Students may bring calculators, but they may not use symbolic math, programming, or graphing features of these calculators. Calculators may not be shared, and their memory must be cleared of data and programs. Cell phones or other electronics may not be used during the exam or while the exam papers are present. Students may not use books or other references.
To start the exam, collect signed Honesty Policy, give students the Part A questions, and allow 90 minutes to complete Part A. Do not give students Part B during this time, even if they finish with time remaining. At the end of the 90 minutes, collect the Part A questions and answer sheets.
Then give students a 5 to 10 minute break. Then give them the Part B questions, and allow 90 minutes to complete Part B. Do not let students go back to Part A.
At the end of the exam, collect everything, including the questions, the instructions, the answer sheets, and the scratch paper. Give them the Honor Code Certification and collect signed Codes. Students may not keep the exam questions.
After the exam, sort each student’s answer sheets by page number. Scan every answer sheet, including blank ones. Everybody may discuss the questions after April 3rd. Until April 19th, hold on to all of the answer sheets in the event that your scans are lost or illegible.

We acknowledge the following people for their contributions to this year’s exam (in alphabetical order):

Tengiz Bibilashvili, Kellan Colburn, Natalie LeBaron, Rishab Parthasarathy, Elena Yudovina, and Kevin Zhou.

Instructions for the Student

You should receive these instructions, the reference table on the next page, answer sheets, and blank paper for scratch work. Read this page carefully before the exam begins.
You may use a calculator, but its memory must be cleared of data and programs, and you may not use symbolic math, programming, or graphing features. Calculators may not be shared. Cell phones or other electronics may not be used during the exam or while the exam papers are present. You may not use books or other outside references.
When the exam begins, your proctor will give you the questions for Part A. You will have 90 minutes to complete three problems. Each question is worth 25 points, but they are not necessarily of the same difficulty. If you finish all of the questions, you may check your work, but you may not look at Part B during this time.
After 90 minutes, your proctor will collect the questions and answer sheets for Part A. You may then take a short break.
Then you will work on Part B. You have 90 minutes to complete three problems. Each question is worth 25 points. Do not look at Part A during this time. When the exam ends, you must return all papers to the proctor, including the exam questions.
Do not discuss the questions of this exam, or their solutions, until April 4. Violations of this rule may result in disqualification.

Below are instructions for writing your solutions.

All of your solutions must be written on the official answer sheets. Nothing outside these answer sheets will be graded. Before the exam begins, write your name, student AAPT number, and proctor AAPT number as directed on the answer sheets.
There are several answer sheets per problem. If you run out of space for a problem, you may use the extra answer sheets, which are at the end of the answer sheet packet. To ensure this work is graded, you must indicate, at the bottom of your last answer sheet for that problem, that you are using these extra answer sheets.
Only write within the frame of each answer sheet. To simplify grading, we recommend drawing a box around your final answer for each subpart. You should organize your work linearly and briefly explain your reasoning, which will help you earn partial credit. You may use either pencil or pen, but sure to write clearly so your work will be legible after scanning.

Fundamental Constants

$g = 9.8\ \mathrm{N/kg}$ $\qquad$ $G = 6.67 \times 10^{-11}\ \mathrm{N \cdot m^2/kg^2}$

$k = 1/4\pi\epsilon_0 = 8.99 \times 10^{9}\ \mathrm{N \cdot m^2/C^2}$ $\qquad$ $k_{\mathrm{m}} = \mu_0/4\pi = 10^{-7}\ \mathrm{T \cdot m/A}$

$c = 3.00 \times 10^{8}\ \mathrm{m/s}$ $\qquad$ $k_{\mathrm{B}} = 1.38 \times 10^{-23}\ \mathrm{J/K}$

$N_{\mathrm{A}} = 6.02 \times 10^{23}\ (\mathrm{mol})^{-1}$ $\qquad$ $R = N_{\mathrm{A}} k_{\mathrm{B}} = 8.31\ \mathrm{J/(mol \cdot K)}$

$\sigma = 5.67 \times 10^{-8}\ \mathrm{J/(s \cdot m^2 \cdot K^4)}$ $\qquad$ $e = 1.602 \times 10^{-19}\ \mathrm{C}$

$1\ \mathrm{eV} = 1.602 \times 10^{-19}\ \mathrm{J}$ $\qquad$ $h = 6.63 \times 10^{-34}\ \mathrm{J \cdot s} = 4.14 \times 10^{-15}\ \mathrm{eV \cdot s}$

$m_e = 9.109 \times 10^{-31}\ \mathrm{kg} = 0.511\ \mathrm{MeV/c^2}$

Useful Approximations

(1+x)^n \approx 1 + nx + n(n-1)x^2/2 \text{ for } |nx| \ll 1

e^x \approx 1 + x + x^2/2 + x^3/6 \text{ for } |x| \ll 1

\sin\theta \approx \theta - \theta^3/6 \text{ for } |\theta| \ll 1

\cos\theta \approx 1 - \theta^2/2 \text{ for } |\theta| \ll 1

You may use this sheet for both parts of the exam.

End of Instructions for the Student

DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD TO BEGIN

Условие

In 2023, American particle physicists recommended developing a muon collider to investigate the nature of fundamental particles. Such a collider requires much less space then other options because of the muon’s high mass $m$ , which makes it easier to accelerate to a very high energy $E \gg mc^2$ .

a. When a muon collides head-on with an antimuon, which has the same energy and mass, a new particle of mass $2E/c^2$ can be produced. If the antimuon was instead at rest, what energy would the muon need to produce such a particle?

Unfortunately, muons and antimuons are unstable, with lifetime $\tau$ . That is, if one such particle exists at time $t = 0$ , then in its rest frame, the probability it has not decayed by time $t$ is $e^{-t/\tau}$ .

b. Suppose the muons begin at rest, and are accelerated so that each muon’s energy increases at a very large, constant rate $\alpha$ in the lab frame. Find the fraction $f$ of muons that have decayed by the time each muon has energy $E$ , assuming $f$ is small.

The collider produces a “bunch” of muons with energy $E$ , uniformly distributed in a thin disc of radius $R = 10^{-6}\ \mathrm{m}$ . It also simultaneously produces a similar “antibunch” of antimuons. For simplicity, model each muon and antimuon as a sphere of radius $r = 10^{-21}\ \mathrm{m}$ , and suppose a muon-antimuon collision occurs whenever two such spheres touch.

c. Initially, the bunch and antibunch each contain $N = 10^{14}$ particles. If they immediately collide head-on, what is the average number of muon-antimuon collisions, to one significant figure?

d. The bunch travels clockwise along a ring of circumference $\ell = 10\ \mathrm{km}$ , while the antibunch travels along the same path in the opposite direction. Assume all particles maintain a constant energy $E = 10^5\,mc^2$ , and that the muon lifetime is $\tau = 2.2 \times 10^{-6}\ \mathrm{s}$ . To one significant figure, what is the average number of muon-antimuon collisions that occur before all of the particles decay?

Отговори

Решение

a. The shortest solution involves setting $c = 1$ and using four-vectors. In the original situation, the muon and antimuon have four-momenta $p_1^\mu$ and $p_2^\mu$ , where

p_1 \cdot p_2 = (E, p) \cdot (E, -p) = E^2 + p^2.

To have an equivalent collision where the antimuon is at rest, we boost this configuration to the antimuon’s rest frame, where $p_1'^\mu = (E', p')$ and $p_2'^\mu = (m, 0)$ . The inner product of the four-vectors stays the same, so

p_1' \cdot p_2' = E'm = p_1 \cdot p_2.

Since $E \gg m$ , we have $p^2 \approx E^2$ , so solving for $E'$ and restoring the factors of $c$ gives

E' = \frac{2E^2}{mc^2}.

This is much greater than $E$ , so it is practical to accelerate the antimuon as well. Incidentally, you can also solve the problem exactly, in which case you’ll get

E' = \frac{2E^2}{mc^2} - mc^2 \approx \frac{2E^2}{mc^2}.

Either answer is acceptable.

b. Continuing to set $c = 1$ , the energy of the muon in the lab frame is $E(t) = \alpha t + m$ , but we also know that $E(t) = \gamma(t)m$ , so that

\gamma = 1 + \frac{\alpha t}{m}.

The acceleration process begins at time $t = 0$ , and ends at time $t_f = (E - m)/\alpha$ . Accounting for time dilation, the amount of time that elapses in the muon’s frame is

\tau_f = \int_0^{t_f} \frac{dt}{\gamma} = \frac{m}{\alpha}\int_0^{t_f} \frac{1}{t + m/\alpha} = \frac{m}{\alpha}\ln\frac{E}{m}.

To relate this to $f$ , we note that

f = 1 - e^{-\tau_f/\tau} \approx \frac{\tau_f}{\tau}

where the second step uses the assumption that $f$ is small. Therefore,

f = \frac{mc^2}{\alpha\tau}\ln\frac{E}{mc^2}

where we restored the factors of $c$ .

c. Consider one muon and antimuon. A collision occurs when their centers are separated by less than $2r$ . Fixing the location of the muon, the probability that the antimuon is within the appropriate area is approximately $\pi(2r)^2/(\pi R^2) = (2r/R)^2$ , since $r \ll R$ . Each pair of muons and antimuons has the same chance to collide, so the expected number of collision events is

\left(\frac{2rN}{R}\right)^2 = 0.04.

This is much smaller than $N$ , which justifies our assumption that the collision events are independent. It might seem odd for the answer to be less than 1, but this is desired, as having many collisions occur at once would make it hard to see what happens in each collision.

d. If the number of particles remaining in the bunch and the antibunch is $N_k$ , where $k = 0$ for the first collision, then the expected number of collisions is

\left(\frac{2rN}{R}\right)^2 \sum_{k=0}^{\infty}\left(\frac{N_k}{N}\right)^2.

We found above that only a small number of collisions occurs per bunch-antibunch crossing, so the decrease in $N_k$ over time is almost entirely due to decay. A crossing occurs every time each bunch or antibunch traverses half of the ring, corresponding to a proper time increment

\Delta\tau = \frac{\ell/(2c)}{10^5} = 1.67 \times 10^{-10}\ \mathrm{s}.

Therefore, since $\tau \gg \Delta\tau$ , we have

\sum_{k=0}^{\infty}\left(\frac{N_k}{N}\right)^2 = \sum_{k=0}^{\infty} e^{-2k\,\Delta\tau/\tau} = \frac{1}{1 - e^{-2\,\Delta\tau/\tau}} \approx \frac{\tau}{2\,\Delta\tau} = 6600.

The expected total number of collisions is 260, which rounds to 300.

The rough numbers given here correspond to a muon collider which would be able to probe new particles 10 times as heavy as those probed at the existing Large Hadron Collider. Roughly one in a million muon-antimuon collisions yield a Higgs boson, so that an enormous number of them can be produced for detailed study. Remarkably, such a muon collider could be smaller in size than the LHC, while other proposals involving electrons or protons would need to be about 10 times longer. However, given the muon’s short lifetime, it may be very hard to create the required focused muon beams. The feasibility of this “muon shot” is currently being investigated by particle physicists around the world.

One of us (TB) thanks Nathaniel Craig, Andrew Fee and Sergo Jindariani for discussions during our work on this problem.

Оригинал в Архива: USAPhO_Exam_24.pdf · официални решения: 2024-USAPhO-Exam_solutions.pdf