com Problems and Solutions on Atomic, Nuclear and Particle Physics www.com This Page Intentionally Left Blank www.com Major American Universities Ph. Qualifying Questions and Solutions Problems and Solutions on Atomic, Nuclear and Particle Physics Compiled by The Physics Coaching Class University of Science and Technology of China Edited by Yung-Kuo Lim National University of Singapore World Scientific Singapore • New Jersey • London • Hong Kong www.com Published by World Scientific Publishing Co. P 0 Box 128, Farrer Road, Singapore 912805 USA office: Suite lB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Major American Universities Ph.
Qualifying Questions and Solutions PROBLEMS AND SOLUTIONS ON ATOMIC, NUCLEAR AND PARTICLE PHYSICS Copyright © 2000 by World Scientific Publishing Co. All rights reserved. This book, or parts, thereof may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA.
In this case permission to photocopy is not required from the publisher. ISBN 981-02-3917-3 981-02-3918-l (pbk) This book is printed on acid-free paper. Printed in Singapore by Uto-Print www.com PREFACE This series of physics problems and solutions, which consists of seven volumes — Mechanics, Electromagnetism, Optics, Atomic, Nuclear and Particle Physics, Thermodynamics and Statistical Physics, Quantum Me- chanics, Solid State Physics and Relativity, contains a selection of 2550 problems from the graduate-school entrance and qualifying examination papers of seven U. universities — California University Berkeley Cam- pus, Columbia University, Chicago University, Massachusetts Institute of Technology, New York State University Buffalo Campus, Princeton Uni- versity, Wisconsin University — as well as the CUSPEA and C.
Ting’s papers for selection of Chinese students for further studies in U., and their solutions which represent the effort of more than 70 Chinese physicists, plus some 20 more who checked the solutions. The series is remarkable for its comprehensive coverage. In each area the problems span a wide spectrum of topics, while many problems overlap several areas. The problems themselves are remarkable for their versatil- ity in applying the physical laws and principles, their uptodate realistic situations, and their scanty demand on mathematical skills.
Many of the problems involve order-of-magnitude calculations which one often requires in an experimental situation for estimating a quantity from a simple model. In short, the exercises blend together the objectives of enhancement of one’s understanding of physical principles and ability of practical application. The solutions as presented generally just provide a guidance to solving the problems, rather than step-by-step manipulation, and leave much to the students to work out for themselves, of whom much is demanded of the basic knowledge in physics. Thus the series would provide an invaluable complement to the textbooks.
The present volume consists of 483 problems. It covers practically the whole of the usual undergraduate syllabus in atomic, nuclear and particle physics, but in substance and sophistication goes much beyond. Some problems on experimental methodology have also been included. In editing, no attempt has been made to unify the physical terms, units and symbols.
Rather, they are left to the setters’ and solvers’ own prefer- ence so as to reflect the realistic situation of the usage today. Great pains has been taken to trace the logical steps from the first principles to the final solution, frequently even to the extent of rewriting the entire solution.com vi Preface In addition, a subject index to problems has been included to facilitate the location of topics. These editorial efforts hopefully will enhance the value of the volume to the students and teachers alike. Yung-Kuo Lim Editor www.com INTRODUCTION Solving problems in course work is an exercise of the mental facilities, and examination problems are usually chosen, or set similar to such prob- lems.
Working out problems is thus an essential and important aspect of the study of physics. The series Major American University Ph. Qualifying Questions and Solutions comprises seven volumes and is the result of months of work of a number of Chinese physicists. The subjects of the volumes and the respective coordinators are as follows: 1.
Atomic, Nuclear and Particle Physics (Jin Huai-cheng, Yang Bao- zhong, Fan Yang-mei) 5. Thermodynamics and Statistical Physics (Zheng Jiu-ren) 6. Solid State Physics and Miscellaneous Topics (Zhang Jia-lu, Zhou You-yuan, Zhang Shi-ling). These volumes, which cover almost all aspects of university physics, contain 2550 problems, mostly solved in detail.
The problems have been carefully chosen from a total of 3100 prob- lems, collected from the China-U. Physics Examination and Applica- tion Program, the Ph. Qualifying Examination on Experimental High Energy Physics sponsored by Chao-Chong Ting, and the graduate qualify- ing examinations of seven world-renowned American universities: Columbia University, the University of California at Berkeley, Massachusetts Insti- tute of Technology, the University of Wisconsin, the University of Chicago, Princeton University, and the State University of New York at Buffalo. Generally speaking, examination problems in physics in American uni- versities do not require too much mathematics.
They can be character- ized to a large extent as follows. Many problems are concerned with the various frontier subjects and overlapping domains of topics, having been selected from the setters own research encounters. These problems show a “modern” flavor. Some problems involve a wide field and require a sharp mind for their analysis, while others require simple and practical methods vii www.com viii Introduction demanding a fine “touch of physics”.
Indeed, we believe that these prob- lems, as a whole, reflect to some extent the characteristics of American science and culture, as well as give a glimpse of the philosophy underlying American education. That being so, we considered it worthwhile to collect and solve these problems, and introduce them to students and teachers everywhere, even though the work was both tedious and strenuous. About a hundred teachers and graduate students took part in this time-consuming task. This volume on Atomic, Nuclear and Particle Physics which contains 483 problems is divided into four parts: Atomic and Molecular Physics (142), Nuclear Physics (120), Particle Physics (90), Experimental Methods and Miscellaneous topics (131).
In scope and depth, most of the problems conform to the usual un- dergraduate syllabi for atomic, nuclear and particle physics in most uni- versities. Some of them, however, are rather profound, sophisticated, and broad-based. In particular they demonstrate the use of fundamental prin- ciples in the latest research activities. It is hoped that the problems would help the reader not only in enhancing understanding of the basic principles, but also in cultivating the ability to solve practical problems in a realistic environment.
This volume was the result of the collective efforts of forty physicists involved in working out and checking of the solutions, notably Ren Yong, Qian Jian-ming, Chen Tao, Cui Ning-zhuo, Mo Hai-ding, Gong Zhu-fang and Yang Bao-zhong.com CONTENTS Preface v Introduction vii Part I. Atomic and Molecular Physics 1 1. Molecular Physics (1123–1142) 173 Part II. Basic Nuclear Properties (2001–2023) 207 2.
Nuclear Binding Energy, Fission and Fusion (2024–2047) 239 3. The Deuteron and Nuclear forces (2048–2058) 269 4. Nuclear Reactions (2108–2120) 382 Part III. Interactions and Symmetries (3001–3037) 403 2.
Weak and Electroweak Interactions, Grand Unification Theories (3038–3071) 459 3. Structure of Hadrons and the Quark Model (3072–3090) 524 Part IV. Experimental Methods and Miscellaneous Topics 565 1. Kinematics of High-Energy Particles (4001–4061) 567 2.
Interactions between Radiation and Matter (4062–4085) 646 3. Detection Techniques and Experimental Methods (4086–4105) 664 4. Error Estimation and Statistics (4106–4118) 678 5. Particle Beams and Accelerators (4119–4131) 690 Index to Problems 709 ix www.com PART I ATOMIC AND MOLECULAR PHYSICS www.com This Page Intentionally Left Blank www.
ATOMIC PHYSICS (1001 1122) 1001 Assume that there is an announcement of a fantastic process capable of putting the contents of physics library on a very smooth postcard. Will it be readable with an electron microscope? Explain. (Columbia) Solution: Suppose there are 106 books in the library, 500 pages in each book, and each page is as large as two postcards. For the postcard to be readable, the planar magnification should be 2 × 500 × 106 ≈ 109 , corresponding to a linear magnification of 104.
As the linear magnification of an electron microscope is of the order of 800,000, its planar magnification is as large as 1011 , which is sufficient to make the postcard readable. 1002 At 1010 K the black body radiation weighs (1 ton, 1 g, 10−6 g, 10−16 g) per cm3. (Columbia) Solution: The answer is nearest to 1 ton per cm3. The radiant energy density is given by u = 4σT 4 /c, where σ = 5.67 × 10 Wm−2 K−4 is the Stefan–Boltzmann constant.
From Einstein’s mass- −8 energy relation, we get the mass of black body radiation per unit volume as u = 4σT 4 /c3 = 4×5. 1003 Compared to the electron Compton wavelength, the Bohr radius of the hydrogen atom is approximately (a) 100 times larger. (c) about the same.com 4 Problems and Solutions in Atomic, Nuclear and Particle Physics Solution: The Bohr radius of the hydrogen atom and the Compton wavelength 2 h a of electron are given by a = me2 and λc = mc respectively. Hence λ c = 1 e2 −1 2π ( c ) = 137 2π = 22, where e2 /c is the fine-structure constant.
Hence the answer is (a). 1004 Estimate the electric field needed to pull an electron out of an atom in a time comparable to that for the electron to go around the nucleus. (Columbia) Solution: Consider a hydrogen-like atom of nuclear charge Ze. The ionization energy (or the energy needed to eject the electron) is 13.
The orbit- ing electron has an average distance from the nucleus of a = a0 /Z, where a0 = 0.53 × 10−8 cm is the Bohr radius. The electron in going around the nucleus in electric field E can in half a cycle acquire an energy eEa. Thus to eject the electron we require eEa 13.53 × 10−8 1005 As one goes away from the center of an atom, the electron density (a) decreases like a Gaussian. (c) oscillates with slowly decreasing amplitude.com Atomic and Molecular Physics 5 Solution: The answer is (c).
1006 An electronic transition in ions of 12 C leads to photon emission near λ = 500 nm (hν = 2. The ions are in thermal equilibrium at an ion temperature kT = 20 eV, a density n = 1024 m−3 , and a non-uniform magnetic field which ranges up to B = 1 Tesla. (a) Briefly discuss broadening mechanisms which might cause the tran- sition to have an observed width ∆λ greater than that obtained for very small values of T , n and B. (b) For one of these mechanisms calculate the broadened width ∆λ using order-of-magnitude estimates of needed parameters.
(Wisconsin) Solution: (a) A spectral line always has an inherent width produced by uncertainty in atomic energy levels, which arises from the finite length of time involved in the radiation process, through Heisenberg’s uncertainty principle. The observed broadening may also be caused by instrumental limitations such as those due to lens aberration, diffraction, etc. In addition the main causes of broadening are the following. Doppler effect: Atoms or molecules are in constant thermal motion at T > 0 K.
The observed frequency of a spectral line may be slightly changed if the motion of the radiating atom has a component in the line of sight, due to Doppler effect. As the atoms or molecules have a distribution of velocity a line that is emitted by the atoms will comprise a range of frequencies symmetrically distributed about the natural frequency, contributing to the observed width.