Problems And Solutions

Problems and Solutionson Atomic, Nuclear andParticle Physics

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Major American Universities Ph.D.Qualifying Questions and SolutionsProblems and Solutionson Atomic, Nuclear andParticle PhysicsCompiled byThe Physics Coaching ClassUniversity of Science andTechnology of ChinaEdited byYung-Kuo LimNational University of SingaporeWorld ScientificSingapore New Jersey London Hong Kong

Published byWorld Scientific Publishing Co. Pte. Ltd.P 0 Box 128, Farrer Road, Singapore 912805USA office: Suite lB, 1060 Main Street, River Edge, NJ 07661UK office: 57 Shelton Street, Covent Garden, London WC2H 9HEBritish Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.Major American Universities Ph.D. Qualifying Questions and SolutionsPROBLEMS AND SOLUTIONS ON ATOMIC, NUCLEAR AND PARTICLE PHYSICSCopyright 2000 by World Scientific Publishing Co. Pte. Ltd.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 retrievalsystem 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 CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission tophotocopy is not required from the publisher.ISBN981-02-3917-3981-02-3918-l(pbk)This book is printed on acid-free paper.PrintedinSingaporebyUto-Print

PREFACEThis series of physics problems and solutions, which consists of sevenvolumes — Mechanics, Electromagnetism, Optics, Atomic, Nuclear andParticle Physics, Thermodynamics and Statistical Physics, Quantum Mechanics, Solid State Physics and Relativity, contains a selection of 2550problems from the graduate-school entrance and qualifying examinationpapers of seven U.S. universities — California University Berkeley Campus, Columbia University, Chicago University, Massachusetts Institute ofTechnology, New York State University Buffalo Campus, Princeton University, Wisconsin University — as well as the CUSPEA and C.C. Ting’spapers for selection of Chinese students for further studies in U.S.A., andtheir 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 areathe problems span a wide spectrum of topics, while many problems overlapseveral areas. The problems themselves are remarkable for their versatility in applying the physical laws and principles, their uptodate realisticsituations, and their scanty demand on mathematical skills. Many of theproblems involve order-of-magnitude calculations which one often requiresin an experimental situation for estimating a quantity from a simple model.In short, the exercises blend together the objectives of enhancement of one’sunderstanding of physical principles and ability of practical application.The solutions as presented generally just provide a guidance to solvingthe problems, rather than step-by-step manipulation, and leave much tothe students to work out for themselves, of whom much is demanded of thebasic knowledge in physics. Thus the series would provide an invaluablecomplement to the textbooks.The present volume consists of 483 problems. It covers practically thewhole of the usual undergraduate syllabus in atomic, nuclear and particlephysics, but in substance and sophistication goes much beyond. Someproblems on experimental methodology have also been included.In editing, no attempt has been made to unify the physical terms, unitsand symbols. Rather, they are left to the setters’ and solvers’ own preference so as to reflect the realistic situation of the usage today. Great painshas been taken to trace the logical steps from the first principles to thefinal solution, frequently even to the extent of rewriting the entire solution.v

viPrefaceIn addition, a subject index to problems has been included to facilitate thelocation of topics. These editorial efforts hopefully will enhance the valueof the volume to the students and teachers alike.Yung-Kuo LimEditor

INTRODUCTIONSolving problems in course work is an exercise of the mental facilities,and examination problems are usually chosen, or set similar to such problems. Working out problems is thus an essential and important aspect ofthe study of physics.The series Major American University Ph.D. Qualifying Questions andSolutions comprises seven volumes and is the result of months of workof a number of Chinese physicists. The subjects of the volumes and therespective coordinators are as follows:1. Mechanics (Qiang Yan-qi, Gu En-pu, Cheng Jia-fu, Li Ze-hua, YangDe-tian)2. Electromagnetism (Zhao Shu-ping, You Jun-han, Zhu Jun-jie)3. Optics (Bai Gui-ru, Guo Guang-can)4. Atomic, Nuclear and Particle Physics (Jin Huai-cheng, Yang Baozhong, Fan Yang-mei)5. Thermodynamics and Statistical Physics (Zheng Jiu-ren)6. Quantum Mechanics (Zhang Yong-de, Zhu Dong-pei, Fan Hong-yi)7. Solid State Physics and Miscellaneous Topics (Zhang Jia-lu, ZhouYou-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 problems, collected from the China-U.S.A. Physics Examination and Application Program, the Ph.D. Qualifying Examination on Experimental HighEnergy Physics sponsored by Chao-Chong Ting, and the graduate qualifying examinations of seven world-renowned American universities: ColumbiaUniversity, the University of California at Berkeley, Massachusetts Institute 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 universities do not require too much mathematics. They can be characterized to a large extent as follows. Many problems are concerned with thevarious frontier subjects and overlapping domains of topics, having beenselected from the setters own research encounters. These problems show a“modern” flavor. Some problems involve a wide field and require a sharpmind for their analysis, while others require simple and practical methodsvii

viiiIntroductiondemanding a fine “touch of physics”. Indeed, we believe that these problems, as a whole, reflect to some extent the characteristics of Americanscience and culture, as well as give a glimpse of the philosophy underlyingAmerican education.That being so, we considered it worthwhile to collect and solve theseproblems, and introduce them to students and teachers everywhere, eventhough the work was both tedious and strenuous. About a hundred teachersand graduate students took part in this time-consuming task.This volume on Atomic, Nuclear and Particle Physics which contains483 problems is divided into four parts: Atomic and Molecular Physics(142), Nuclear Physics (120), Particle Physics (90), Experimental Methodsand Miscellaneous topics (131).In scope and depth, most of the problems conform to the usual undergraduate syllabi for atomic, nuclear and particle physics in most universities. Some of them, however, are rather profound, sophisticated, andbroad-based. In particular they demonstrate the use of fundamental principles in the latest research activities. It is hoped that the problems wouldhelp the reader not only in enhancing understanding of the basic principles,but also in cultivating the ability to solve practical problems in a realisticenvironment.This volume was the result of the collective efforts of forty physicistsinvolved in working out and checking of the solutions, notably Ren Yong,Qian Jian-ming, Chen Tao, Cui Ning-zhuo, Mo Hai-ding, Gong Zhu-fangand Yang Bao-zhong.

CONTENTSPrefacevIntroductionviiPart I. Atomic and Molecular Physics11. Atomic Physics (1001–1122)2. Molecular Physics (1123–1142)3173Part II. Nuclear Physics2051.2.3.4.5.6.207239269289323382Basic Nuclear Properties (2001–2023)Nuclear Binding Energy, Fission and Fusion (2024–2047)The Deuteron and Nuclear forces (2048–2058)Nuclear Models (2059–2075)Nuclear Decays (2076–2107)Nuclear Reactions (2108–2120)Part III. Particle Physics4011. Interactions and Symmetries (3001–3037)2. Weak and Electroweak Interactions, Grand UnificationTheories (3038–3071)3. Structure of Hadrons and the Quark Model (3072–3090)403Part IV. Experimental Methods and Miscellaneous Topics5651.2.3.4.5.567646664678690Kinematics of High-Energy Particles (4001–4061)Interactions between Radiation and Matter (4062–4085)Detection Techniques and Experimental Methods (4086–4105)Error Estimation and Statistics (4106–4118)Particle Beams and Accelerators (4119–4131)Index to Problems459524709ix

PART IATOMIC AND MOLECULARPHYSICS

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1. ATOMIC PHYSICS (1001 1122)1001Assume that there is an announcement of a fantastic process capable ofputting the contents of physics library on a very smooth postcard. Will itbe readable with an electron microscope? Explain.(Columbia)Solution:Suppose there are 106 books in the library, 500 pages in each book, andeach page is as large as two postcards. For the postcard to be readable,the planar magnification should be 2 500 106 109 , corresponding toa linear magnification of 104.5 . As the linear magnification of an electronmicroscope is of the order of 800,000, its planar magnification is as large as1011 , which is sufficient to make the postcard readable.1002At 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 5.67 810 Wm 2 K 4 is the Stefan–Boltzmann constant. From Einstein’s massenergy relation, we get the mass of black body radiation per unit volume asu 4σT 4 /c3 4 5.67 10 8 1040 /(3 108 )3 108 kg/m3 0.1 ton/cm3 .1003Compared to the electron Compton wavelength, the Bohr radius of thehydrogen atom is approximately(a) 100 times larger.(b) 1000 times larger.(c) about the same.(CCT )3

4Problems and Solutions in Atomic, Nuclear and Particle PhysicsSolution:The Bohr radius of the hydrogen atom and the Compton wavelength 2haof electron are given by a me 2 and λc mc respectively. Hence λc1 e2 12π ( c )2 1372π 22, where e / c is the fine-structure constant. Hencethe answer is (a).1004Estimate the electric field needed to pull an electron out of an atom ina time comparable to that for the electron to go around the nucleus.(Columbia)Solution:Consider a hydrogen-like atom of nuclear charge Ze. The ionizationenergy (or the energy needed to eject the electron) is 13.6Z2 eV. The orbiting electron has an average distance from the nucleus of a a0 /Z, wherea0 0.53 10 8 cm is the Bohr radius. The electron in going around thenucleus in electric field E can in half a cycle acquire an energy eEa. Thusto eject the electron we requireeEa 13.6 Z2 eV ,orE 13.6 Z3 2 109 Z3 V/cm .0.53 10 81005As one goes away from the center of an atom, the electron density(a) decreases like a Gaussian.(b) decreases exponentially.(c) oscillates with slowly decreasing amplitude.(CCT )