" Problems and Solutions on Atomic, Nuclear and Particle Physics
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Major American Universities Ph.D. 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
Published by World Scientific Publishing Co. Pte. Ltd. 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
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Major American Universities Ph.D. Qualifying Questions and Solutions PROBLEMS AND SOLUTIONS ON ATOMIC, NUCLEAR AND PARTICLE PHYSICS Copyright © 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 retrieval system now known or to be invented, without written permission from the Publisher.
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ISBN
981-02-3917-3 981-02-3918-l
(pbk)
This book is printed on acid-free paper.
Printed
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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 Mechanics, Solid State Physics and Relativity, contains a selection of 2550 problems from the graduate-school entrance and qualifying examination papers of seven U.S. universities — California University Berkeley Campus, Columbia University, Chicago University, Massachusetts Institute of Technology, New York State University Buffalo Campus, Princeton University, Wisconsin University — as well as the CUSPEA and C.C. Ting’s papers for selection of Chinese students for further studies in U.S.A., 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 versatility 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 preference 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.
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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
INTRODUCTION
Solving 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 of the study of physics. The series Major American University Ph.D. 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. Mechanics (Qiang Yan-qi, Gu En-pu, Cheng Jia-fu, Li Ze-hua, Yang De-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, 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 problems, collected from the China-U.S.A. Physics Examination and Application Program, the Ph.D. Qualifying Examination on Experimental High Energy Physics sponsored by Chao-Chong Ting, and the graduate qualifying examinations of seven world-renowned American universities: Columbia University, 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 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
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Introduction
demanding a fine “touch of physics”. Indeed, we believe that these problems, 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 undergraduate syllabi for atomic, nuclear and particle physics in most universities. Some of them, however, are rather profound, sophisticated, and broad-based. In particular they demonstrate the use of fundamental principles 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.
CONTENTS
Preface Introduction Part I. Atomic and Molecular Physics 1. Atomic Physics (1001–1122) 2. Molecular Physics (1123–1142) Part II. Nuclear Physics 1. 2. 3. 4. 5. 6. Basic 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)
v vii 1 3 173 205 207 239 269 289 323 382 401 403 459 524 565 567 646 664 678 690 709
Part III. Particle Physics 1. Interactions and Symmetries (3001–3037) 2. Weak and Electroweak Interactions, Grand Unification Theories (3038–3071) 3. Structure of Hadrons and the Quark Model (3072–3090) Part IV. Experimental Methods and Miscellaneous Topics 1. 2. 3. 4. 5. Kinematics 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 Problems
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PART I
ATOMIC AND MOLECULAR PHYSICS
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1. 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 104.5 . 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 × −8 10 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 as u = 4σT 4 /c3 = 4×5.67×10−8×1040 /(3×108 )3 ≈ 108 kg/m3 = 0.1 ton/cm3 .
1003 Compared to the electron Compton wavelength, the Bohr radius of the hydrogen atom is approximately (a) 100 times larger. (b) 1000 times larger. (c) about the same. (CCT )
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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 λc = = 137 = 22, where e2 / c is the fine-structure constant. Hence 2π the answer is (a).
1 e2 −1 2π ( c )
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 13.6Z2 eV. The orbiting 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 or E 13.6 Z3 ≈ 2 × 109 Z3 V/cm . 0.53 × 10−8 13.6 Z2 eV ,
1005 As 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 )
Atomic and Molecular Physics
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Solution: The answer is (c).
1006 An electronic transition in ions of 12 C leads to photon emission near λ = 500 nm (hν = 2.5 eV). 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 transition 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. Collisions: An atomic system may be disturbed by external influences such as electric and magnetic fields due to outside sources or neighboring atoms. But these usually cause a shift in the energy levels rather than broadening them. Broadening, however, can result from atomic collisions which cause phase changes in the emitted radiation and consequently a spread in the energy.
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Problems and Solutions in Atomic, Nuclear and Particle Physics
(b) Doppler broadening: The first order Doppler frequency shift is given v by ∆ν = ν0c x , taking the x-axis along the line of sight. Maxwell’s velocity distribution law then gives dn ∝ exp −
2 M vx 2kT
dvx = exp −
M c2 2kT
∆ν ν0
2
dvx ,
where M is the mass of the radiating atom. The frequency-distribution of the radiation intensity follows the same relationship. At half the maximum intensity, we have (ln 2)2kT ∆ν = ν0 . M c2 Hence the line width at half the maximum intensity is 2∆ν = In terms of wave number ν = ˜
1 λ
1.67c λ0
ν c
2kT . M c2
=
we have 1.67 λ0 2kT . M c2
ν ΓD = 2∆˜ =
With kT = 20 eV, M c2 = 12 × 938 MeV, λ0 = 5 × 10−7 m, ΓD = 1.67 5 × 10−7 2 × 20 = 199 m−1 ≈ 2 cm−1 . 12 × 938 × 106
Collision broadening: The mean free path for collision l is defined by nlπd2 = 1, where d is the effective atomic diameter for a collision close enough to affect the radiation process. The mean velocity v of an atom can ¯ be approximated by its root-mean-square velocity given by 1 M v 2 = 3 kT . 2 2 Hence 3kT v≈ ¯ . M Then the mean time between successive collisions is t= l 1 = v ¯ nπd2 M . 3kT
Atomic and Molecular Physics
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The uncertainty in energy because of collisions, ∆E, can be estimated from the uncertainty principle ∆E · t ≈ , which gives ∆νc ≈ or, in terms of wave number, Γc = 1 2 nd 2 3 × 10−3 3kT ∼ M c2 λ0 2kT , M c2 1 , 2πt
if we take d ≈ 2a0 ∼ 10−10 m, a0 being the Bohr radius. This is much smaller than Doppler broadening at the given ion density.
1007 (I) The ionization energy EI of the first three elements are Z 1 2 3 Element H He Li EI 13.6 eV 24.6 eV 5.4 eV
(a) Explain qualitatively the change in EI from H to He to Li. (b) What is the second ionization energy of He, that is the energy required to remove the second electron after the first one is removed? (c) The energy levels of the n = 3 states of the valence electron of sodium (neglecting intrinsic spin) are shown in Fig. 1.1. Why do the energies depend on the quantum number l? (SUNY, Buffalo)
Fig. 1.1
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Problems and Solutions in Atomic, Nuclear and Particle Physics
Solution: (a) The table shows that the ionization energy of He is much larger than that of H. The main reason is that the nuclear charge of He is twice than that of H while all their electrons are in the first shell, which means that the potential energy of the electrons are much lower in the case of He. The very low ionization energy of Li is due to the screening of the nuclear charge by the electrons in the inner shell. Thus for the electron in the outer shell, the effective nuclear charge becomes small and accordingly its potential energy becomes higher, which means that the energy required for its removal is smaller. (b) The energy levels of a hydrogen-like atom are given by En = − For Z = 2, n = 1 we have EI = 4 × 13.6 = 54.4 eV . (c) For the n = 3 states the smaller l the valence electron has, the larger is the eccentricity of its orbit, which tends to make the atomic nucleus more polarized. Furthermore, the smaller l is, the larger is the effect of orbital penetration. These effects make the potential energy of the electron decrease with decreasing l. Z2 × 13.6 eV . n2
1008 Describe briefly each of the following effects or, in the case of rules, state the rule: (a) Auger effect (b) Anomalous Zeeman effect (c) Lamb shift (d) Land´ interval rule e (e) Hund’s rules for atomic levels (Wisconsin) Solution: (a) Auger effect: When an electron in the inner shell (say K shell) of an atom is ejected, a less energetically bound electron (say an L electron)
Atomic and Molecular Physics
9
may jump into the hole left by the ejected electron, emitting a photon. If the process takes place without radiating a photon but, instead, a higherenergy shell (say L shell) is ionized by ejecting an electron, the process is called Auger effect and the electron so ejected is called Auger electron. The atom becomes doubly ionized and the process is known as a nonradiative transition. (b) Anomalous Zeeman effect: It was observed by Zeeman in 1896 that, when an excited atom is placed in an external magnetic field, the spectral line emitted in the de-excitation process splits into three lines with equal spacings. This is called normal Zeeman effect as such a splitting could be understood on the basis of a classical theory developed by Lorentz. However it was soon found that more commonly the number of splitting of a spectral line is quite different, usually greater than three. Such a splitting could not be explained until the introduction of electron spin, thus the name ‘anomalous Zeeman effect’. In the modern quantum theory, both effects can be readily understood: When an atom is placed in a weak magnetic field, on account of the interaction between the total magnetic dipole moment of the atom and the external magnetic field, both the initial and final energy levels are split into several components. The optical transitions between the two multiplets then give rise to several lines. The normal Zeeman effect is actually only a special case where the transitions are between singlet states in an atom with an even number of optically active electrons. (c) Lamb shift: In the absence of hyperfine structure, the 22 S1/2 and 22 P1/2 states of hydrogen atom would be degenerate for orbital quantum number l as they correspond to the same total angular momentum j = 1/2. However, Lamb observed experimentally that the energy of 22 S1/2 is 0.035 cm−1 higher than that of 22 P1/2 . This phenomenon is called Lamb shift. It is caused by the interaction between the electron and an electromagnetic radiation field. (d) Land´ interval rule: For LS coupling, the energy difference between e two adjacent J levels is proportional, in a given LS term, to the larger of the two values of J. (e) Hund’s rules for atomic levels are as follows: (1) If an electronic configuration has more than one spectroscopic notation, the one with the maximum total spin S has the lowest energy. (2) If the maximum total spin S corresponds to several spectroscopic notations, the one with the maximum L has the lowest energy.
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Problems and Solutions in Atomic, Nuclear and Particle Physics
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