Problems and Solutions on Atomic, Nuclear and Particle Physics for U.S. PhD qualifiers

718 Pages·2000·2.933 MB·English

by  Yung-Kuo Lim

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

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Major American Universities Ph.D. Qualifying Questions and Solutions
PROBLEMS AND SOLUTIONS ON ATOMIC, NUCLEAR AND PARTICLE PHYSICS
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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 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.S. 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.C. Ting’s
papers for selection of Chinese students for further studies in U.S.A., and
theirsolutionswhichrepresenttheeffortofmorethan70Chinesephysicists,
plus some 20 more who checked the solutions.
The series is remarkable for its comprehensive coverage. In each area
theproblemsspanawidespectrumoftopics,whilemanyproblemsoverlap
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
inanexperimentalsituationforestimatingaquantityfromasimplemodel.
Inshort,theexercisesblendtogethertheobjectivesofenhancementofone’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
thestudentstoworkoutforthemselves,ofwhommuchisdemandedofthe
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.
Inediting, noattempthasbeenmadetounifythephysicalterms,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
finalsolution,frequentlyeventotheextentofrewritingtheentiresolution.
v
vi Preface
Inaddition, a subjectindexto problemshas beenincluded to facilitatethe
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 prob-
lems. 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 Bao-
zhong, 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 prob-
lems, collected from the China-U.S.A. Physics Examination and Applica-
tion Program, the Ph.D. Qualifying Examination on Experimental High
EnergyPhysics sponsoredby Chao-ChongTing, and the graduatequalify-
ingexaminationsofsevenworld-renownedAmericanuniversities: Columbia
University, the University of California at Berkeley, Massachusetts Insti-
tute ofTechnology,theUniversityofWisconsin,theUniversityofChicago,
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
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
thoughtheworkwasbothtediousandstrenuous. Aboutahundredteachers
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),ParticlePhysics(90), ExperimentalMethods
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 researchactivities. It is hoped that the problems would
helpthereadernotonlyinenhancingunderstandingofthebasicprinciples,
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 v
Introduction vii
Part I. Atomic and Molecular Physics 1
1. Atomic Physics (1001–1122) 3
2. Molecular Physics (1123–1142) 173
Part II. Nuclear Physics 205
1. 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 Models (2059–2075) 289
5. Nuclear Decays (2076–2107) 323
6. Nuclear Reactions (2108–2120) 382
Part III. Particle Physics 401
1. Interactions and Symmetries (3001–3037) 403
2. Weak and ElectroweakInteractions, 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
PART I
ATOMIC AND MOLECULAR
PHYSICS
1. ATOMIC PHYSICS (1001 1122)
1001
Assumethatthereis anannouncementofa fantasticprocesscapableof
putting the contents of physics library on a very smooth postcard. Will it
be readable with an electron microscope? Explain.
(Columbia)
Solution:
Supposethere are106 booksinthelibrary,500pagesineachbook,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
microscopeisoftheorderof800,000,its planarmagnificationis aslargeas
1011, which is sufficient to make the postcard readable.
1002
At1010 Kthe blackbodyradiationweighs(1 ton,1g,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 = 4σT4/c, where σ = 5.67×
10−8 Wm−2 K−4 istheStefan–Boltzmannconstant. FromEinstein’smass-
energyrelation,wegetthemassofblackbodyradiationperunitvolumeas
u=4σT4/c3 =4×5.67×10−8×1040/(3×108)3 ≈108kg/m3 =0.1ton/cm3.
1003
Comparedto the electronCompton wavelength,the Bohr radius of the
hydrogen atom is approximately
(a) 100 times larger.
(b) 1000 times larger.
(c) about the same.
(CCT)
3
4 Problems and Solutions inAtomic, Nuclear and Particle Physics
Solution:
The Bohr radius of the hydrogen atom and the Compton wavelength
of electron are given by a = (cid:1)2 and λ = h respectively. Hence a =
me2 c mc λc
1 (e2)−1 = 137 = 22, where e2/(cid:1)c is the fine-structure constant. Hence
2π (cid:1)c 2π
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(ortheenergyneededtoejecttheelectron)is13.6Z2 eV.Theorbit-
ing electron has an average distance from the nucleus of a = a /Z, where
0
a =0.53×10−8 cm is the Bohr radius. The electron in going around the
0
nucleus in electric field E can in half a cycle acquire an energy eEa. Thus
to eject the electron we require
eEa(cid:1)13.6 Z2 eV,
or
13.6 Z3
E (cid:1) ≈2×109 Z3 V/cm.
0.53×10−8
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)