an sities Ph. Qualifying Qe acafinnaetdlssieiy SOLID STATE PHYSICS, RELATIVITY AND MISCELLANEOUS TOPICS ‘Major American Universities Ph.D. ‘Qualifying Questions and Solutions PROBLEMS and. SOLUTIONS on SOLID STATE PHYSICS, RELATIVITY AND MISCELLANEOUS TOPICS Compile by The University of Science and Technology of China rysiex Coaching Class feted Zhang Jia-ta, Zhou Youryuan & Zhang Shi-ling, give ated by gs 43 Lim Yung-kuo_ wP i ra s World Scientific. Newersey » London « Singapore « Hong Kona toffee Sue 202.0 Man Se River lg, ICTS (i ec 57 Sten Sut, Cove a Lan CR HE Le a Majer American Untretles P.QultyingQuetannoStn [PubaLiaus aND SOLUTIONS ON SOLID STA FAYSICS, open 1955 Word Scent aang Ce a ‘Ai iv To a pn 0 pote asim yy ‘enon orca dng pening tre erason toe ac ‘Sen inoue vente itt wren eon am Pasir ‘Gusts Cut. 25 Rwnoe Deve Damen MA ISB Sh tee poms phoceory ented foe psc nN Set) ed by, g 4 ee Prosi Spry Mang Sevens 2 PREFACE ‘This suis of physies problems and solutions, which coussts of seven ‘wlimer — Mechanirs, Flectromagnatim, Opti, Atomic, Nuclear ar Particle Physics, Thermodynamics nnd Statistical Physics, Quantum Me- shavie, Solid State Physin and Relativity, contains a aclertion of 2550 troblems from the graduate achool entrance and qualifying examination papensof ween U.S. wniveities— California Uriverity Bereiey Cas, Coluabia University, Cheago University, Massachusett Institute of Tech ology, New York State University Buflalo Campus, Princeton Univer, Wiseousin University — as well asthe CUSPEA aad C. C. Ting’s papers for seletion of Chinese students fr further studies in U.S.A. and their soluvins which represent the effort of more than 70 Chinese physics plas tome 29 more who cheked the solutions, The series is remarkable for ite comprehensive coverage, In each area the pralems span a wide sectnim of tplen while many penblersa overlay several areas. The problems themicoes aze rere fr thelr versal. ity i applying the physical lane and principlog, their uptodate realnic situations, aad ther wcanty demand on mathematical skill. Many of the problem involve order of uneitude aleulaions whieds one olen replat ‘nan experimental siuatan for eaimating 2 qaatity from a simple model Inshort, the exerses bind together the objectives of enhancement of net _undertanding ofthe physical principles and ability of practical aplication. ‘The solutions as presented generally Just provide a guidance to solving ‘the problems, rather than step by step manipulation, and lease much to ‘the strats Co werk out for themselves, of whom rich i demanded of the ‘asic knowledge in physic Thus the series would provide an Invaluable complement to the textbooks. "The present volume, under the Utle “Za State Physics, Relativity ‘and Miccllaneou Topics” conse of 165 probleise Both special and ‘general rdativity probleme nee included, although some other special rela Uvity problems have boun solved elsewhere, particularly ln the volumes 02 Mochonics and on Electromagnetam. Problems that are not appropriace for “ Preface ‘the ater sia voiunes, auch ae mathematical tecuiues, ate ete gona ‘the soction Missllanoous Tops, In editing, no attempt bas boca made to unify tho physical corms, ‘nits aud symbols, Kathe, they arc let to the setters’ und solvers" 2a peefoenen sas to met the realistic bution of the wage today. Great ins as been sake to trave the logical stes rm the ist principles to the fiual solution, roqueaty wen to the extent of rewriting th entire sclution. In addition a subject index to problems hs been iacuded :ofocitate the lweition ef topies. These itr efforts hopefully will bance the wale fof tae sme tn His alent and Wachers alk Yon Kuo Linn waiter INTRODUCTION Soleing problems in school werk ie exercise of mental facies, and ‘camipation problems ae sally piled out fom problems in sol wor. ‘iorking ont probleme je thue an sosuril snd important aspuet of Ube sud of Physics Major Auescan University PhD. Qualifying Questions sal Solutions lea seres of seven volumes. The subjets ofeach wo ume and he respective 'efrees (n panties are as follow 1. Mechanics (Qing Yuan-i, Gu Expy, Cheng Jiao, Li Ze-bua, Yang Dotian) Bleetromagueis (Zhoo Shusping, You Jon-ian, Zha Jone} Opts (Bai Gu-ra, Guo Guang-can) ‘Atami, Nuclear and Partie Piysies (Tin. Hust stg, Fa Yanai) “herteodyaatnce and Statinieal Plysice (bang Jw-ru) Quast Mechs (Zhong, Vong-de, Zhn Dong-pe, Tan Hlons35) Sobel State Physi, Relatvity and Mlseallancous Topics (Zhang Jao, ‘acu You-yuna, Zheng Shing) i Yang Bao- This srs nvees alms all topic of Universiy Physics ad contain 260 problems, most of whch ace solved 1 det ‘The problems have been car‘ully chosen from a collection of S100 problems, of which acme came ftom: the China-U S.A, Physits Examination and Applea:ion Program, some wore seloted fom the Ph.D. Quelifing xxniuation oa Expermencal igh Bucrey Piysies sponsored by Choo Choog Ting, and the rest caze from the graduate peliminary or queifing reaminasons ofthe flowing aster wrold-renemmed American ynieersiee Columbia Univesity: Univerny of Califomis at Berkeley, Massachusetts Ine:tre of Techrology, Univemity of Wisconsin, Univerniey of Chicwgn, Trincston University State University of New York at Dl Generally speaking, ceaminsticn problems of physics in American un'- verses do not involve too much mathematics. Rather, they afe to 8 large extent characterized by the following chic aspects. Some probleme involving varions Ronter subjects and evelapping domains of scence ad bon selectod ty profesors directly from their own reserch work and havo an “up-to-date vor": some problems involve broad Holds wad require a ‘ick mind to analyec, while the methods noted for solving the ocher protlens are simple and practical but require a fall “touch of physi Indeed. we venture lo opine that the problems, as a whole, embody some extent the characteristics of American scleace and culture as wel ws the philospiy underlying American auction, ‘Therein, wie t wether and induce thera to sbudests and teachers in ganera, even Gg the fort involved wat formidable. Ax many at ands teases and ‘graduate stvlots took part this neconauaing take ‘A total of 105 problems mabe up this voluuo of tne parte: sli wale physles (81). relativity (28) and miscollanzous copes (6). Ta seope and. ‘depth, most ofthe problems conform to the undergraduate physica ayabi for slid sata physics and relativity in most universities. The misellneow Dart teste che student's comprahension of general principles, famiiviy ‘with magoitudes snd scales, xnowledge ofthe history of physic, ability «> ‘timate experimental errors, as woll as ie/her mathematical sila, Some ‘questions are onthe latest development in physies, purporing, no dost, to tet the scope af tho studant’s scopo of ganeral reading. Yet afew otners relate tothe aident's university and ita research; hese ane nce sled it ‘inched for rferenoe ony ‘The problems ia this volume were ohed by: Cuo Zhichun, Chen Bing, Wang Ping, Zhang Shi-ng, Zhu Bing, Wang Yoog, Zhou Dong fang, Ning Bo, Wang Shar-xa, $i Qiao, Lu Jar-xi Qin Riv, Wing Aci, Sun “6, Jing Yepeng, Lin Yurzhen, Liu Fong-xin apd Zinng Zico-quan, and the inial translation froma Chineve to English was exeried out by Zhang refereed by Zhu Xing fen and Ding Zejun, CONTENTS Preface y [etroduction a Part I Solid State Physics a 1A. Crystal Structures and Properses (1001-3027) 8 1.2. Election Theory, Energy Bands and Semiconductors (1025-1061) 46 113. Electomagietc Properties, Optical Properties ‘and Supereondueivty (1052-1075) 105 1.4, Other tops (1077-1081) 155 Part IT Relativity 105 2.4. Special Relativity (2001-2007) iT 22. Ganeral Relativity (2008-2023) 1a 25. Relatvste Coemology (2024-2028) 26 Part IIL Miscellaneous Topics m1 421, History af Physien and General Questions (2001-2025) 2 ‘32. Mexeurements, Eatimetions and rare (3028-3048) 2s ‘83, Mathematical Techniques (2049-8056) a [edex to Probleme at PART I SOLID STATE PHYSICS