Introduction to Solid State Physics (Wiley; 8th Edition, 2005)(ISBN 047141526X), Charles Kittel - Solution Manual

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1、CHAPTER11.Thevectorsxˆˆ++yzˆand−−xˆyˆ+zˆareinthedirectionsoftwobodydiagonalsofacube.Ifθistheanglebetweenthem,theirscalarproductgivescosθ=–1/3,whence−1θ=cos1/3=90°+19°28'=109°28'.2.Theplane(100)isnormaltothexaxis.Itinterceptsthea'axisat2a'andthec'axisat2c';thereforetheindicesreferredt

2、otheprimitiveaxesare(101).Similarly,theplane(001)willhaveindices(011)whenreferredtoprimitiveaxes.3.Thecentraldotofthefourisatdistancecos60°aaa=°ctn60=cos30°3fromeachoftheotherthreedots,asprojectedontothebasalplane.Ifthe(unprojected)dotsareatthecenterofspheresincontact,then222⎛⎞ac⎛⎞a,

3、=⎜⎟+⎜⎟⎝⎠3⎝⎠2or2122c8a==c;1.633.34a31-1CHAPTER21.ThecrystalplanewithMillerindiceshk^isaplanedefinedbythepointsa1/h,a2/k,anda3/^.(a)Twovectorsthatlieintheplanemaybetakenasa1/h–a2/kanda1/h−a3/^.ButeachofthesevectorsgiveszeroasitsscalarproductwithGa=hk+a+^a,sothatGmustbeperpendiculartoth

4、eplane123hk^.(b)Ifnˆistheunitnormaltotheplane,theinterplanarspacingisnaˆ⋅/h.ButnGˆ=/

5、G

6、,1whenced(hk^)G=⋅aG/h

7、

8、=2π/

9、G

10、.(c)ForasimplecubiclatticeG=(2π+/a)(hxˆˆky+^zˆ),1whence22221Gh++k^==.222d4πa113aa022112.(a)Cellvolumeaa⋅×a=−3aa01232200c12=3ac.2xˆˆyzˆaa×π41123(b)b=π2=−3aa012

11、

12、aa⋅×a3a

13、c2212300c21π=+(xyˆˆ),andsimilarlyforb,b.23a3(c)Sixvectorsinthereciprocallatticeareshownassolidlines.Thebrokenlinesaretheperpendicularbisectorsatthemidpoints.TheinscribedhexagonformsthefirstBrillouinZone.3.Bydefinitionoftheprimitivereciprocallatticevectors33(a23×a)⋅(a3×××a1)(a1a2)V(=π

14、2)=π(2)/

15、(a⋅a×a)

16、BZ3123

17、(a12⋅×aa3)

18、3=π(2)/V.CForthevectoridentity,seeG.A.KornandT.M.Korn,Mathematicalhandbookforscientistsandengineers,McGraw-Hill,1961,p.147.4.(a)Thisfollowsbyforming2-121−−exp[iM(a⋅∆k)]1−exp[iM(a⋅∆k)]

19、F

20、=⋅1−−exp[i(a⋅∆k)]1−exp[i(a⋅∆k)]211−⋅cosM(a∆k)sinM(a⋅∆k)2==.211−

21、⋅cos(a∆k)sin(a⋅∆k)21(b)ThefirstzeroinsinMεoccursforε=2π/M.Thatthisisthecorrectconsiderationfollowsfrom2111sinM(π+hε)=sinπMhcosMε+c%os(&πMh('sinMε.22%(&('2zero,±1asMhisaninteger−π2i(xj1v+yjv2+zjv3)5.S(vvv)=Σfe123j111Referredtoanfcclattice,thebasisofdiamondis000;.Thusintheproduct444S(v

22、vv)=S(fcclat