V Envelope wave functions

Transcription

1 V Envlop wav funtions 5. atoriation of Bloh stat W onsidr htrountion of two matrials A and B. Th disussion applis as suh also for quantum wll strutur in whih A is th arrir and B th wll matrial. To mak things simpl w assum dirt gap smiondutor matrials. A on partil stat in matrial A an alwas writtn as ) r = Â f ) r u ) r m m A m A k C) whr k = for dirt gap matrial. As long as th sum is takn ovr omplt st of atomi-stats C) rmains an xat solution of wav quation A) - s th disussion of k p mthod. n matrial B w hav ) r = Â f ) r u ) r m m B m B k.. C) W will now onsidr th total wav funtion xtnding trough th htrountion. This will oviousl impl som spifi onditions to AB, fulfilld th position dpndnt wight fators f m to alld nvlop wav funtions. W will mak a vr spial assumption that th atomi parts of th total wav funtions ar idntial i.. w st k = ) u m A = um B C) n spit of ing idntial th wav funtions u diffrnt ignvalus., u ar assoiatd with m A m B 8

2 n th following w first mak a tmporar ngltd all spin-dpndnt trms. Th ro-ordr atomi amiltonian = k = ) is thrfor idntial with th total amiltonian of th total wav funtion). Th atomi parts ar thn solutions of p m V A u m A m A u m A + ) rp ) r = ) r p m V B u m B m B u m B m B u m A + ) rp ) r = ) r = ) r. C) Equations C) an fulfilld simultanousl onl if V - V = - i.. if th potntials ar idntial in shap ut diffr a onstant fator. A B m A m B ot that V AB, ar priodi potntials assoiatd with th atomi stats. Th do not inlud an slowl varing on unit ll sal) potntials indud for instan onfind arrirs. n th following w will driv a wav quation for th nvlop funtions AB, f m in th as of a W. Sin th atomi stats in C) ar ortonormal th ontinuit of th wav funtion aross th intrfas at = and = a of th W rquirs that 9

3 f m A r, ) = f r, ) f r, a) = f r, a). m B m A m B r r is th omponnt of th position vtor in th W plan and a th width of th W. Th priodi oundar ondition prpndiular to th W plan). givn is no mor valid in th - dirtion Thrfor th nvlop wav funtions ar f r, ) = xp ik rg m AB ), C4) S m AB,, whr k is th omponnt of th wav vtor in th W plan. S is th ara of th sampl rgion in th W plan. B using th stp funtions th atomi as wll as total) amiltonian an writtn in th form p = + m V A) r Y A + V B) r Y B, C5) whr Y ) =, Y ) = ; Œ, a A B a f and YA) =, YB) = ; œa, af. Using amiltonian C5) w an rwrit Eqs. C) C6) u ) r = Y + Y u ) r wll is loatd in l l A A l B B l Œa, af; A = arrir matrial, B = wll matrial) Comining Eqs. C and C4, th total planar Bloh stat is thn givn g l. C7) l, ) r = xp ik r  l AB ) u S 4

4 5. Condution and W first onsidr th singl unoupld ondution and. n this as thr is onl on trm l = in th sum of Eq. C7). Thwavquationis ) r = ) r. C8) n th arrir w hav: p + m V A) r P ) r = ) r C9) nsrting th Bloh stat p m V r A i A ) + ) P xp + k r) ) u r) S i A ) = ) r xp + k r) ) u) r S C) W start from th kinti nrg trm p p p = + m m m. Th planar omponnt of linar momntum: g ) ) A A p xp + ik r ) ) u r) = k xp + ik r ) ) u r) + fi ) A xp + ik r ) ) p u r) A ) A ) p xp + ik r ) ) u ) r = k xp + ik r ) ) u ) r + A ) g C) k xp + ik r ) ) p u r) + A ) xp + ik r ) ) p u r) g C) 4

5 -omponnt of linar momntum: g A ) A ) p u xp + ik r ) ) u r) = xp + ik r ) p ) u r) u + fi A ) xp + ik r ) ) p u r) u A ) A ) A ) g p xp + ik r ) ) u r) = xp + ik r ) p ) u r) + xp + ik r ) p ) p u r) + A ) xp + ik r ) ) p u r) n th wll th kinti nrg trm is idntial xpt A has to rplad B. g C) C4) ow multipl oth sids of wav quation C8) from lft S u * r)xp - i k r) * AB, ) ) and intgrat ovr th hol rstal. Th suprsript A,B) indiats that in th intgration ovr th rstal volum on should us nvlop funtion A in th arrirs and B in th wll. W onsidr th lft hand sid of th wav quation first. rom th planar kinti nrg trm w hav S m u )xp r -ik r ) ) p xp + ik r ) ) u ) r d r = * * AB, ) AB, ) All spa 4

6 k Sm k Sm Sm Sm * * AB, ) AB, ) All spa u r)xp -ik r ) ) xp + ik r ) ) u r) d r = * * AB, ) r AB, ) All spa u ) ) ) u ) r d r+ * * AB, ) AB, ) All spa * * u ) r AB, ) AB, ) ) ) ) r All spa u ) r ) ) k p u ) r d r + p u d r C5) W valuat th intgrals assuming that th nvlop funtion is slowl varing on unit ll sal: k Sm * * AB, ) r AB, ) All spa u ) ) ) u ) r d r = k Sm  i all lls i * AB, ) i AB, ) * i r W ) ) u ) u r) d r C6) W assum that th atomi Bloh stats ar normalid as follows * * un) r um) r d r = un) r um) r d r =d nm, C7) W W all spa ll whr W is th volum of th rstal and W volum of a unit ll. Using Eq. C7) th rhs of Eq. C6) an writtn as k k AB i AB *, ), ) * * i u u d r AB, )  =  i AB, ) ) ) ) r ) r ) i ) Wi Sm Sm i all lls W i i all lls 4

7 C8) W us again th fat that th nvlop funtion is slowl varing on unit ll sal and onvrt th sum ak into an intgral: k k AB i AB Â *, ) ), ) i ) W i = * AB, ) i) AB, ) i ) d r Sm Sm i all lls all spa C9) Th nvlop funtions do not dpnd on varial r. Thrfor th intgration ovr r givs onl th ara S of th W: k k AB i AB *, ) ), ) i ) d r = Sm Sm S * AB, ) i) AB, ) i ) d ; C) all spa whr th S anl. all spa A similar produr for th othr trms in th planar kinti nrg givs S m * * AB, ) AB, ) All spa u ) r ) ) k p u ) r d r = m * AB, ) AB, ) * All spa ll ) ) d k u ) r p u ) r d r C) and 44

8 Sm * * AB, ) r AB, ) All spa u ) ) ) p u ) r d r = m * AB, ) AB, ) * All spa ll ) ) d u ) r p u ) r d r C) or th omponnt of th kinti nrg w hav Sm Sm Sm Sm All spa All spa All spa All spa * * AB, ) AB, ) u )xp r -ik r ) ) p xp + ik r ) ) u ) r d r = * * AB, ) AB, ) r u ) ) p ) u ) r d r+ * * AB, ) AB, ) r u ) ) p ) p u ) r d r+ u g * *, ), ) r AB AB ) ) ) p u ) r d r C) Whih givs in analog to - Sm m m m * * AB, ) AB, ) * AB, ) AB, ) All spa * AB, ) AB, ) * r All spa ll All spa u )xp r -ik r ) ) p xp + ik r ) ) u ) r d r = ) p ) d+ ) p ) d u ) p u ) r d r + * AB, ) AB, ) * ) ) d u ) r pu ) r dr All spa g ll Potntial nrg matrix lmnt is givn C4) 45

9 Sm Sm * * AB, ) AB, AB, ) All spa u r)xp -ik r ) ) V r)xp + ik r ) ) u r) d r = * * AB, ) AB, AB, ) All spa u ) r ) V ) r ) u ) r d r C5) W an omin this with th matrix lmnt C) and th last trm in Eq. C4) to giv : * * AB, ) AB, AB, ) All spa u ) r ) V ) r ) u ) r d r + m m All spa * * AB, ) AB, ) u ) r ) ) p u ) r d r + * * AB, ) AB, ) All spa u ) r ) ) p u ) r d r = * AB, ) AB, ) * ) ) u) r S m p + p + V A, B) r u ) r d r S All spa * AB, ) AB, ) * r AB, ) All spa ) ) u ) u ) r d r = AB, ) * AB, ) AB, ) * ) ) d u) r u) r d r = W All spa A, B) *, ) AB, ) AB All spa ) ) d ll Th right hand sid of C8) is givn S * * AB, ) AB, ) All spa * AB, ) AB, ) ) ) d All spa = P C6) u r)xp -ik r ) ) xp + ik r ) ) u r) d r = C7) 46

10 inall w ollt th potntial and kinti nrg trms. W us th following notation: l pm = ul * ) r pul) r d r. C8) W ll Th wav quation is givn *, ) A A B k i Y YB AB m m m m p, ) k p - P d = AB Allspa C9) * AB, ) AB, ) Allspa This has th standard form an ignvalu prolm d - =, C) whr k k i =- + V m m m m p ) p. C) n th amiltonian C) w hav writtn V)= A YA + B YB. C) This is th msosopi potntial rsulting from th and disontinuit. Th diagonal dipol matrix lmnts ar ro and C) rdus to k =- + V ) +. C) m m 47

11 5. Coupld ands and nvlop wav funtions W now onsidr 8 strongl oupld ands and som distant wakl oupld ands. W modif th prvious tratmnt for ulk -V smiondutor asd on simultanous us of onfiguration intration and prturation thor mthods to inlud nvlop funtions instad of position indpndnt onstants l strongl oupld ands ) and n distant ands). W now assum that th priodi amiltonian inluds also th spin-orit intration p m V r = k = ) = + AB, ) + s VAB, p 4m h. D) To mak th notations mor ompat w dnot th sum of th priodi amiltonian and spin-orit intration S VAB, = VAB, ) r + VAB, 4m s h p D) With this notation w an writ th quations C) as p m V A s u m A m A u m A + ) rp ) r = ) r p m V B S u m B m B u m B m B u m A + ) rp ) r = ) r = ) r. D) Equations C) ar again fulfilld simultanousl onl if V - V = - i.. if th potntials ar idntial in shap ut diffr a onstant fator. A S B S m A m B Th atomi Bloh stats u. m AB, m ) r u ) r ar now qual to thos listd in Tal n analog to singl and as th planar Bloh stat fulfills now th ignvalu quation 48

12 p m V S r AB i AB AB + P l ul u S +, ),  +, ) ) xp k r ) ) ) r Ân ) n ) r l AB, ) AB, ) = ) r xp + ik r) Âl ) ul ) r + Ân ) un) r S D4) whr th sums ovr strongl oupld and distant ands ar writtn sparatl. l n n W now driv a oupld st of ignvalu quations whih ar similar to Eqs. A). ow multipl oth sids of wav quation D4) from lft S u m * r)xp - i k r) * m AB, ) ) D5) and intgrat ovr th hol rstal. r m is on of th strongl oupld stats and as aov th suprsript A,B) indiats that in th intgration ovr th rstal volum on should us nvlop funtion A in th arrirs and B in th wll. ollowing th sam stps as in singl-and as w otain  l R S T k i + Y + - -d P m l m m + k m - p m mp l m A A m B AB, ) Y B ml l k i +  m p n - n m m mp n R S T U P V W AB, ) n Corrspondingl multipling from lft S u m )xp r - i k r) m ) whr m is on of th distant ands w hav = * * AB, ) U P V W D6) 49

13  n  l R S T R S T k i + Y + + -d P m n mp n m m + k m - p m A B AB, ) ny A n B mn n k m i m p l - m m p l U P V W AB, ) l = D7) U P V W + B omparing D6-7) with Eq. A of Bastard it an sn that th matrix lmnts of oprator W in A ar rplad and mw l k i Æ+ m l - m m mp l p mπl D8a) mw l k Æ + m= l. D8) m m f w rdfin W rplaing k Æ- i, whr th diffrntial oprats on th nvlop wav funtions onl w an rwrit Eqs. D6-7) in th sam form as A) ut with th offiints l,n rplad with th nvlop funtions AB, ) l, n A B AB n Y A + n Y B - d mn + mwnn mw l  P A B AB, ) AB, ) m A m B l  Y + Y - d lm + mw l +  mwn n = l n, ) AB, )  + l = n l D9a) D9) whr w hav writtn W instad of W to indiat that th formr also inluds a diffrntial opration on nvlop funtions. ow w an follow th sam argumnts that wr usd for ulk smiondutor. Sin th l ands ar onl wakl oupld with n ands w hav AB, ) AB, ) l n ) >> ). D) 5

14 A B m A m B Thn th Eq. D9) givs w an assum Y + Y - >> mwn aus th nrgis m ar dfinition far largr than whihisassumdto los to th and dg of th l stats):  mwl l AB, ) AB, ) m ª+ A B l -ny A -ny B D) W insrt this ak in Eq. D9a) R S T n n m A A m B AB Y YB d lm mw l mw W l  + - +, ) +  A B l = D) l n -ny A -ny B whih aftr som rarrangmnt givs R S T n n m A A m B AB  Y + YB - d lm + mw l + mw  A B l n n Y A n Y Wl, ) = l - - B n analog to A dfin U V W U V W D) ~ n n W = W + W - Y - Y Wl A B n n A n B D4) Aftr this dfinition th drivation of th ulk uttingr-kohn formalism an rpatd without an modifiations, xpt that in all matrix lmnts th - omponnt of wav vtor has to rplad k Æ- i D5) An important fatur of th oupld quations is th position dpndn of th fftiv mass and th uttingr paramtrs. Th origin of position dpndn an sn as follows. Considr th fator 5

15 ~ n n W - W = mwâ - Y - Y Wl A B n n A n B D6) in Eq. D). n th nondiagonal matrix lmnts w an us Eq. D8a). W dfin m = a, ml m  mp a n n p l A B -n Y A -n Y B n D7) whr is th avrag nrg of th strongl oupld ands - s th ulk as. This givs ~ W W i - = - k + k +  a a a a ml  a, = x, a = x, k k a a ml ml ml D8) Sin th ondution and fftiv mass and th uttingr paramtrs ar xprssd in trms of quantitis Eq. D7) it follows from D8) that th diffrntial oprators must applid also on ths paramtrs. or th ondution and th diagonaliation of th 8 and amiltonian rplas th prvious nvlop wav quation k AB, V AB, ) D9),, m m AB AB = whr w now hav position dpndnt fftiv mass instad of th ar ltron mass. or th W disussd hr th rsult for 4x4 sumatrix dsriing th and ands is givn 5

16 hh - * lh - * lh * * hh whr P + hh x p = g + g ) k + k )- - ) V ) m g g P + lh x p = g - g ) k + k )- + ) V ) m g g k ) =+ i kx + iki + m g g d k ) = g kx -k)-ikxk. m K D) ot that in driving wav quation for nvlop funtions w hav not assumd slow variation of and dg on latti ll sal. t is oviousl nough to assum that th atomi Bloh stats ar unhangd aross th rstal and that thr is onl a finit numr of htrountions in th sampl. Thrfor w an appl our rsults also for W s, whr th and dg modifiation is not smooth ut arupt and taks pla within on monolar. 5

17 V Appliations of nvlop wav funtions 6. Basis st transformations and axial approximation Two diffrnt asis sts, whih ar rlatd a unitar transformation ar gnrall usd in th disussion of th Kohn-uttingr formalism. Both ths asis sts ar slightl diffrnt from thos usd Bastard). n this stion all quations in ltron pitur valn and fftiv mass is ngativ) Dfin th asis st split-off and ngltd),,-,,- g X + iy A 6 i 6 i X - iy ga+ Z B X + iygb- Z A X g - iy B E) Th orrsponding uttingr-kohn ulk amiltonian is givn hh * lh - * lh - * * hh 54

18 E) whr hh =- g - g gk + kx + kg + g g E) m lh =- g + g gk + kx + kg -g g E4) m x x E5) k ) =- g k -k -ig k k m d xi g k ) = m k + ik k g. E6) Th uttingr paramtrs g, g, g hav n dfind prviousl. Basis st is dfind, g X + iy A ga,- - X - iy + Z 6 B E7), 6 X + iy gb- Z A,- - - B X iy g 55

19 Th nw amiltonian is givn ' = A = G - i -i J Th matrix ' is thn givn i ' = Aiii A ~ ~ AA whr E8) E9) Aordingl, diagonal matrix lmnts hh nondiagonal matrix lmnts on otains and lh ar unhangd. or th Æ - Æ - Æ -i * * * * Æ i. E) As a final rsult amiltonian an writtn as hh ' * lh - ' * lh -' * ' * hh E) n E th diagonal trms ar givn E and E4 ut th nondiagonal trms om 56

20 k ) = g k -k -ig k k m x x d i g k ) = m k x - ik g k. 6. uantum wll or quantum wll sustitution asis st E7) lads to a ral amiltonian whn th k Æ-i is mad in matrix lmnts. Adding also th onfinmnt potntial V diagonal trms This givs P + k k V g g g g g g g g g P + ) =- g - g g k + g + g g k - + V m g g g ) hh x p = m lh x p p )to th and for th nondiagonal lmnts k ) = kx + - kx + m g g g g m k k k ) = xg - - g g Th disadvantag of this hoi of oordinat sstm is that th and stats ar oupld vn at G point. Th doupling taks pla onl if th -axis is takn orthogonal to th W. P 57

21 plan. f ours th hoi of oordinat sstm dos not modif th planar fftiv masss or an othr masural phsial proprt) at th and dg. 6. uantum wir or quantum wir th rplamnt k Æ- i x x =, givs = g - g + -k g + V g m x P + hh p g g P ) E) g g g k - V g g g P ) E) k x x k + g g P E4) k g + m x E5) = m x lh p ' k ) =- + m ' k ) =- + and for th omplx onugat lmnts ' * k ) =- g + k m x - g x k P E6) ' * k ) =- -k g + m x g. E7) g + g Th axial approximation implis stting g, g Æ in offiints K whihthnom and ' k ) =- g + g + m K x k K E8) 58

22 ' * k ) =- g + g - m K x k K. E9) 59

23 6.4 uttingr-kohn amiltonian in axiall smmtri D Th D uttingr-kohn amiltonian in Cartsian oordinat sstm is hh ' * lh - ' * lh -' * ' * hh whr aftr making th rplamnt k w hav hh lh Æ-i ; k Æ-i ; k Æ-i x x Vp r g g g g g g P,) ) i - g g x P ) g i. m x 4) = Vp r m g g g g g g x P +,) ) = m x '=- - m x '=- - uttingr paramtrs ar assumd to position indpndnt in th following disussion) or th omplx onugats w hav 6

24 ' * =- g - m x + ig ' * =- g + i m x Th axial approximation implis g, g Æ thn om '=- m ' * =- m g + g - i K x g + g + i K x x P 5). 6) g + g K in offiints whih 7). 8) uttingr-kohn amiltonian in lindr oordinats Dfin th oordinat transformation xæ rosf Æ rsinf Æ 9) Th partial drivativs ar givn f sinf Æos - x r r f f osf Æ sin +. ) r r f Æ Aordingl 6

25 and ± if ± i = ± x r i r, ) f + = + + ) x r r r r f ± if i ± i = - - x r r r r f ± r f r - r f P. ) Dfin R =- g + g m K 4) R =- m g. 5) Th non-diagonal trms ar givn -if i '= R - - r r r r - r r - f f r f P 6) and -if i '= R - r r f. 7) Th diagonal lmnts of amiltonian in lindr oordinats hh = Vp r m g g g g g g r r r r P + hh,)8) f 6

26 lh = Vplh r m g g g g g g r r r r P +,)9) f whr w hav assumd diffrnt onfinmnt + dformation potntials for hav- and light-hols. Th sparation of th angl dpndn Th total nvlop funtion is givn Y,) r = 4 r, ) r, ) r, ) r, ) P P = rom a) it follows that -im -) f -imf -im -) f -im -) f 4 r, ) r, ) r, ) r, ) P. a) =-im f =-m f =-im -) f =- m -) f ; ; f f 4 f f 4 =-im -) =- m -) =-im -) =- m -) 4 4. ) Th diagonal lmnts av-hols R S T hh = m g - g g + g g + g + - V r m r r r r P +,) P U V W phh ) 6

27 4 hh 4 = R ) S T g g m - 4 g - g + g g m r r r r ight-hols lh = R ) S T g g m - g + g + g g m r r r r lh = R ) S T g g m - g + g + g g m r r r r P V P +,) r U V W phh 4 P V P +,) r U V W plh P V P +,) r U V W ) plh Th oupld four and ignvalu prolm Equation hh -if i + R r r r r f r f r r f R -if i - r r f = E P P + ) Using Eq. B) to liminat th f dpndn hh m - ) m - ) + R r r r r r r r R r m - ) - r = E K P +. 4) 64

28 Equation hh + if i + R r r r r f r f r r f R -if 4 - r i 4 r f = E P P - 5) Using Eq. B) to liminat th f dpndn lh R r m - ) - r 4 4 E K m m R P - r r r r r r r = 6) Equation lh -if i 4 + R r r r r f r f r r f R + if i + r r f = Using Eq. B) to liminat th f dpndn lh E 4 4 m - ) m - ) 4 + R r r r r r r r R E r m + r = P P + 7) K P + 8) 65

29 Equation V hh 4 + if i + R r r r r f r f r r f R + if + r i r f = E 4 P P -. 9) Using Eq. ) to liminat th f dpndn hh m - ) m - ) 4 + R P - r r r r r r r. ) m - ) R + E 4 r r = 6.5 Strain indud D s - on-ro magnti fild as K n this stion w onsidr D s indud slf assmld strssors. Th strain indud onfinmnt fft is disussd in dtail in th nxt haptr. B making th minimal sustitution kα = pα + Aα th uttingr and Kohn amiltonian for Γ 8 -smmtri ands, an dsrid th fftiv amiltonian R S T k = + 5 k J k k J J m m J B + + K m q J γ B γ γ ) γ α, β α, β κ α n α βs d α β β αi α β n sn s α α ) whr k, k = k k + k k, α, β = xand,, J α ar th 4D matris rprsnting th omponnts of th angular momntum. Equation ) also ontains a st of dimnsionlss matrial paramtrs g, k and q. or a quantum onfind smiondutor th onfinmnt potntial is addd to Eq. ) and th omponnts of th linar momntum ar rplad p α = i. To xpliitl writ down Eq. ), w hoos a rprsntation for th J-matris α of th form U V W, 66

30 J J J x = = = - i i i -i - i - - P, P i P, ) nsrting Eq. ) into Eq. ), and using th spifi vtor potntial introdud aov, w otain th valn and amiltonian in th matrix form = + V hh hh + V lh lh + V - lh lh - + V hh hh P m BJ m P + k + qj B + / + / -/ - /, ) whr hh lh R S T R S T ω = ) x + ) ω γ γ γ γ ) m x ω = + ) + + x + ) ω γ γ γ γ ) m x U V W U V W 4a) 4) ω = γ i + x i ) 4) m x 67

31 ω =+ γ i + x m x x ω ω iγ + i x x x x + + K J ω d x. i 4d) Sin th nondiagonal trms of th Pikus-Bir strain amiltonian wr found to small, spiall for th fw lowst stats, th onfinmnt potntial was takn to diagonal. Th and hav diffrnt potntials Vhh, lh,) r = Vh ) + Vh,) r ± Vh,) r,whrth+ -) sign orrsponds to ). r V hw )is th hol and dg onfinmnt potntial, V,) r = a + + ) th hdrostati dformation potntial, and h v xx V, r ) = - + )/ th shar dformation potntial. Th smmtr axis of th h S v xx onfinmnt potntial is prpndiular to th W plan and paralll to th magnti fild. W S To furthr simplif th alulations w will assum th axial approximation in th uttingr paramtrs, making th rplamnt g, g Æ g + g / g. W also st th offiint q =, sin this paramtr is vr small for th prsnt D strutur. Sin th potntials ar assumd lindriall smmtri, th -omponnt of th total angular momntum, J = J +, is a onsrvd quantit. Th ignfuntions of Eq. ) ar thus also tot ignfuntions of J tot and th four omponnt nvlop funtion in lindrial oordinats must thn of th form Y r,, f) =,, rf) P,, rf),, rf) P = -,, rf) - i -J ) f Z,) r P,) r,) r, r) P = - - i -) f i -) f i + ) f - i + ) f -, r ), r ), r ), ) r P, 5) whr is th ignvalu of th -omponnt of th total angular momntum. Sin th wav funtion must uniqu undr a rotation of π around th -axis, must a positiv or ngativ half intgr ut not an intgr). Using Eqs. 5) and 6) w otain th following xprssion for th amiltonian 68

32 = B C hh + lh + + lh C B hh B C C B P JB P + κ + / + / / /, 6) whr th diagonal trms ar givn g g r hh ± ± = ) w - ± w g g g - Vhh r g m r r r r ) P +, ), P 7a) th diagonal trms g g r lh ± ± = ) w - ± w g g g - Vlh r g m r r r r ) P +, ) P and th off-diagonal trms 7) B ± + ± =± γ + ) + ω r m r r C ± = 7) K J + P + γ + γ ± + ± ) + ω r ) + ω r. 7d) m r r r r P 69

33 ot that in lindrial oordinats =, aus of th r-dpndnt volum r r r lmnt. Th ignvalu prolm aov lads to four ral oupld partial diffrntial quations whih hav to solvd numriall. 7