Distance dependence of fluorescence resonance energy transfer

Transcription

1 J hm Sc, Vol, o 5, Sptmbr 009, pp Indn cdmy of Scncs stnc dpndnc of fluorscnc rsonnc nry trnsfr R S SWTHI nd K L SEBSTI prtmnt of Inornc nd Physcl hmstry, Indn Insttut of Scnc, Bnlor ml: ls@pcscrntn bstrct vtons from th usul R 6 dpndnc of th rt of fluorscnc rsonnc nry trnsfr (FRET on th dstnc btwn th donor nd th ccptor hv bn common scnro n th rcnt tms In ths ppr, w prsnt crtcl nlyss of th dstnc dpndnc of FRET, nd try to llustrt th non-r 6 typ bhvour of th rt for th cs of trnsfr from locld lctronc xctton on th donor, dy molcul to thr dffrnt nry ccptors wth dlocld lctronc xcttons nmly, rphn, two-dmnsonl smconductn sht nd th cs of such smconductn sht rolld to obtn nnotub W us smpl nlytc modls to undrstnd th dstnc dpndnc n ch cs Kywords FRET; rphn; nnotub; tht-bndn modl Introducton Rsonnc nry trnsfr s th procss of nonrdtv trnsfr of nry from n xctd stt donor to round stt ccptor Th rsonnc condton mpls tht th msson spctrum of th donor hs to hv snfcnt ovrlp wth th bsorpton spctrum of th ccptor Whn th donor s fluorscnt spcs, th procss s clld fluorscnc rsonnc nry trnsfr (FRET ccptor my or my not b fluorscnt In FRET, th rt of nry trnsfr follows n R 6 dpndnc whr R s th dstnc btwn th donor nd th ccptor Th rt of nry trnsfr cn b vlutd usn th Frm oldn rul of tm-dpndnt untum mchncs: π = ρ ρ Ψ Ψ H Ψ Ψ, f v u v u I v u v u v u δ ( E E E E ( nd rfr to th donor nd th ccptor rspctvly Th ntl stt of th donor s n xctd stt dnotd s Ψ v, wth n nry E v Th donor s ntlly n n xctd stt wth n lctron tht ws sttn n th molculr orbtl ψ promotd to th orbtl ψ Intlly, th ccptor s n th round stt, nd ftr th nry trnsfr, n dctd to th mmory of th lt Profssor S K Rnrjn For corrspondnc lctron whch ws sttn n th molculr orbtl ψ s promotd to ψ In th bov procss, thr r chns n th vbrtonl stts of both th donor nd th ccptor Th vbrtonl unt r dnotd by ν, ν, u nd u ρ v nd ρ u r th ntl dstrbutons of th donor n th xctd stt nd th ccptor n th round stt rspctvly H s th prt of th Hmltonn rsponsbl for th nry trnsfr nd s oulombc n ntur, s no othr ntrcton s possbl f th ovrlp btwn th orbtls of th donor nd th ccptor s smll (whch s usully th cs Wthn th dbtc pproxmton, th mtrx lmnt my thn b wrttn s 3 v u HI v u Ψ Ψ Ψ Ψ ψ ψ ψ ψ vu r = v u ( ( ( (, ( whr ε s th prmttvty of th mdum sprtn th donor nd th ccptor s rsult of th oulombc ntrcton, n lctron n th donor ts d-xctd, wth th smultnous xctton of n lctron of th ccptor lctn th nuclr poston dpndnc of th mtrx lmnt r ψ ( ψ ( ψ ( ψ ( 777

2 778 R S Swth nd K L Sbstn nd pproxmtn t by ts vlu vlutd t th ulbrum postons of th nucl s = ( drψ r rψ ( r r ψ ( ψ ( ψ ( ψ ( r ψ ( ψ ( ψ ( ψ ( ˆR s th unt vctor n th drcton of th vctor R connctn th cntrs of chr of th two spcs u to th dpolr pproxmton of (4, th rt of trnsfr hs n R 6 dpndnc Förstr 3 ws th frst to nlys ths procss thortclly nd h rrvd t th follown xprsson for th rottonlly vrd rt: lds to = v u HI v u Ψ Ψ Ψ Ψ v v u u r ψ ( ψ ( ψ ( ψ ( ot tht th ntrl r ψ ( ψ ( ψ ( ψ ( (3 (w shll from now onwrds nlct th subscrpt s just th lctrosttc ntrcton btwn th trnston dnsts ψ ( ψ ( nd ψ ( ψ ( For lr ntrprtcl sprtons, ths ntrcton my b pproxmtd s th ntrcton btwn th corrspondn trnston dpols Ths s nown s th dpolr pproxmton Wthn ths pproxmton, r ψ ( ψ ( ψ ( ψ ( 3 ˆˆ RR =, 3 R (4 whr, r th trnston dpols of th donor nd th ccptor rspctvly nd r dfnd by nd = ( drψ r rψ ( r, 6 R0 =, τ 0 R (5 whr τ 0 s th lftm of th donor n th bsnc of th ccptor nd R 0 s th wll-nown Förstr rdus 4 Snc th xprmntl dmonstrton of FRET s usful spctroscopc rulr by Stryr nd Hulnd, 5 t hs bn usd xtnsvly for undrstndn th conformtonl dynmcs of bolocl molculs l protns, R, tc Whn th donor nd th ccptor r two dy molculs, FRET s found to b ffctv n th rn 0 Å 00 Å Enry trnsfr provds n ddtonl dxctton pthwy for th donor thrby ldn to dcrs n th lftm of th xctd donor n prsnc of th ccptor comprd to tht n th bsnc of th ccptor ftr bout 00 Å, n FRET, th xctd donor dcys ccordn to ts nturl lftm, rthr thn by nry trnsfr Whn thr th donor or th ccptor or both r xtndd systms wth dlocld chr dnsts, brdown of th dpolr pproxmton lds to dvtons from th usul R 6 dpndnc In vw of th rcnt studs ldn to non-r 6 dpndncs of th rt of nry trnsfr, w nlys th procss of nry trnsfr from dy molcul, whch hs locld xctton to thr dffrnt ccptors wth dlocld xcttons: rphn, two-dmnsonl smconductn sht nd lon smconductn nnotub W prsnt crtcl nlyss of th dstnc dpndnc of th rt n ll th css Th ppr s ornd s follows: n th scond scton, w prsnt nrl formlsm for vlutn th rt for th cs of trnsfr from locld donor to dlocld ccptors Thrd, fourth nd ffth sctons dl wth trnsfr to rphn, two-dmnsonl smconductn sht nd lon smconductn nnotub rspctvly In th sxth scton, w prsnt nrl dscusson of th non-r 6 bhvour of th rt for vrous systms

3 stnc dpndnc of RET 779 Rsonnc nry trnsfr from locld donor to ccptors wth dlocld lctron dnsts W consdr th procss of rsonnc nry trnsfr from locld donor, nmly dy molcul to ccptors wth dlocld chr dnsts Th mtrx lmnt for ntrcton s vn by U d d ψ ψ ψ r r r ψ r r r r r ( ( ( ( = (6 Th orbtls ψ ( r nd ψ ( r on th ccptor r xtndd n spc (n comprson wth th dstnc of th donor from th ccptor In such stuton, w cn thn of th ntrcton btwn th donor nd th ccptor s tht btwn th trnston dpol of th donor, loctd t th cntr of chr vn by = ( drψ r rψ ( r nd th trnston chr dnsty ψ ( r ψ ( r of th ccptor Thrfor, th mtrx lmnt for ntrcton bcoms U = Φ, (7 whr Φ s th lctrosttc potntl t th pont r (th poston of th donor du to th chr dnsty ψ ( r ψ ( r W shll thn of th ccptor s hvn prodcty, s rsult of whch th stts r chrctrsd by th wv vctor s rsult of nry trnsfr from th donor, n lctron n n nry lvl wth wv vctor s xctd to lvl wth wv vctor f For th trnsfr of n nry Ω from th donor to th ccptor, th rt s vn by: π U E E ( Ω=, δ ( Ω f f f (8 W dfn f =, whr s th momntum trnsfrrd to th ccptor Th rt cn thrfor b wrttn s: π U E E ( Ω=, δ ( Ω (9 Th trnston dnsty vn by ψ ( r ψ ( r = ψ ( r ψ ( r thn hs prodcty wth wv vctor If on dopts th smplst possbl pproxmton, th lctrosttc potntl t pont outsd such chr dstrbuton s functon only of U, = U( (ths s not tru for rphn nd crbon nnotubs Thrfor, th rt cn b wrttn s π U F whr ( Ω= ( (, (0 F( = δ ( E E Ω ( Th dstnc dpndnc of th rt of nry trnsfr s thrfor ovrnd by th functonl forms of th ntrcton nry, U( nd F( W us th nrl formlsm vn bov to undrstnd th procss of nry trnsfr from locld donor to th follown nry ccptors: rphn, twodmnsonl smconductn sht nd th cs of such smconductn sht rolld to obtn nnotub 3 Rsonnc nry trnsfr from dy to rphn W now consdr th procss of nry trnsfr from fluorscnt dy to rphn (s fur 6,7 W us th tht-bndn modl for rphn 8 Th wv functons for rphn r ψ s χ ( r s ( r=, δ s B χ ( r sb B B ( whr th sn holds for th vlnc bnd (π bnd nd th sn holds for th conducton bnd (π bnd nd B rfr to th two crbon toms of th unt cll of rphn s th numbr of toms of ch typ n th lttc χs r th p tomc orbtl wv functons of th crbon toms of th lttc Th phs fctor δ s dfnd by th follown rlton: δ H B ( =, H ( B (3

4 780 R S Swth nd K L Sbstn whr / 3 / 3 x x H ( = t[ cos( /] (4 B In th bov uton, t s th hoppn ntrl nd = l 3 Th nrs of th bnds r vn by E = t[ 4cos( /cos( 3 / y / y 4cos ( /], (5 whr th sn s for th conducton bnd nd th sn s for th vlnc bnd Th vlnc nd th conducton bnds mt t th K-ponts r th K- ponts, 9 th nry dsprson bcoms 3 E = t = vf, (6 whr = x y Whn th donor, fluorscnt dy molcul s xctd, n lctron from n occupd orbtl s xctd to n unoccupd orbtl W consdr th x y trnston dpol momnt of th dy, = ˆ ˆ ˆ x yj to b ntrctn wth th lctrons of rphn s rsult of such n ntrcton, th dy rturns bc to ts round stt nd n lctron n rphn s xctd Exctton of n lctron from ψ ( r to ψ ( r lds to trnston chr dnsty vn by ρ ψ ψ (= r ( ( r r ( δ δ ( s χ r s χ ( r s, (7 whr th summton s ovr crbon toms of thr typ n th lttc Furthr, w hv nlctd th product of χs whch blon to dffrnt toms of th lttc, s thy r nlbl Th lctrosttc potntl du to such trnston dnsty s vn by ρ( r Φ r r (8 r r (= d 4 πε Snc th dnsty χ ( r s χ( r s s locld nr th th tom, w cn us th multpol xpnson to clcult ts lctrosttc potntl t th pont r Th lowst ordr trm s th monopol trm ldn to s ( δ δ (= Φ r 8 πε r s (9 For smll vlus of (=, th sum n th bov uton my b rplcd by n ntrl so tht s ( δ δ (= Φ r d, 8πε s r s u (0 whr u s th r of th unt cll of rphn Th two-dmnsonl ntrl wrttn bov cn b vlutd to t Fur schmtc of th rphn lttc nd th donor dy Th poston vctors nvolvd n th nlyss r lso shown ( δ δ X Φ(= r, ( 4ε whr w hv usd r = (X,, wth X bn prlll to th pln of rphn s th r of th

5 stnc dpndnc of RET 78 rphn lttc W vlut th ntrcton nry usn (7 nd fnd t to b ( δ δ = ( ˆ ˆ U X, 4ε ( whr ˆ = / s th unt vctor n th drcton of nd ˆ s th unt vctor n th drcton Th sur of th ntrcton nry s U = [ cos( ϕ ϕ ] 8ε ˆ ( ˆ (3 W now us (9 to obtn th follown xprsson for th rt of trnsfr: π G whr ˆ ( Ω= ( ˆ (, 4ε (4 G( = [ cos( ϕ ϕ ] δ( E E Ω (5 ot tht th xtr trm n G( comprd to F( dfnd n ( s du to th phs fctor n th wv functons of rphn W frst vlut G( W now us (6 for th nry lvls of rphn, rplc th sum ovr by n ntrl to t π cosψ ψ 0 0 ψ G( = d d 4π cos f δ[ v ( cos ψ Ω] (6 Th bov ntrl cn b vlutd xctly 0 to obtn Θ( Ω v G( = 8 π ( Ω f vf (7 W now substtut th bov rsult nto (4, rplc th sum ovr by n ntrl to t π ( Ω= d dθ[ μ 64πhε 0 0 v f vf θ ( Ω ( μxcosθ μysn θ ] ( Ω (8 Th ntrl ovr θ cn b sly prformd to t ( Ω = ( μ x μy μ 64hε Ω/ v f 0 3 d ( Ω vf (9 For lr vlus of, only vlus of /( contrbut to th bov ntrl In such cs, w cn nlct v f n comprson wth ( Ω, nd xtnd th uppr lmt of th ntrl n (9 to nfnty, to t μx μy μ 4 3π ( ( Ω= (30 56Ωh ε Th bov xprsson for rt cn b rwrttn s sn cos 4 3π ( μ θ μ θ ( Ω=, (3 56Ωh ε whr = nd θ s th nl tht ms wth th -xs W now prform n vrn ovr ll possbl ornttons of th donor trnston dpol momnt to t μ 4 π ( Ω = (3 64Ωh ε Thrfor, th rt of nry trnsfr hs powr lw dpndnc [(dstnc 4 ] for lr vlus of 4 Rsonnc nry trnsfr from dy to two-dmnsonl smconductn sht W now consdr th procss of nry trnsfr from dy molcul to th lctrons confnd to two-dmnsonl smconductn sht W ssum

6 78 R S Swth nd K L Sbstn tht thr s bnd p ε nd th nry lvls bov nd blow th p r vn by E ε = m (33 Th sn s for th conducton bnd nd th sn s for th vlnc bnd s two-dmnsonl wv vctor W hv ssumd th ffctv mss m to b th sm for stts bov nd blow th bnd p Th corrspondn wv functons r ψ (= r ( r R, (34 R χ = whr th sn s for th conducton bnd nd th sn s for th vlnc bnd R dnots th poston of th th tom n th sht s th totl numbr of toms of th sht χ (r R l nd χ (r R dnot th locld orbtls on th th tom tht contrbut to th conducton nd th vlnc bnds rspctvly Wth th bov, xctton of n lctron from ψ ( r to ψ ( r lds to trnston chr dnsty vn by ( = r ( ( r r ρ ψ ψ (35 ( R χ r R χ ( r R, = whr w hv nlctd th product of χs whch blon to dffrnt toms of th lttc, s thy r nlbl Th lctrosttc potntl du to such trnston dnsty s vn by ρ( r Φ r r (36 r r (= d 4 πε Snc th dnsty χ ( r R χ ( r R s locld nr th th tom, w cn us th multpol xpnson to clcult ts lctrosttc potntl t th pont r Th monopol trm of th multpol xpnson vs ro snc th two orbtls χ nd χ r orthoonl Thrfor, th lowst ordr non-ro trm s dpolr n ntur nd on ts whr = dr χ ( r R ( r R χ ( r R s th trnston dpol for th χ χ trnston For smll vlus of (=, th sum n th bov uton my b rplcd by n ntrl so tht R Φ(= r r d, 4 πε R (38 r R u whr u s th r of th unt cll of th twodmnsonl lttc W consdr tht th unt cll of th lttc contns on tom such tht = u, whr s th r of th sht Th twodmnsonl ntrl wrttn bov cn b vlutd to t X Φ(= r r, (39 ε whr w hv usd r =( X,, wth X bn prlll to th pln of th sht Th bov uton cn b smplfd to obtn (= ( ˆ ˆ X Φ r (40 ε W now us (7 to t ˆ = ( ˆ ( ˆ ˆ U, ε X (4 whr s th trnston dpol momnt of th dy vn by = ˆ ˆ ˆ x y j ot tht U s functon only of Thrfor, w substtut (4 nto (0 to t th rt of trnsfr vn by π ( Ω= ( ˆ ˆ ε ˆ ( ˆ F( (4 W now vlut F( usn (33 nd ( to t Φ(= r, 4 πε r R r = R (37 m m F( = Θ ( Ω ε π (43

7 stnc dpndnc of RET 783 W now substtut ths bc nto (4 nd rplc th sum ovr by n ntrl to obtn th follown xprsson for th rt: 3 m 3 ε 0 m ( Ω= d Θ ( Ω 6π ε π 0 ˆ ˆ d ( ˆ ˆ θ ( (44 Th two-dmnsonl vctor s xprssd n ts polr coordnts nd θ For lr, only smll vlus of r mportnt Thn, m m Θ ( Ω ε ( =, Θ Ω ε f Ω > ε For such stuton, th ntrl ovr cn b xctly vlutd to obtn th follown xprsson for th rt 3 m ( Ω= π ε π ˆ ˆ d θ ( ˆ ( ˆ (45 0 Th ntrl ovr θ cn now b prformd to t n xprsson nvolvn th componnts of th dpol momnts, nd W thn prform n vrn ovr ll possbl ornttons of th donor dpol, ldn to th follown xprsson for th rt: m ( Ω = 64 πε 3 4 vry lon tubul of rdus (whch s of nnodmnsons nd lnth L ( (s fur W dopt th smplst possbl dscrpton for th lctrons of th tubul W ssum tht w hv smconductn nnotub, hvn structur smlr to th boron ntrd nnotub, wth bnd p ε (s fur 3 Th cs of crbon nnotub s ntrstn nd s dscussd n sprt publcton Th stts bov nd blow th bnd p my b chrctrd by th momntum prlll to th xs of th tub, whch w dnot by nd th nulr momntum bout th tub xs, whch s untd (nulr momntum bn ul to m, wth m = 0,,, 3 W t th lctron wv functons to b ψ ( ρ, φ, = χ ( r R m ϕ, m = Th suprscrpts n ψ, m( ρ, φ, dnot stts bov ( or blow ( th p χ (r R nd χ (r R dnot th locld orbtls on th th lttc st tht contrbut to th conducton nd th vlnc bnds rspctvly s th numbr of lttc sts of th tubul W t th nrs of th stts to b vn by 4 ε =, m m m ε m, (48 whr m s th ffctv mss Wth th bov, xctton of n lctron from ψ, m( r to ψ, m ( r f f lds to trnston chr dnsty vn by ( x y f Ω> μ μ μ μ ε (46 =0 f Ω <, (47 ε whr w hv usd = nd ˆ = ˆ ˆ x yj Th rt of trnsfr n ths cs too hs (dstnc 4 dpndnc 5 Rsonnc nry trnsfr from dy to lon tubul W now consdr th procss of nry trnsfr from dy molcul to th lctrons confnd to Fur schmtc of th trnston dpol momnt of th donor dy nd th tub

8 784 R S Swth nd K L Sbstn whr u s th surfc r of th unt cll of th cylndrcl surfc W consdr tht th unt cll of th lttc contns on tom ldn to = u, whr s th surfc r of th lon cylndr W now us th follown multpol xpnson for th /r R trm n cylndrcl coordnts: 5 = d r R π m = m ( ϕ ϕ ( Fur 3 schmtc of th on-dmnsonl nry bnds of th nnotub shown th untum numbrs lon th utorl drcton, th nry p nd th mount of nry trnsfrrd f, mf, m ρ( r= ψ ( r ψ ( r m ϕ ( χ r R χ ( r R, = (49 whr w hv put f =, m f = m m nd nlctd th product of χs whch blon to dffrnt sts of th lttc, s thy r nlbl (, ϕ, r th cylndrcl polr coordnts of R Th lctrosttc potntl du to such trnston dnsty s vn by ρ( r Φ r r (50 r r (= d 4 πε Snc th dnsty χ ( r R χ ( r R s locld nr th th tom, on cn wrt th potntl s Φ(= r, r R r = m ϕ (5 whr = drχ ( r R( r R χ ( r R s th trnston dpol for th χ χ trnston For smll vlus of, th sum n th bov uton my b rplcd by n ntrl so tht m ϕ Φ(= r r d, R r R u (5 I ( K ( d R, (53 m m whr w hv usd r =dcosϕˆ dsnϕˆj ˆ nd R = cos ˆ sn ˆ ˆ ϕ ϕj On substtutn ths bc nto (5 nd vlutn th subsunt ntrls, on ts m (= I ( [ m ϕ Φ r K m ( d ] ε r (54 W now us μ = ˆ μρρ μϕϕˆ μ ˆ to obtn mϕ Φ(= r I m ( ε ρ ( K m ( m ( d K d m ϕ K m ( d d (55 W now vlut th ntrcton nry usn (7 nd t s vn by mϕ U = I m ( ε K m ( d K m ( d ρ ρ d d m ( ϕ ρ ϕ ρ ( ρ ρ d, m m ϕ ρ ϕ ϕ K m ( d d d m ( ϕ ϕ d (56

9 stnc dpndnc of RET 785 whr w hv usd μ = ˆ ˆ μρρ μϕϕ μ ˆ W now vlut F( usn ( W us th nry lvls from (48 to obtn th follown uton for F(: Lm F( = d π δ m ( m m m ( m ( Ω ε On vlutn th ntrl, w obtn Lm F ( = π { m } 4 m ( Ω ε {( m m m } (57 (58 Th nulr brcts ovr m ndct tht th summton ncluds thos vlus of m for whch th untty wthn th sur root trm s postv For smll vlus of momntum trnsfr (whch dtrmn th lon dstnc bhvour, w cn nlct th trm n comprson wth Ω ε 4 m ( {( m m m } Wth ths condton, Lm F ( π 4 m( Ω ε { m } {( m m m } (59 Substtutn (59 nto (0 nd rplcn th sum ovr by n ntrl vs ( Ω= Lm 3 π { m }, m {( m m m } 4 m ( Ω ε d U (, (60 whr U( s vn by (56 To vlut th ntrl ovr, w chn th vrbl of ntrton to t dfnd by t = d nd subsuntly us th symptotc form of I m vn by I m t d, m t t d d Γ ( m Th ntrl ovr t cn now b sly vlutd for vrous vlus of m Th ldn ordr trm n th xprsson for th rt corrsponds to th cs whn m = 0 W frst vlut th ntrl ovr t for such cs nd thn vr ovr ll possbl ornttons of th donor trnston dpol momnt to t th follown xprsson for th rt: 3 m (33μ 7 μ ρ ( Ω= π ε d { m } μ m m ( Ω ε, (6 whr w hv usd = Thrfor, th ldn ordr trm n th xprsson for th rt hs (dstnc 5 dpndnc 6 scusson In th rcnt tms, vrty of mtrls l polymrs, 6 nnoprtcls, 7 untum wlls, 8 untum dots, untum wrs, 9 mtl surfcs 0 tc hv bn usd s nry donors/ccptors nd thy hv ld to dvtons from th R 6 dpndnc Whn th nry trnsfr occurs from locld xctton on th donor to locld xctton on th ccptor, th dpolr pproxmton wors nd thrfor on hs n R 6 dpndnc On th contrry, f th nry trnsfr procss nvolvs n xtndd lctronc systm whr th xcttons r dlocld l n th cs of polymrs, untum wrs, untum wlls, th dpolr pproxmton s not vld nd on cn hv dvtons from th R 6

10 786 R S Swth nd K L Sbstn rts non-r 6 dpndnc s of rt ntrst du to th nd for dvlopn nnoscopc rulrs, tht cn msur dstncs wll byond 00 Å hnc t l hv thortclly nlysd th procss of dcy of n xctd molcul fluorscn nr mtllc flm Thy found d 3 dpndnc of th rt of nry trnsfr on th dstnc btwn th molcul nd th mtl for vry thc flms But, for th cs of thn mtllc flms, thy found d 4 dpndnc mpon t l xprmntlly studd th procss of lctronc xctton nry trnsfr from pyrn to snl crystl mtl surfc nd found d 3 dpndnc of th rt d 4 dpndnc hs lso bn obsrvd for th dcy of bctyl bov slvr surfc 0 Furthr, Prsson nd Ln 3 studd th procss of nry trnsfr from vbrtn dpol to th conducton lctrons of mtl Thy fnd tht th nry trnsfr to th surfc mods of th mtl lds to d 4 dpndnc of th rt on th dstnc, whl th trnsfr to th bul mods lds to d 3 dpndnc For th cs of nry trnsfr to th surfc plsmon mods of flt mtl surfc from proxml dpolr mttr, 4 urtc dstnc dpndnc hs bn rportd Th bov dpndncs cn b sly undrstood In FRET, th trnsfr occurs from ro-dmnsonl donor to ro-dmnsonl ccptor ldn to n R 6 dpndnc For th cs of trnsfr to th bul mods of th mtl, th ccptor xcttons r n thr-dmnsons Thrfor, on hs d 3 dpndnc Whn th trnsfr occurs to th ccptors whr th xcttons r confnd to two dmnsons, on hs d 4 dpndnc Lyo 5 hs studd th procss of xctonc nry trnsfr from nrrow untum wr to wd untum wr Th rt of trnsfr ws found to hv d 4 dpndnc for ntrmdt vlus of d nd d 5 dpndnc n th symptotc lmt for lr vlus of d H hs lso nlysd th cs of trnsfr from nrrow untum wr to unform rry of dntcl prlll wd untum wrs nd found d 4 dpndnc n such cs Vry rcntly, Govorov t l 9 rportd study of th procss of xctton nry trnsfr from nnoprtcl to nnowr Whn th rdus of th nnoprtcl s smll comprd to th dstnc btwn th nnoprtcl nd th nnowr, th xcttons n nnoprtcls cn b consdrd locld nd hnc dpolr For lr nnoprtclnnowr dstncs, thy fnd d 5 dpndnc of th rt on th dstnc Snc th ccptor xcttons r confnd to on-dmnson, th rt hs d 5 dpndnc Enry trnsfr from n nornc untum wll to n ornc polyfluorn flm hs bn found to occur 6 du to th oulombc coupln btwn th Mott Wnnr xctons on th untum wll to th Frnl xctons on th polymr Th rt ws found to hv d dpndnc Smlr dstnc dpndnc hs bn found rlr for th cs of lctronc xctton nry trnsfr from polyfluorn to ttrphnyl porphyrn 6 Hll t l 7 hv lso rportd smlr dstnc dpndnc for th cs of nry trnsfr btwn two fluorscnt polymrs tht r rown s lyrd structurs Snc th xcttons n polymr flms nd untum wlls r dlocld ovr two-dmnsons, th rt hs d dpndnc Th procss of nry trnsfr from untum wll 8 to monolyr of smconductor nnocrystls ws rportd rcntly Th xprmnts rvld vry hh ffcncs of nry trnsfr nd th procss ws ttrctv for us n vrty of tchnolocl pplctons l LEs, lsrs, tc Kos t l 8 hv studd ths procss thortclly ssumn tht th s of th untum dot nd th wdth of th untum wll r smll n comprson wth th dstnc btwn thm Thy consdrd two typs of xctd stts n th untum dots s rsult of trnsfr, nmly uscontnuum of hh nry stts nd dscrt low nry stts For th cs of trnsfr to uscontnuum of stts, thy fnd d 4 dpndnc of th rt, whl for xctton to dscrt low nry stts, thy fnd n xponntl dpndnc rnovch t l 9 hv consdrd th procss of nry trnsfr from n nornc untum wll to n ornc lyr nd found n xponntl dstnc dpndnc In ll th bov css, thr th donor or th ccptor or both wr xtndd systms wth dlocld chr dnsts Thrfor, brdown of th dpolr pproxmton to th rt ld to dvtons from th usul R 6 dpndnc Strous t l 30 hv studd th procss of nry trnsfr from th dy fluorscn to 4 nm dmtr old nnoprtcl oubl-strndd molculs of vrous lnths wr usd to fx th dstncs btwn th donor nd th ccptor Surprsnly, thouh th dstncs btwn th donor nd th ccptor r much lrr thn th dmnsons of th donor nd th ccptor, thy fnd d 4 dpndnc of th rt Thy rfr to ths procss s nnoprtcl surfc nry trnsfr (SET nd th rn of

11 stnc dpndnc of RET 787 dstncs tht cn b msurd usn SET s mor thn doubl tht of th trdtonl FRET xprmnts Thrfor, t s ntrstn to s why th dstnc dpndnc n SET s d 4 But, thortcl clcultons for such systm consdrn th trnsfr to th plsmons nd th lctron-hol pr xcttons of th nnoprtcl prdomnntly fnd n R 6 dpndnc 3,3 7 onclusons W hv studd th dstnc dpndnc of th procss of rsonnc nry trnsfr from locld donor, dy molcul to vrous typs of nry ccptors-rphn, two-dmnsonl smconductn sht nd smconductn nnotub usn smpl nlytc modls For th cs of trnsfr to rphn nd two-dmnsonl smconductn sht of lctronc chr dnsty, w fnd (dstnc 4 dpndnc For th cs of trnsfr to smconductn nnotub, w fnd (dstnc 5 dpndnc W hop tht smpl nlytc modls of nry trnsfr prsntd hr cn ld to ntrstn physcl nshts nto th procss of rsonnc nry trnsfr cnowldmnts R S Swth cnowlds ouncl of Scntfc nd Industrl Rsrch (SIR, Ind nd Brstol Myrs Subb Fllowshp for fnncl support Rfrncs Lowc J R 006 Prncpls of fluorscnc spctroscopy (w Yor: Sprnr My V nd Kuhn O 000 hr nd nry trnsfr dynmcs n molculr systms (Wly-VH 3 Förstr T 948 nn Phys 55 4 Vlur B 00 Molculr fluorscnc (w Yor: Wly-VH 5 Stryr L nd Hulnd R P 967 Proc tl cd Sc US Swth R S nd Sbstn K L 008 J hm Phys Swth R S nd Sbstn K L 009 J hm Phys Wllc P R 947 Phys Rv Ktsnlson M I 007 Mtr Tody Shun Knnth W K 986 Phys Rv B Mruls V, Muyumn E E nd Gdu E 008 Phys Rv B Swth R S nd Sbstn K L (unpublshd 3 Blsss nd Gumbs G 006 Phys Rv B Gumbs G nd Blsss 005 Phys Rv B Jcson J 975 lsscl lctrodynmcs (w Yor: Wly Estrn Lmtd 6 Won K F, Bch B nd Rossy P J 004 J Phys hm Sönnchsn, Rnhrd B M, Lphrdt J nd lvstos P 005 t Botchnol chrmnn M, Ptrus M, Kos S, Smth L, Kols nd Klmov V I 004 tur Mrtn P L H nd Govorov O 008 Phys Rv B lvstos P, Vldc H nd Hrrs B 985 J hm Phys 8 54 hnc R R, Proc nd Slby R 978 dv hm Phys 37 mpon, Gllo R, Hrrs B, Robot H J nd Whtmor P M 980 hm Phys Ltt Prsson B J nd Ln 98 Phys Rv B Lrn I, Stocmn M I, chrmnn M nd Klmov V I 004 Phys Rv B69 403(R 5 Lyo S K 006 Phys Rv B Itsos G, Hlots G, Louds P G, Lupton J, Brrds P, lvs E, Prr S, Wtson I M, wson M, Fldmnn J, Murry R nd Brdly 007 Phys Rv B Hll J, Hrot S Y, Worsfold O, Rchrdson T H nd Fox M 004 Phys Rv B (R 8 Kos S, chrmnn M, Klmov V I nd Smth L 005 Phys Rv B rnovch V M, Bso M, L Rocc G nd Bssn F 998 J Phys: ondns Mttr Yun S, Jvr, Jnnns T, Fshr M, Hr S, Ptrson S, Hopns B, Rch O nd Strous G F 005 J m hm Soc Swth R S nd Sbstn K L 007 J hm Phys Bhowmc S, Sn S, Shnoy V B nd Bch B 006 J hm Phys 5 80