5.33 Lecture Notes: Magnetic Resonance Spectroscopy ( )

Transcription

1 5.33 Lctur Nots: Magntic Rsonanc Spctroscopy In our discussion of spctroscopy, w hav shown that absorption of E.M. radiation occurs on rsonanc: Whn th frquncy of applid E.M. fild matchs th nrgy splitting btwn two quantum stats. Magntic rsonanc diffrs from ths othr mthods in th sns that w nd to immrs th sam in a magntic fild in ordr to s th lvls that w prob with an xtrnal (rf or µwav) fild. (Two filds: Static magntic and E.M.) W will b probing th nrgy lvls associatd with th spin angular momntum of nucli and lctrons: NMR--nuclar magntic rsonanc and ESR/EPR--lctron spin rsonanc. Angular momntum: In our tratmnt of rotational nrgy lvls, w said that th nrgy lvls dpndd on th rotational angular momntum, L, which was quantizd: L = J J+ J = 0,, rot. quant. numbr Dgnracy of J was ( m J = 0,, ±J) ( J +) W rlatd L to th nrgy lvls L Erot = BJ J+ I Actually, all angular momntum is quantizd. If a particl can spin, it has A.M. and quantizd E lvls. In particular, w also hav to b concrnd with th spin of individual nucli and lctrons Lctur Nots: Magntic Rsonanc Spctroscopy Pag

2 You alrady know that lctrons hav ORBITAL angular momntum M = + = 0,, orbital angular momntum quantum numbr dgnracy of orbitals: + from m =, + magntic quantum numbr M Z = m m rprsnts th quantization of th componnts of M : (How w choos ˆ z dosn t mattr until w apply a magntic fild.) + 0 z M = M = Now, th angular momntum that w ar concrnd with is: Elctron Spin Angular Momntum = + s: lctron spin quantum numbr = ½ S s s for ach unpaird S z = m s m: s ± ( S, S +,, + S) on unpaird 5.33 Lctur Nots: Magntic Rsonanc Spctroscopy Pag

3 Nuclar spin angular momntum I = II+ I: nuclar spin quantum numbr I z = m I m I : I, I +,, I What is I? Each proton/nutron has a spin quantum numbr of /. Spin of many nuclons add up vctorily to giv I. Spins pair up so that vn numbr of spins ar paird I = 0.. For vn numbr of protons plus vn numbr of nutrons: I = 0. 6 C, O. For on unpaird nuclon I = m I = ± dgnracy I + = H, 3 C, 5 N So th proton and lctron ar similar both spin particls. W ll talk about ths two particls mor spcifically 3. Two unpaird nuclons I = m I = 0,± H, 4 N 5.33 Lctur Nots: Magntic Rsonanc Spctroscopy Pag 3

4 For spin ½ particls (I or S = ½) thr ar two dgnrat nrgy stats. Whn you put ths in a magntic fild you gt a splitting. A low nrgy stat alignd with th fild, and A high nrgy stat alignd against th fild Classical pictur: Think of our lctron or nuclus as a chargd particl with angular momntum, M. A circulating charg producs a magntic fild. This charg posssss a magntic dipol momnt µ that can b affctd by an applid magntic fild. m,q M µ M = m( v r) µ= Q( v r ) /c Th dipol lis along M and th strngth of µ is proportional to µ : v r Q µ = M γ M mc γ is th gyromagntic ratio Quantum: For lctrons: µ = g S m c = γ S g factor (.003) For nucli: + µ N = g m c =γ N N I N I g N = 5.6 for H g is a rlativistic quantum mchanical corrction Lctur Nots: Magntic Rsonanc Spctroscopy Pag 4

5 If w immrs this systm in a magntic fild, B, which is orintd along th z axis Th intraction potntial is Eint = µ B Th spins align along th magntic fild: Eint = µ zb Sinc µ z = γszand µ = γ z N I z, using Sz = msand Iz = mi w hav For lctrons: Eint = γ msb ms = ± For nucli: Eint = γn mib mi = ± So, as w incras th magntic fild strngth, th two nrgy lvls originally dgnrat split, on incrasing in nrgy and on dcrasing in nrgy. This is known as th Zman ffct. Nucli Elctrons Eint m I = / m s = +/ 0 E E = γn B E =γ B 0 B m I = +/ m s = / Now w hav a systm that can absorb E.M. radiation on rsonanc: E=hν ν is th applid frquncy (in th radio frquncy rang) Slction Ruls: m s = ± ; m I = ± Typically swp B and hold ν constant. γb ν= (Larmor frquncy) π 5.33 Lctur Nots: Magntic Rsonanc Spctroscopy Pag 5

6 Typical numbrs: Nuclar Magntic Rsonanc Elctron Spin Rsonanc µ H = (± ½) rg/gauss µ = (± ½) rg/gauss for B = 0 kg: ν = 4.6 MHz ν = 8 GHz Typical: 300 MHz/70 kg 9.5 GHz/3.4 kg Typical population diffrnc N = (Ν N + )/(Ν + N + ) 0.008% Th Chmical Shift Thus far you would think that all H absorb at sam frquncy. Yt, in practic diffrnt nucli absorb at diffrnt frquncis. Th rsonanc frquncy dpnds on th ffctiv magntic fild that a proton fls. This can diffr for diffrnt typs of H du to local lctron currnts that countract th applid fild. Shilding ff app B = B σ σ: scrning constant typically 0-6 Eint 0 E No shilding With shilding (shiftd to lowr ν) E = γ m B int N I app ( σ) B app γ ν= σ π ( ) 0 Bapp Shift of frquncy du to scrning: Chmical Shift 5.33 Lctur Nots: Magntic Rsonanc Spctroscopy Pag 6

7 Typically you masur th chmical shift du to scrning rlativ to a standard (TMS). γb ν ν = σ σ π app i rf rf i 6 i rf Masur positions in δ: ppm. δ = ( σ σ ) 0 ν ν i rf i νrf Th magnitud of th shilding dpnds on how th motion of lctrons modifis th local fild. (Intraction btwn fild and lctron angular momntum) Qualitativly, proton NMR spctra can b intrprtd by considring lctrongativity of bound functional groups. Gratr E.N. draws lctrons away from H, lowring th rsonanc fild and giving a largr δ. Othr nucli hav largr chmical shifts bcaus thr ar mor lctrons. Couplings btwn Nuclar and Elctron Spins Basd on th ffcts of scrning (chmical shift), w would xpct on lin for ach nuclus in an NMR spctrum. Actually thr is usually additional structur, with ach lin split into svral othrs. (Both in NMR and ESR) Splittings aris whn diffrnt magntic spins on nucli and/or lctrons intract with ach othr. Th intractions chang th spin nrgis to giv nw lins. Undrstanding th intraction allows you to rval structural information such as connctivity. Th couplings btwn two magntic dipols can b writtn as Ecoupling µ µ Couplings btwn unpaird lctrons (a,b,c ) and nuclar spins (i,j,k ) ar: nuclar-nuclar ( J ) coupling N N lctron-lctron ( fin ) coupling µ ( a) µ ( b) = γγsa Sb lctron-nuclar ( hyprfin ) coupling µ µ = γγ µ i µ j = γ i γ j I I N N i j a i i S I N N a i 5.33 Lctur Nots: Magntic Rsonanc Spctroscopy Pag 7

8 NMR Intraction btwn protons: () () E = J m i m j m = ± coupling ij I I J: nuclar spin coupling constant; typically -0 Hz. I Total nuclar intraction nrgy with spins: E = E + E int nuclar Zman J coupling () () () () = B γ i m i σ + J m i m j app N I i ij I I i i j ESR Dominant ffct: hyprfin coupling coupling bj s E = a m b m j I a: hyprfin coupling constant; typically -00 MHz Total intraction nrgy of lctron and nuclar spins: E = E + E + E + E int lctron Zman hyprfin coupling nuclar Zman J coupling = B γ b m b σ + a m b m i + s s app b bj I b i, j,... a,b... Sum of lctron Zman ffct and hyprfin intraction. Nuclar Zman nrgy and J coupling is rlativly small in comparison. ESR Exampl: H atom proton and lctron Eint = Elctron Zman + Ehyprfin coupling + Enuclar Zman = Bγ m + a m m s s With no coupling: two stats, with splitting as bfor. I 5.33 Lctur Nots: Magntic Rsonanc Spctroscopy Pag 8

9 With coupling: ms =± ; mi =± and mm s I = ± 4 Eint ms +/ +/ mi +/ / /4 /4 E ms= mi=0 / / / +/ 0 B B B At zro fild: two stats E=+ a and 4 E= a 4 With incrasing fild, dgnrat stats split. By swping fild with constant rsonanc frquncy E=hν, w s rsonancs at two filds, B and B. E a B = γ B and E+ a = γ so that th fild splitting givs th hyprfin coupling: a = γ ( B B ) Altrnativly, you could stay in a fixd fild B and swp th rf frquncy E, and you would obsrv two rsonancs: E/ =γ B± a Again th frquncy splitting givs th hyprfin coupling Lctur Nots: Magntic Rsonanc Spctroscopy Pag 9

10 Splitting Pattrns For multipl nucli coupld to an unpaird lctron, w can xpct ach hyprfin intraction to split th rmaining transition into a pair of paks split by a. Th ovrall spctrum can b prdictd diagrammatically by a pattrn of splittings in which on lctron rsonanc is squntially split in frquncy by ach hyprfin coupling intraction: Two inquivalnt couplings a and a : Two quivalnt couplings (a = a ): 5.33 Lctur Nots: Magntic Rsonanc Spctroscopy Pag 0