Spin Interactions Giuseppe Pileio 24/10/2006
Magnetic moment µ = " I ˆ µ = " h I(I +1) " = g# h
Spin interactions overview
Zeeman Interaction
Zeeman interaction Interaction with the static magnetic field B 0 Zeeman E Z = " r µ # B r = "µ z # B 0 H ˆ Z = "$B 0 I ˆ z = % 0 I ˆ z = "$B 0 hm = % 0 hm
Chemical Shift Interaction
Magnetic shielding magnetic screening arises from orbital electronic currents induced by the external field; Static magnetic field Secondary magnetic field due to these currents B 0 "# B 0 Total field on nucleus B = B 0 (1"# ) magnetic screening constant
tomic decomposition " = " d +" p + $ " B +", ring +" other B# B 0 tomic electron density " d Diamagnetic contribution due to Langevin-type currents on atom itself nucleus Negative contribution that nucleus would have if its electrons were free to circulate about the direction of the field Lamb formula for isolated atoms " d = e2 3mc 2 $ i #1 r i verage distance of the electron i from the nucles
tomic decomposition " = " d +" p + $ " B +", ring +" other B# " p Paramagnetic contribution due to paramagnetic-type currents on atom B 0 Positive contribution that arises from the fact that electrons on atom are not free to circulate about the field (because of bonds or, more generally, because of other atoms)
tomic decomposition " = " d +" p + $ " B +", ring +" other B# B 0 " B contribution to due to currents on atom B B In the dipole approximation only local anisotropy of local susceptibility on atom B is important
tomic decomposition " = " d +" p + $ " B +", ring +" other B# ", ring contribution due to delocalized ring currents B 0 I
tomic decomposition " = " d +" p + $ " B +", ring +" other B# contribution due to other effect like: " other 1. Solvent properties 2. Other molecules in the proximity 3. etc
Shielding tensor For molecules with no particular symmetry both the screening constant and the magnetic moment will not be parallel to the external field and will depend on orientation # % % $ " xx " yx " zx " xy " yy " zy " xz " yz " zz *#" x'x' 0 0,% &, 0 " y'y' 0 diag (, % $ 0 0 " z'z' ) + (,# x 1 ' y 1 ' z 1 '& ',% ( x 2 ' y 2 ' z 2 ', % ( - $ x 3 ' y 3 ' z 3 '' & ( ( ' Principal values Principal axes
Isotropic chemical shift Chemical shift tensor Chemical shift " = #$ Screening z Principal axes # % % $ " xx " yx " zx " xy " yy " zy " xz " yz " zz & ( ( ' 2 x 1 " iso = 1 3 (" x'x' +" y'y' +" z'z' ) Isotropic Chemical shift
Liquid State spectra: I The rapid isotropic overall rotational motion averages out all the anisotropic interactions and the spectrum consists of well resolved, sharp lines. typical range of 1 H signals
Liquid State spectra: II The rapid isotropic overall rotational motion averages out all the anisotropic interactions and the spectrum consists of well resolved, sharp lines. typical range of 13 C signals
Liquid State spectra: III The rapid isotropic overall rotational motion averages out all the anisotropic interactions and the spectrum consists of well resolved, sharp lines.
Chemical shift anisotropy " aniso = " z'z' # " iso nisotropy of chemical shift Principal axes assignment z : the one for which the principal value is the furthest from δ iso y : the one for which the principal value is the closest to δ iso x : the other one " z'z' # " iso $ " y'y' # " iso $ " x'x' # " iso symmetry parameter " = # y'y' $ # x'x' # aniso
Static Solid State spectra: I Powder Pattern Each crystallite in the solid powder contribute to a broad spectral lineshape since they all have different Ω MR Euler angles
Static Solid State spectra: II " aniso < 0 " iso " x'x' " y'y' " z'z' " aniso " aniso = " z'z' # " iso " = # y'y' $ # x'x' # aniso
Static Solid State spectra: III " aniso < 0 " = 0 " = 0.5 " =1 " aniso = " z'z' # " iso " = # y'y' $ # x'x' # aniso
CS Hamiltonian B 0 Interaction with the static field EZ = " r µ # B r = "µ z # B 0 H ˆ Z = "$B 0 I ˆ z = % 0 I ˆ z H ˆ Z = "$B 0 hm = % 0 hm B ind #" XX = " # B % 0 " YX % $ " ZX H ˆ CS = " r µ # B ind " ZY " YY " ZY " XZ " YZ " ZZ & # 0 ( % ) 0 ( % ' $ B 0 & ( ( = ' # % " YZ B 0 % " $ ZZ B 0 " XZ B 0 = "$ % XZ B 0 I ˆ x " $ % YZ B 0 I ˆ y " $ % Secular approximation ZZ B 0 ˆ & ( ( ' I z
Spin-spin coupling Interaction
Scalar coupling Interaction between spins mediated by electrons B 0 B Electrons have spin 1/2 with γ H /γ e 660 thus they interact through a dipole-dipole mechanism with nuclei that have spin. This interaction is purely anisotropic and thus averaged to 0 by molecular motions except in the vicinity of the nucleus ( 10-14 m)
Fermi contact contribute.in the vicinity of the nucleus, in facts, the dipole approximation no longer holds and the dipolar interactions is better described by an isotropic interaction known as Fermi contact interaction: electron magnetogyric ratio (<0) 1 if r e,n = 0 0 otherwise nuclear spin operator J FC = " 2 3 µ 0# n # e $ re,n I ˆ % S ˆ Electron Spin operator For γ n >0: stabilized if e and n are antiparallel For γ n <0: stabilized if e and n are parallel vanishes if the wavefunction has no s character (p,d,f wavefunction have no amplitude at r e,n = 0)
Other contributions The Fermi contact contribution although is commonly the most important part of the interaction is not the only one acting, usually we can write: Fermi contact diamagnetic spin-orbit paramagnetic spin-orbit J = J FC + J DSO + J PSO + J SD spin-dipole FC: couples nuclear spin with the orbital motion of the electrons at r=0; DSO, PSO: couples nuclear spin with the orbital motion of the electrons at r SD: couples nuclear spin with electron spin (dipole approx)
Pauli principle The total wavefunction that describe an orbital must be antisymmetric with respect to the exchange of two electrons (i.e. fermions, it must be symmetric for bosons) tomic or molecular wavefunction Generally speaking the two electrons that occupy one orbital should have unpaired spin (antiparallel spin configuration)
Scalar coupling mechanism I B 0 Pauli B Fermi Fermi Positive J coupling constant
Scalar coupling mechanism II Hund B 0 Pauli C Pauli B Fermi Fermi negative J coupling constant
J tensor s for the chemical shift even the J interaction is described by a second rank tensor " $ $ # J xx J yx J zx J xy J yy J zy J xz J yz J zz % ' ' & J iso J aniso isotropic part anisotropic part J iso = 1 3 (J + J xx yy + J zz ) J aniso = J zz " J iso " = J # J yy xx J aniso
Isotropic J: properties The strength depends strongly on the orbital s character The coupling is note affected by external fields Magnitudes H 1 J HH 280Hz (but another mechanism makes it very larger ~GHz) 2 J HH -25/40Hz 3 J HH 2 - cos(θ) + 10 cos 2 (θ) 4,5 J HH 0.1/3Hz (in conugated systems) θ H 1 J CC 60Hz 1 J CH 5*%(s-character) X - CH 3 "125Hz X = CH 2 "150Hz X # CH " 250Hz
Spectra: I if a nucleus sees a spin I particle its signal will appear centered at the isotropic chemical shift value but split in 2I+1 signals Thus, if it sees a spin-1/2 nucleus will split into 2 signals Thus, if it sees n spin-1/2 nuclei will spit into n+1 signals X MX X 2 X 3
Spectra: II The intensities will follow the Pascal s triangle rule: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 X X 2 X 3 X 4 X 5 ν-ν /J 0
J Hamiltonian H ˆ J = 2" ˆ $ <k I # J # I ˆ k " $ $ $ # ˆ I x ˆ I y ˆ I z % ' ' ' & T " $ ( $ # J XX J YX J ZX J XY J YY J ZY J XZ J YZ J ZZ % " ˆ ' $ ( $ ˆ ' & $ # ˆ I x k I y k I z k % ' ' ' & Secular approximation ˆ H J = 2" * <k $ %& J ZZ I ˆ z I ˆ zk # 1 4 J XX Secular approximation + J if ν YY -ν k << J ( ) ˆ ( k I ) + I + I ˆ #k + I ˆ # ˆ ' ()
Dipolar Interaction
Dipolar coupling Interaction between spins through space B 0 B It represents the direct (through space) magnetic interactions between spins (magnetic moments).
Equations I r B r ( ) = µ 0 4" r (3( r µ 3 # r ˆ )# r ˆ $ r µ ) Magnetic field generated by spin in the space B 0 " r k E = "µ k # r B ( r ) E = " µ 0 4# r (3( r µ 3 $ r ˆ )( r µ k $ r ˆ ) " r µ $ r µ k )
Equations II E = " µ 0 4# r (3( r µ 3 $ r ˆ )( r µ k $ r ˆ ) " r µ $ r µ k ) µ = " hˆ I b H DD = " µ 0h 2 # # k 4$ r 3 (3( I ˆ % r ˆ )( I ˆ k % r ˆ ) " I ˆ % I ˆ k )
Equations III H DD = b (3( I ˆ " r ˆ )( I ˆ k " r ˆ ) # I ˆ " I ˆ k ) Z B 0 " r k H DD = b "[ (1# 3c 2 ($ ))( ˆ I z, ˆ I z,k ) # 1 4 (1# 3c2 ($ ))( ˆ I + ˆ I k# + ˆ I # ˆ I k+ ) Y # 3 2 (s($ )c($ )e #i% )( ˆ I + ˆ I z,k + ˆ I + ˆ I z,k ) X " x = r sin(" )sin(# ) # 3 2 (s($ )c($ )e i% )( I ˆ # I ˆ z,k + I ˆ # I ˆ z,k ) Secular approximation # 3 4 (s2 ($ )e #i2% )( I ˆ + I ˆ k+ ) y = r sin(" )cos(# ) z = r cos(" ) # 3 4 (s2 ($ )e i2% )( ˆ I # ˆ I k# )]
DD tensor I H DD = I ˆ " D " I ˆ d " b (# $% & 3e $ e % ) x y z,r & " #$ = ' 1 # = $ ( 0 # % $ e # = cos() z# ) #,$ = X,Y,Z " $ $ # d xx d yx d zx d xy d yy d zy d xz d yz d zz % ' ' & d iso d aniso d iso = 1 3 (d xx + d yy + d zz ) d aniso = d zz " d iso " = d yy # d xx d aniso
DD tensor II d iso d iso = 1 3 (d xx + d yy + d zz ) = b 3 (3 " 3(cos2 (# zx ) + cos 2 (# zy ) + cos 2 (# zz ))) = 0 No effect on liquid spectra d aniso d aniso = d zz " d iso = d zz " = d yy # d xx = 0 d aniso
Solid state spectra d = " µ 0 h2 # # k 4$ r 3 (1" 3cos 2 (% )) = b (1" 3cos 2 (% )) " = 0 0 # d = 2b 0 < " < 360 0 # $b < d < 2b " = 90 0 # d = $b Powder (i.e. all the possible θ )
Quadrupolar Interaction
Quadrupolar coupling Electric interaction between spins>1/2 and surrounding electric fields B 0 It represents the direct electric interactions between spins that have quadrupolar moment (I>1/2) and an electrical field gradient
Mechanism I nuclear charge distribution dipole C(r) = C 0 (r) + C 1 (r) + C 2 (r) +... monopole quadrupole V(r) = V 0 (r) + V 1 (r) + V 2 (r) +... E e0 = E e1 = E e2 = " C 0 (r)v 0 (r) dr " C 1 (r)v 1 (r) dr " C 2 (r)v 2 (r) dr Wigner-Eckart C k (r) = 0 if k > 2I Electrostatic force between n and e 0 (no nuclear electric dipole observed) Quadrupole coupling
Mechanism II E = Charge density # "( r )V( r )d$ electrical potential V( r ) = V(0) + #"V " + 1 #"$V "$ +... with V " = %V V "$ = % 2 V " 2! ",$ %" r=0 %"%$ r=0 Electrostatic force Quadrupolar interaction ' &( r )V( r ) d( = ' &( r )V(0) d( + # V " ' "&( r ) d( + 1 # V "$ ' "$&( r ) d( +... " 2! ",$ 0
Mechanism III E Q = 1 2! ' V "# ",# "#$( r ) % d& Q "# = ' [ 3"# $ r 2 % "# ]&( r ) d( ) ' "#&( r )d( = 1 3 Q + 1 "# 3 ' r 2 % "# &( r )d( E Q = 1 6 ( ",# V "# Q "# + 1 6 0 because Laplace s ' 2 V=0 rule ( ",# $ "# V "# r 2 %( r )d& 3"# $ r 2 % "# & 3 2 ( ˆ I " ˆ I # + ˆ I # ˆ I " ) $ % "# I 2 H Q = eq 6I(2I "1)h % #,$ V #$ ' 3 2 ( I ˆ # I ˆ $ + I ˆ $ I ˆ * # ) " & #$ I 2 () +,
Q Hamiltonian H ˆ 3eQ Q = 12I(2I "1)h [V (3ˆ I zz z2 " I ˆ # I ˆ ) + (V xx " V yy )( I ˆ x2 " I ˆ y2 ) +2V xy ( I ˆ x I ˆ y + I ˆ y I ˆ Secular x ) + 2V xz ( I ˆ x I ˆ z + I ˆ z I ˆ x ) + 2V yz ( I ˆ y I ˆ z + I ˆ z I ˆ y )] Quadrupolar splitting approximation ˆ H Q = " Q 3 [(3ˆ I z2 # ˆ I $ ˆ I ) + % Q ( ˆ I x2 # ˆ I y2 )] " Q = 3e 2 Qq 4I(2I #1)h eq is the Quadrupolar moment and q = V zz e
Q tensor Quadrupole moment Electric Field Gradient (EFG) Q = eq 4I(2I "1)h # % % $ V xx V yx V zx V xy V yy V zy V xz V yz V zz & ( ( ' V "# " V "a = V #" = $ 2 V( r ) $"$# # = 0 as $ 2 V = 0 (Laplace) Q aniso Q aniso = Q zz " = Q yy # Q xx Q aniso Q iso = 1 3 (Q xx + Q yy + Q zz ) = 0 No effect on liquid spectra Q iso
Solid state spectra
Interaction with an rf field
RF interaction B rf (t) = 2B 1 cos(" rf t + #) E RF = "µ # B rf (t) B rf (t) = B 1 cos(" rf t + # )ˆ i + B 1 cos($" rf t + # )ˆ i Cannot induce + B 1 cos(" transitions rf t + # ) ˆ + B 1 cos($" rf t + # ) ˆ
Interaction Strength H = H ext + H int H CS + H DD + H J + H Q