Chem8028(1314)  Spin Dynamics: Spin Interactions


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1 Chem8028(1314)  Spin Dynamics: Spin Interactions Malcolm Levitt see also IK m106 1
2 Nuclear spin interactions (diamagnetic materials) 2
3 Chemical Shift 3
4 Direct dipoledipole coupling 4
5 Jcoupling 5
6 Nuclear quadrupole coupling (1) spin > 1/2 only 6
7 Spin System spinspin coupling chemical shift 7
8 Spin System Ensemble (perfectly rigid solid) all spin interactions are coherent (the same all the time, and the same for each member) 8
9 Spin System Ensemble (liquid) most spin interactions are incoherent (fluctuating, and different for each member) 9
10 Spin System Ensemble (liquid) most spin interactions are incoherent (fluctuating, and different for each member) 9
11 Spin Interactions coherent same for all spin systems in the ensemble NMR peak frequencies incoherent fluctuating in time a snapshot would show differences between different spin systems in the ensemble linewidths and relaxation 10
12 Coherent interactions Solids: chemical shifts (both isotropic and CSA) DDcouplings Jcouplings Isotropic liquids: isotropic chemical shifts Jcouplings 11
13 Coherent spin interactions (isotropic liquids) 12
14 Coherent spin interactions (solids) 13
15 Isotropic and anisotropic spin interactions Isotropic: independent of molecular orientation with respect to the magnetic field Anisotropic: dependent on molecular orientation with respect to the magnetic field often represented by a tensor (drawn as a 3D ellipsoid) 14
16 Tensors Spinach representation 15
17 Tensors principal axes Spinach representation 15
18 Jcoupling 16
19 Jcoupling Hamiltonian: homonuclear case H J k = 2π J k I I k I I k = I x I kx + I y I ky + I z I kz
20 Jcoupling Hamiltonian: heteronuclear case H J k = 2π J k I z I kz
21 Direct dipoledipole coupling 19
22 Direct dipoledipole coupling The magnetic field generated by one spin influences its neighbour 20
23 Dipoledipole coupling constant b k = ( µ 0 4π )!γ γ k r k 3 (in rad s 1 ) Note: The DD coupling in Hz is b k / 2π cube of the distance between the spins Examples: Two protons at 3Å separation: DD coupling = 4.5 khz Two 13 C s at 1.5Å separation: DD coupling =2.2 khz Two 13 C s at 5Å separation: DD coupling = 61 Hz Two 13 C s at 8Å separation: DD coupling = 15 Hz 21
24 DD Hamiltonian: homonuclear b k = ( µ 0 4π )!γ γ k r k 3 H k DD = d k ( 3I z I kz I I ) k d k = b k 1 ( 2 3cos2 θ k 1) 22
25 DD Hamiltonian: heteronuclear b k = ( µ 0 4π )!γ γ k r k 3 H k DD = d k 2I z I kz d k = b k 1 ( 2 3cos2 θ k 1) 23
26 Principal axes of DD coupling The principal zaxis of the DD coupling between two spins is along the internuclear vector 1 2 x y P 12 z 24
27 Chemical Shift 25
28 Mechanism of chemical shift 26
29 CSA 27
30 CSA tensors field field Chemical shift depends on molecular orientation with respect to the field CSA tensor 28
31 Oriented 15 Nlabelled spider silk field vary angle between the silk fibre and the magnetic field Zhao and Asakura (2001) 29
32 Chemical shift tensor induced field at spin (a threecomponent vector) δ = B induced = δ B applied δ xx δ yx δ zx δ xy δ yy δ zy δ xz δ yz δ zz applied field (a threecomponent vector) CS tensor The induced field is not (usually) parallel to the applied field. 30
33 Principal axes There are three special directions in which the induced field is parallel to the applied field. These are called the principal axes of the chemical shift tensor (denoted (X,Y,Z) ). The principal axes are in general different for different chemical sites. Z Y P P X Y principal axes shown as the axes of 3D ellipsoids Z X 31
34 Principal values When the applied field is along a principal axis, the induced field is proportional to the applied field, multiplied by a number, called the principal value of the chemical shift tensor X P Z Y B induced B induced B induced ( along X ) = δ XX ( along Y ) = δ YY ( along Z) = δ ZZ B applied B applied B applied ( along X ) ( along Y ) ( along Z) chemical shift principal values of spin 32
35 Isotropic chemical shift The mean of the three principal chemical shift values is called the isotropic chemical shift. This is the chemical shift reported in solution NMR. P Z δ iso = 1 ( δ 3 XX + δ YY + δ ) ZZ X Y isotropic chemical shift of spin 33
36 Assignment of the principal axes By convention, the principal axes are assigned so that: The Zaxis is the one for which the principal value is furthest from the isotropic shift; P Z The Yaxis is the one for which the principal value is the closest to the isotropic shift; The Xaxis is the other one X Y 34
37 Chemical shift anisotropy The difference between the isotropic shift and the principal value which is furthest from it is called the chemical shift anisotropy (CSA) Z δ aniso = δ ZZ δ iso P X Y 35
38 Biaxiality The difference between the x and y principal values is usually quantified by the biaxiality* η: Z η = δ YY δ XX δ aniso X P Y If η is zero, the x and y principal values are the same, and the CSA tensor is said to be uniaxial *Sometimes called (misleadingly) the asymmetry 36
39 Static spectrum (powder pattern) field δ aniso δ δ XX δ YY δ iso δ ZZ 37
40 Powder patterns δ aniso < 0 δ aniso > 0 uniaxial CSA tensor η = 0 0 < η <1 biaxial CSA tensor δ η =1 38
41 Shielding convention Bad news: In many papers and books, CSA is specified using a shielding parameter σ, with opposite sign to the (deshielding) chemical shift δ σ aniso = δ aniso Even worse news: some authors write σ when they mean δ! 39
42 Nuclear quadrupole coupling (1) spin > 1/2 only 40
43 Quadrupolar spin Hamiltonian Full form of spin interaction: 1st order term (usually sufficient for 2 H): For other nuclei, the 2nd order term is usually needed as well. 41
44 I=3/2: Nuclear quadrupole coupling (2) 1storder: shifts spin energy levels according to M 2 2ndorder: shifts energy levels in a more complicated way 3/2 > 1/2 > +1/2 > +3/2 > 1storder Q 2ndorder Q 42
45 Relative Magnitudes of Interactions (before motional averaging) spins > 1/2 only 43
46 Interactions after motional averaging spins > 1/2 only 44
47 Motionallyaveraged spin interactions: isotropic liquids 45
48 Motionallyaveraged spin interactions: spins1/2 in solids 46
49 Motionallyaveraged spin interactions: spins > 1/2 in solids 47
50 Summary Only spins >1/2 experience electric spin interactions (electric quadrupole interaction) The chemical shift is a threebody interaction between the applied field, the electrons, and the nuclei The CSA is described by three principal values and three principal axes The DD coupling is directly proportional to the inverse cube of the distance between the nuclei The DD principal axis system is aligned along the internuclear vector In high magnetic field, all interactions may be replaced by their secular (1storder) parts, except the large quadrupole coupling, where the 2nd or higher terms must be included as well. 48
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