NMR spectroscopy Matti Hotokka Physical Chemistry Åbo Akademi University
Angular momentum Quantum numbers L and m (general case) The vector precesses
Nuclear spin The quantum numbers are I and m Quantum number I =, ½,, 3/,,... The most important nuclei Proton, H, I = ½ Carbon-3, 3 C, I = ½ Value of angular momentum, z component I m z
Rule of thumb Atomic number Z Atomic mass A Spin quantum number I Even Even Zero Even or odd Odd Half integer, ½, 3/, 5/,... Odd Even Integer,,, 3,...
Hydrogen Frequency MHz Spin I Abundance % H / H 3 H 5.3 76. /.5-4
Periodic table 7 Li 38.8 3/ 93 9 e 4. 3/ 3. 3/ 8.4 3 C 5. /. 4 N 7. 99.6 7 O 3.6 5/.37 9 F 94. / Ne 7.8 3/.6 3 Na 6.5 3/ 5 Mg 6. 5/. 7 Al 6. 5/ 9 Si 9.9 / 4.7 3 P 4.5 / 33 S 7.7 3/.8 35 Cl 9.8 3/ 75.5 4 Ar 39 K 4.7 3/ 93 43 Ca 6.7 7/.4 45 Sc 4.3 7/ 47 Ti 5.6 5/ 7.3 5 V 6.3 7/ 99.8 53 Cr 5.7 3/ 9.6 55 Mn 4.7 5/ 57 Fe 3. /. 59 Co 3.7 7/ 6 Ni 8.9 3/. 63 Cu 6.5 3/ 69. 68 Zn 6.3 5/ 4. 69 Ga 4. 3/ 6.4 73 Ge 3.5 9/ 7.8 75 As 7. 3/ 77 Se 9. / 7.6 79 r 5. 3/ 5.5 83 Kr 3.8 9/.6 85 Rb 9.7 5/ 7 87 Sr 4.3 9/ 7. 89 Y 4.9 / 9 Zr 9.3 5/. 93 Nb 4.5 9/ 95 Mo 6.5 5/ 5.7 98 Tc Ru 5. 5/ 7. 3 Rh 3. / 5 Pd 4.6 5/. 7 Ag 4. / 5.8 Cd. /.8 5 In.9 9/ 95.7 9 Sn 37.3 / 8.6 Sb 3.9 5/ 57. 5 Te 3.6 / 7. 7 I. 5/ 9 Xe 7.7 / 6.4 33 Cs 3. 7/ 37 a. 3/.3 38 La 3. 5.9 77 Hf 4. 7/ 8.5 8 Ta. 7/ 99.9 83 W 4. / 4.4 85 Re.5 5/ 37.5 89 Os 7.8 3/ 6. 89 Ir.9 3/ 6.6 95 Pt.5 / 33.8 97 Au.7 3/ 99 Hg 9.9 / 6.8 5 Tl 57.8 / 7.5 7 Pb.9 /.6 9 i 6. 9/ 9 Po At Rn www.chem.tamu.edu/services/nmr/periodic/index.shtml
Magnetic moment For every nuclear spin angular momentum there is a related magnetic moment Its magnitude can be written in two ways I Component along the z-axis m z gn I N /
Constants for common nmr nuclei
Energy levels The Zeemann effect gives an energy in a magnetic field E The magnetic moment and thus the energy is quantitized E z m Selection rule m=± E m
Transition energy Proton, I=/, external field =.49 T E( ) (.675.988 6 J 8 HzT ) 6.66 34 J Hz (.49T ) E( ).988 6 J E E( 6 ) E( ) 3.975 J 6. MHz
A spin ½ particle E E -/ E E +/
A spin particle N.. Spin ½ particles are assumed if not otherwise stated E m=- E - E E E m= E + m=+
One nucleus z The x- and y-components of the magnetic moment are blotted out because the vector precesses.
Several nuclei
Populations g E N N m kt g N N e N N / x e x kt g N N N N N A N N N kt g N N N N A kt g N N N N A kt g N N N n N N A Proton,.49 T, 3 K => n/n A = -5
Larmor frequency A nucleus without electrons (fictious case) The nucleus feels the field Transition energy E Larmor The nucleus is actually shielded It feels a lower field
Chemical shift The electron cloud shields the nucleus The nucleus feels a lower field ( ) Transition energy E Obs Displacement from Larmor frequency E E Chemical shift Larmor Obs E Larmor
Practical scale E Ref Ref E Obs E Larmor Obs ( 6 Ref Obs) [Hz] The more electrons the better shielding and lower.
Practical scale N.. Fictious spectrum, no coupling Ar-H CH CH CH 3 TMS [ppm] 7 6 5 4 3
Practical scale Negative shielding No shielding Positive shielding Downfield, negative σ Paramagnetic shielding [T] Upfield, positive σ Diamagnetic shielding
Practical scale The scale is δ [ppm] for all nuclei C 6 H 6 CHCl 3 H C H 4 OC(CH 3 ) C H 6 TMS ML 5 H 7. 5.5..9-5 to -3 3 C CS OC(Me)OH C 6 H 6 CHCl 3 OC(CH 3 ) TMS 9.8 78.3 8.6 77. 3.4 5 N NO - CH 3 NO RSCN RNC NH4 + 5-9 - -355
Classical theory Definitions r
Classical theory oth the nuclei and the electrons precess under an external field. Precession frequency of electrons in their orbit e mec Electrons moving in their orbit create a current I e e 4m c e
Classical theory Current gives rise to a magnetic field This field has the mgnitude ' I sin cr e sin m rc It lies opposite to the external field The total field felt by the nucleus is the external field minus the field generated by the electrons. ' e
Classical theory The total field is obtained by integrating Assume a spherical charge distribution (r) ) ( 3 4 ) ( )sin (cos 3 ' r rdr m c e r rdr d m c e e e ) ( 3 4 dr r r m c e e
Ramsey s theory Consider a free molecule in space No field, Hamiltonian H The unperturbed Shrödinger equation H k E k where k = is ground state, k > electronically excited state k
Ramsey s theory Perturbation is the magnetic field The perturbed Hamiltonian z z y y x x M M M H H Pert xyz xyz z z y y x x H chemistry! polarization diamagnetic frequency Larmor induced dipole gives dipole permanent Pert H M
Ramsey s theory Solve using perturbation theory Zeroth order First order Second order E E E E E Pert H E E E E E m m Pert m m Pert E E H H E
Ramsey s theory Chemical shift asic equation Interaction energy Diamagnetic and paramagnetic term E E E p d E E
Ramsey s theory In a detailed treatment σ is a tensor Explicit expression for perturbation gives for nucleus A 3 8 N el i ia ia i ia i e d r r r r r m e 3 3 8 m m m N i i N i ia ia m m N i ia ia m m N i i e p E E L r L E E r L L m e el el el el
Chemical shifts Example: benzene Current I Induced Paramagnetic Diamagnetic
Coupling Consider the field felt by nucleus X Extra electrons A Extra electrons X
Coupling E Energy levels of nucleus X Nucleus A is Nucleus A is Nucleus A is Nucleus A is Without field With field With coupling
Coupling scheme Chemical shift Coupling TMS J [ppm] 7 6 5 4 3 The coupling constant J is given in Hz
Coupling patterns Two spin ½ nuclei, e.g., protons J
Coupling patterns Two spin nuclei, e.g., deuterium J J J J -,- -, -,+ +, +,+,-,,+ +,-
Satellites Different elements do not interact. However Deuterium of heavy water solvent may contaminate the sample, e.g., acetone J(HD)=.4 Hz H 3 C-C(O)-CH 3 D HC-C(O)-CD 3 (oth H spectra!)
Spin vectors A quantum mechanical description of the nuclear spin
Angular momentum operator The operator for a general angular momentum Pauli spinors Ŝ z y x S i i S S
Square of S 4 3 4 4 4 4 4 4 i i S S S S z y x
Eigenvalues z z S S 4 3 4 3 S S
Two particles Wavefunction Note the order 3 4 () () () () () () () ()
Hamiltonian The Hamilton operator for high-resolution nmr is Note: the nuclear angular momentum is denoted by I, H H I I J I g H N i j j i j i ij N i zi i N i
Matrix elements Two spin ½ nuclei zero. are The rest 4 4 3 3 3 3 4 4 I I I I I I I I I I I I
How to calculate () () () () 4 3 k ij i k ij i k j i i i k j i i i I I I I () () () () () () () () z z z z I I I I
Hamilton matrix J J J J J J L L L L 4 4 4 4
Solutions 4 4 4 3 4 4 L L L J C J E C J E C J E J E 4 4 3 3 ' i i i i i c c c c 4 3 c Q Q Q c Q Q Q c c J Q L C
Intensities The intensity is ' ' i j i j I P ) /( ) ( ) /( ) ( ) /( ) ( ) /( ) ( 34 4 3 Q Q P Q Q P Q Q P Q Q P
Example 4 J AX 3 J 4 C J Midpoint 3 A J C ε Midpoint ε
Classification of spectra AX AM A δ δ δ
Classification of spectra A A spectrum schematically A J J J
Classification of spectra A A spectrum schematically A J J J ββα βαβ αββ βαα βββ 5 5 6 6 7 7 3 3 αβα ααβ 4 8 8 ααα 5 6 7 8 3 4
Klassificering av spektra Ett AMX spektrum schematiskt X M A J AX J AM J AX J MX J MX J MX J MX J AM J AM
Relaxation Excess of α spins Macroscopic magnetic moment Relaxation M n Ng N kt Equilibrium excess is n magnetic moment M Excess in a disturbed system n and M N Ng N N kt N N dn dt n n dm dt M M
Perturbations Longitudinal The magnetization M z is affected Alters the ratio of α and β nuclei Energy is absorbed Transversal The magnetizations Mx and My are affected The number of α and β nuclei unchanged Precession phase is affected Energy is not absorbed
Relaxation times After longitudinal perturbation Longitudinal or spin-lattice relaxation time τ After a transversal perturbation Transversal or phase coherence relaxation time τ The relaxation times are seconds or even minutes The relaxation times are additive m m i i i i
Longitudinal perturbation
Transversal perturbation
Relaxation mechanisms A few of the factors that cause relaxation Inhomogeneties in the field rownian motion Dipol-dipol interactions The electric quadrupole moment Spin-rotation interactions Anisotropy of the coupling tensor J
rownian motions Local inhomogeneties in magnetic field There is a characteristic correlation time τ C All local oscillation frequencies from to / τ C are present The perturbing fieldstrength is K( C ) Here ν C is the frequency of the perturbation Time τ C is typically - s From Debye s theory C C C 4a 3k C 3 T
Dipole-dipole interaction Affects both τ and τ 6 4 4 4 C C C C DD r K 6 4 4 5 3 C C C C C DD r K
Electric quadrupole interaction I 3 Q d V C I dz Q Q I
loch s equations asis for all relaxation treatments Macroscopic magnetic moment Total external field 3 M i i cos( t)î sin( t)ĵ k Here is the intense static field and is the radio frequency at ν
loch s equations Relaxation after a perturbation dm z dt dm x dt dmy dt M M M z x y M
loch s equations Equation of motion dm dt M î M x cos( t) ĵ M y sin( t) k M z
loch s equations loch s equations From these M x, M y and M z can be solved ) sin( ) sin( ) cos( ) sin( M t M M dt dm t M M M dt dm t M t M M M dt dm x z y y z y x x y x z z
loch s equations Magnetic susceptibility χ Value in the xy plane depends on the radio signal Value in z direction, χ, is uninteresting because the field is static Susceptibility defined via M
loch s equations Consider only susceptibility in xy plane The x and y are usually given as a complex number Assuming a non-saturated system ' " i L L L ) "( ) '(
Absorption Susceptibility Χ gives dispersion Χ gives absorption χ τ Δν χ
Absorption Assuming a radio field in x direction, x Absorbed energy sin( t) Absorbed effect P E M x x de dt ( ) At resonance χ may be much larger than χ 8 4 4 " L
Macroscopic sample Only an extremely small net magnetisation M z is observed. The radio pulse affects all nuclei and therefore the macroscopic magnetisation M z will also behave in the same way. Only the macroscopic magnetisation can be observed.
Pulse types
Normal FT-NMR spectrum 9 9 9 Accumulate Accumulate Accumulate
ATP pulse sequence Attached Proton Test Run 3 C FT-NMR spectrum H decoupling Decoupling off 3 C
ATP pulse sequence Up: Normal proton decoupled spectrum. Middle: = /J, only signals from the quaternary carbons will show. Down: = /J, even/odd separation. Lines pointing up refer to CH carbons, lines pointing down refer to CH and CH 3 groups Jokisaari, Lecture notes, Oulu
INEPT Insensitive Nuclei Enhanced by Polarization Transfer Transfer intensity from the strong absorbers (e.g., H) to unsensitive nuclei (e.g., 3 C, 4 N, 9 Si).
INEPT (9 o ) x 8 o (9 o ) y H: X 3 C: A Prepare transfer
Spin-echo experiment Pulse sequence 9 8 Wait time t Wait time t Read the spin echo
Spin-echo experiment y y b a x a x Sväng. Alla spinn precesserar koherent i xy planet. b Vänta. Kärnorna förlorar koherens. Sväng 8. Kärnorna återförs till koherens.
Spinnekoexperiment