2 Basis Principles of FT NMR C Gerd Gemmecker 1999 Nuclei in magnetic fields Atomic nuclei are composed of nucleons, i.e., protons and neutrons. Each of these particles shows a property named"spin"(behaving like an angular momentum) that adds up to the total spin of the nucleus(which might be zero, due to pairwise cancellation). This spin interacts with an external magnetic field, comparable to a compass-needle in the Earth's magnetic field( for spin-12 nuclei) I(x, y) c in-/2 Spin- left: gyroscope model of nuclear spin. Right: possible orientations for spin-2 and spin-I nuclei in a homogeneous magnetic field, with an absolute value of n =I(+I). Quantisation of the component I_ results in an angle 0 of54.73(spin-72)or 45(Spin-1)with respect to the axis in a magnetic field, both I and Iz are quantized therefore the nuclear spin can only be orientated in (2 I 1) possible ways, with quantum number mI ranging from -I to I(-1,-I+,-1+2,.D) the most important nuclei in organic chemistry are the spin-2 isotopes'H,C,SN,F,and P(with different isotopic abur as spin-72 nuclei they can assume two states in a magnetic field, a(m=-72)and b(m =+72)
7 2 Basis Principles of FT NMR © Gerd Gemmecker, 1999 Nuclei in magnetic fields Atomic nuclei are composed of nucleons, i.e., protons and neutrons. Each of these particles shows a property named "spin" (behaving like an angular momentum) that adds up to the total spin of the nucleus (which might be zero, due to pairwise cancellation). This spin interacts with an external magnetic field, comparable to a compass-needle in the Earth's magnetic field (for spin-1 /2 nuclei). left: gyroscope model of nuclear spin. Right: possible orientations for spin-1 /2 and spin-1 nuclei in a homogeneous magnetic field, with an absolute value of |I| = I(I + 1). Quantization of the z component Iz results in an angle Q of 54.73° (spin-1 /2 ) or 45° (spin-1) with respect to the z axis. - in a magnetic field, both I and Iz are quantized - therefore the nuclear spin can only be orientated in (2 I + 1) possible ways, with quantum number mI ranging from -I to I (-I, -I+1, -I+2, … I) - the most important nuclei in organic chemistry are the spin-1 /2 isotopes 1H, 13C, 15N, 19F , and 31P (with different isotopic abundance) - as spin-1 /2 nuclei they can assume two states in a magnetic field, a (mI = - 1 /2) and b (mI = + 1 /2)
Usually the direction of the static magnetic field is chosen as z axis, and the magnetic quantum number mr often called mz, since it describes the size of the spins component in units of h/2T [2-1] resulting in a magnetic moment H =yI [2-2] Hz=myh/2 [2-3] y being the isotope-specific gyromagnetic / magnetogyric constant(ratio) The interaction energy of a spin state described by m, with a static magnetic field Bo in z direction can then be described as E=-HBo=uBo cos o [2-4] E=μ2YBoh2π For the two possible spin states of a spin-n nucleus(mz=+/2)the energies are 5yhBo E 0.5hB [2-6b The energy difference E Ei2-E-In=hv =o h/2T corresponds to the energy that can be absorbed or emitted by the system, described by the larmor frequency o E=YBoh2π The Larmor frequency can be understood as the precession frequency of the spins about the axis of the magnetic field Bo, caused by the magnetic force acting on them and trying(Iz is quantized!) to turn them completely into the field's direction (like a toy gyroscope"feeling"the pull of gravity)
8 Usually the direction of the static magnetic field is chosen as z axis, and the magnetic quantum number mI often called mz, since it describes the size of the spin's z component in units of h/2p: Iz = mz h/2p [2-1], resulting in a magnetic moment m: m = g I [2-2] mz = mz gh/2p [2-3] g being the isotope-specific gyromagnetic / magnetogyric constant (ratio). The interaction energy of a spin state described by mz with a static magnetic field B0 in z direction can then be described as: E = -mB0 = mB0 cos Q [2-4] E = mz g B0 h/2p [2-5] For the two possible spin states of a spin-1 /2 nucleus (mz = ± 1 /2) the energies are E1/2 = 05 2 0 . g p hB [2-6a] E-1/2 = - 05 2 0 . g p hB [2-6b] The energy difference DE = E1/2-E-1/2 = hn = w h/2p corresponds to the energy that can be absorbed or emitted by the system, described by the Larmor frequency w: DE = g B0 h/2p [2-7] w0 = gB0 [2-8] The Larmor frequency can be understood as the precession frequency of the spins about the axis of the magnetic field B0, caused by the magnetic force acting on them and trying (Iz is quantized!) to turn them completely into the field's direction (like a toy gyroscope "feeling" the pull of gravity)
9 According to eq. 2-8, this frequency depends only on the magnetic field strength Bo and the spin,s gyromagnetic ratio y. For a field strength of 11.7 T one finds the following resonance frequencies for the most important isotopes Isotope y(relative) resonance fre relative quency at 11.7T abundance sensitivity 500 MHZ 9998% 25 125 MHZ 1.1% 50 MHZ 0.37% 455 MHZ 100% 0.8 99 MHZ 4.7% P 40 203 MHZ 100% 0.07 also taking into account typical linewidths and relaxation rates △E=?hB The energy difference is proportional to the bo field strength B c How much energy can be absorbed by a large ensemble of spins (like our NMR sample)depends on the population difference between the a and B state(with equal population, rf irradiation same number of spins to absorb and emit energy: no net effect observable!) According to the boltzmann equation N(csexp I=exp 2kT △E N(B) For 2.35 T(=100 MHz) and 300 K one gets for H a population difference N(a)-N(B)of ca. 8. 10-6 i.e., less than /1000 of the total number of spins in the sample!
9 According to eq. 2-8, this frequency depends only on the magnetic field strength B0 and the spin's gyromagnetic ratio g. For a field strength of 11.7 T one finds the following resonance frequencies for the most important isotopes: Isotope g (relative) resonance frequency at 11.7 T natural abundance relative sensitivity* 1H 100 500 MHz 99.98 % 1 13C 25 125 MHz 1.1 % 10-5 15 N -10 50 MHz 0.37 % 10-7 19 F 94 455 MHz 100 % 0.8 29 Si -20 99 MHz 4.7 % 10-3 31 P 40 203 MHz 100 % 0.07 · also taking into account typical linewidths and relaxation rates The energy difference is proportional to the B0 field strength: How much energy can be absorbed by a large ensemble of spins (like our NMR sample) depends on the population difference between the a and b state (with equal population, rf irradiation causes the same number of spins to absorb and emit energy: no net effect observable!). According to the BOLTZMANN equation N N E kt hB kT ( ) ( ) exp exp a b g p = = D 0 2 [2-9] For 2.35 T (= 100 MHz) and 300 K one gets for 1H a population difference N(a)-N(b) of ca. 8.10-6, i.e., less than 1 /1000 % of the total number of spins in the sample!
Irradiation of an oscillating electromagnetic field Absorption Resonance condition rf frequency has to match Larmor frequency rf energy has to match energy difference between a and B level cos(at a linear oscillating field B, cos(ot) is identical B, co at+ sin(at) to the sum of two counter-rotating components, one being exactly in resonance with the precessing spins Rotating coordinate system Switching from the lab coordinate system to one rotating "on resonance"with the spins(and B1) about the axis results in both being static. Generally all vector descriptions, rf pulses etc are using this rotating coordinate system Now the effect of an rf irradiation(a pulse) on the macroscopic ()magnetization can be easily described(keeping in mind the gyroscopic nature of spins) Polarisation(M) Coherence(M The flip angle B of the rf pulse depends on its field strength Bi and duration t =yB p
10 Irradiation of an oscillating electromagnetic field Absorption Resonance condition: rf frequency has to match Larmor frequency = rf energy has to match energy difference between a and b level. a linear oscillating field B1 cos(wt) is identical to the sum of two counter-rotating components, one being exactly in resonance with the precessing spins. Rotating coordinate system Switching from the lab coordinate system to one rotating "on resonance" with the spins (and B1) about the z axis results in both being static. Generally all vector descriptions, rf pulses etc. are using this rotating coordinate system! Now the effect of an rf irradiation (a pulse) on the macroscopic (!) magnetization can be easily described (keeping in mind the gyroscopic nature of spins): The flip angle b of the rf pulse depends on its field strength B1 and duration t: b = gB1 tp [2-11] Polarisation (M ) z Coherence (-M ) y
Being composed of individual nuclear spins, a transverse (in the x y plane) macroscopic magnetization Mxv(coherence)starts precessing about the axis with the Larmor frequency (in lab coordinate system)under the influence of the static Bo field, e.g., after a 90 pulse M (t=M cos(ot)+ M sin(ot) [2-12] thus inducing a voltage / current in the receiver coil (which is of course fixed in the probehead in a transverse orientation the Fid(free induction decay) typical H FID of a complex compound(cyclic hexapeptide) t(sec) According to eq. [2-12], the FID can be described as sine or cosine function, depending of its phase Relaxation The excited state of coherence is driven back to BOLTZMANN equilibrium by two mechanisms 1)spin-spin relaxation(transverse relaxation) dMx/dt=-Mxv/T2 [2-13] corresponds to a loss of phase coherence magnetixation is spread uniformly across the x y plane decay of net transverse magnetization/FID(entropic effect) due to the bo field being not perfectly uniform for all spins( disturbance by the presence of other spins), this inherent T2 relaxation is increased by experimental inhomogeneities(bad shim! ) T2* 2)spin-lattice relaxation(longitudinal relaxation
11 Being composed of individual nuclear spins, a transverse (in the x,y plane) macroscopic magnetization Mx,y (coherence) starts precessing about the z axis with the Larmor frequency (in the lab coordinate system) under the influence of the static B0 field, e.g., after a 90ºx pulse: M -y (t) = M-y cos(wt) + Mx sin(wt) [2-12] thus inducing a voltage / current in the receiver coil (which is of course fixed in the probehead in a transverse orientation): the FID (free induction decay) typical 1H FID of a complex compound (cyclic hexapeptide) According to eq. [2-12], the FID can be described as sine or cosine function, depending of its phase. Relaxation The excited state of coherence is driven back to BOLTZMANN equilibrium by two mechanisms: 1) spin-spin relaxation (transverse relaxation) dMx,y / dt = -Mx,y / T2 [2-13] corresponds to a loss of phase coherence Þ magnetixation is spread uniformly across the x,y plane: decay of net transverse magnetization / FID (entropic effect) due to the B0 field being not perfectly uniform for all spins (disturbance by the presence of other spins); this inherent T2 relaxation is increased by experimental inhomogeneities (bad shim!): T2* 2) spin-lattice relaxation (longitudinal relaxation)