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)
M:-M0 dM,/dt T due to the excited state ' s dissipation of energy into the " lattice", i.e., other degrees of freedom (molecular vibrations, rotations etc. ) until the BOLTZMANN equilibrium is reached again(Mi) In the BOLtzMann equilibrium, all transverse magnetization must have also disappeared: T2< TI TI=T2 for "small "molecules; however, TI can also be much longer than T2(important for relaxation delay"between scans)! Measuring TI To avoid T2 relaxation, the system must be brought out of BoLTZMANN equilibrium without creating Mxy magnetization: a 180 pulse converts Mz into M-z, then T1 relaxation can occur during a defined period t. For detection of the signal, the remaining Mz/M-z component is turned into the x,y plane by a 90 pulse and the signal intensity measured 180°-τ-90°- acquisition Inversion-recovery experiment From integration of eq, [2-141, one gets zero signal intensity at time to=T, In 2 =0.7TI M2=(1-2exp(t/T)·M M2=99%M t=5T t=3T 0.5 0.0 t/T M2=0 t=T. In2 0.5
12 dMz / dt = - M M T z - 0 1 [2-14] due to the excited state's dissipation of energy into the "lattice", i.e., other degrees of freedom (molecular vibrations, rotations etc.), until the BOLTZMANN equilibrium is reached again (Mz). In the BOLTZMANN equilibrium, all transverse magnetization must have also disappeared: T2 £ T1 ; T1 = T2 for "small" molecules; however, T1 can also be much longer than T2 (important for "relaxation delay" between scans) ! Measuring T1: To avoid T2 relaxation, the system must be brought out of BOLTZMANN equilibrium without creating Mx,y magnetization: a 180º pulse converts Mz into M-z , then T1 relaxation can occur during a defined period t. For detection of the signal, the remaining Mz / M-z component is turned into the x,y plane by a 90º pulse and the signal intensity measured: 180°-t-90°-acquisition inversion-recovery experiment From integration of eq, [2-14], one gets zero signal intensity at time t0 = T1 ln 2 » 0.7 T1
T22 and linewidth Due to the characteristics of FT, the linewidth depends on the decay rate of the FID (for the linewidth at half-height) The FId being a composed of exponentially decaying sine and cosine signals, eq [2-12] should read M-y(t=M-ycos(ot)+ Mxsin(ot))exp(t/T2) Chemical shift Resonance frequencies of the same isotopes in different molecular surroundings differ by several ppm(parts per million). For resonance fr 100 MHz range these differences can be up to a few 1000 Hz. After creating a Mx,y coherence, each spin rotates with its own specific resonance frequency o, slightly different from the Bi transmitter(and receiver) frequency (o. In the rotating coordinate system, this corresponds to a rotation with an offset frequency Q2=@-0o time domain frequency domain
13 T2 and linewidth Due to the characteristics of FT, the linewidth depends on the decay rate of the FID: lw1/2 = 1 pT2 [2-15] (for the linewidth at half-height) The FID being a composed of exponentially decaying sine and cosine signals, eq. [2-12] should read M-y(t) = {M-ycos(wt) + Mx sin(wt)}exp(-t/T2 ) [2-16] Chemical Shift Resonance freuquencies of the same isotopes in different molecular surroundings differ by several ppm (parts per million). For resonance frequencies in the 100 MHz range these differences can be up to a few 1000 Hz. After creating a Mx,y coherence, each spin rotates with its own specific resonance frequency w, slightly different from the B1 transmitter (and receiver) frequency w0. In the rotating coordinate system, this corresponds to a rotation with an offset frequency W = w - w0 . time domain frequency domain
Sensitivity The signal induced in the receiver coil depends 1. on the size of the polarisation Mz to be converted into Mxy coherence by a 90 pulse, which is (from BOLTZMANN equation 2. and on the signal induced in the receiver coil at detection, depending on the magnetic moment of the nucleus detected yet and its precession frequency @=yet B o, in summa Yet unfortunately the noise also grows with the frequency, i.e., 1YBo The complete equation for sensitivity is thus S/N=nYexc det h2 bo/2(NS)2T-IT2 [2-17] n=number of nuclei in the sample, NS=number of scans acquired Conclusions: importance of detecting the nucleus with the highest y(i.e, H), important in heteronuclear H, X correlation experiments: inverse detection double sample concentration gives double sensitivity, but to get the same result from longer measuring time, one needs four times the number of scans! sensitivity should increase at lower temperatures (larger polarisation), but lowering the temperature usually also reduces T2, leading to a loss of s/n due to larger linewidths
14 Sensitivity The signal induced in the receiver coil depends 1. on the size of the polarisation Mz to be converted into Mx,y coherence by a 90° pulse, which is (from BOLTZMANN equation) µ g excB T 0 2. and on the signal induced in the receiver coil at detection, depending on the magnetic moment of the nucleus detected gdet and its precession frequency w = gdetB0 , in summa S µ gdet 2 B0 unfortunately the noise also grows with the frequency, i.e., gB0 . The complete equation for sensitivity is thus S /N = n gexc gdet 3 /2 B0 3 /2 (NS) 1 /2 T -1 T2 [2-17] n = number of nuclei in the sample, NS = number of scans acquired Conclusions: - importance of detecting the nucleus with the highest g (i.e., 1H), important in heteronuclear H,X correlation experiments: "inverse detection" - double sample concentration gives double sensitivity, but to get the same result from longer measuring time, one needs four times the number of scans! - sensitivity should increase at lower temperatures (larger polarisation), but lowering the temperature usually also reduces T2 , leading to a loss of S/N due to larger linewidths
Basic Fourier-Transform NMR Spectroscopy In FT NMR (also called pulse- FT NMR) the signal is generated by a(90)rf pulse and then picked up by the receiver coil as a decaying oscillation with the spins resonance frequency o Generating an audio frequency signa The rf signal (o) from the receiver coil is"mixed"with an rf reference frequency oo(usually the same used to drive the transmitter ), resulting in an"audio signal"with frequencies Q2=0-Oo. The phase"of the receiver(x or y) is set electronically by using the appropriate phase for the reference frequency @o(usually, a complex signal -1.e, the x and y component is detected simultaneously by splitting the primary rf signal into two mixing stages with 90 phase shifted refernce frequencies The Analog-Digital Converter(ADC) For storage and processing the audio frequency signal has to be digitized first. There are two critical parameters involved dynamic range describes how fine the amplitude resolution is that can be achieved; usually 12 bit or 16 bit. 16 bit corresponds to a resolution of 1: 2(since the FID amplitudes will go from up to 2), meaning that features of the Fid smaller than /32768 of the maximum amplitude will be lost! 2. time resolution corresponds to the minimum dwell time that is needed to digitize a single data point(by loading the voltage into a capacitor and comparing it to voltages within the chosen dynamic range ). This needs longer for higher dynamic range, limiting the range of offset frequencies that can be properly detected(the sweep width sw High resolution spectrometers: dynamic range 16 bit, time resolution ca 6 us(133,333 Hz SW) Solid state spectrometers dynamic range 9 bit, time resolution ca. I us( 1,000,000 Hz sw) NYQUIST frequency: the highest frequency that can be correctly detected from digitized data, corersponding to two(complex) data points per period. After FT, the spectral width will go from NyQuist freq to +NyQUIST freq. Signals with absolute offset frequencies o larger than the NYQUIST freq. will appear at wrong places in the spectrum(folding)
15 Basic Fourier-Transform NMR Spectroscopy In FT NMR (also called pulse-FT NMR) the signal is generated by a (90° ) rf pulse and then picked up by the receiver coil as a decaying oscillation with the spins resonance frequency w. Generating an audio frequency signal The rf signal (w) from the receiver coil is "mixed" with an rf reference frequency w0 (usually the same used to drive the transmitter), resulting in an "audio signal" with frequencies W = w - w0 . The "phase" of the receiver (x or y) is set electronically by using the appropriate phase for the reference frequency w0 (usually, a complex signal –i.e., the x and y component – is detected simultaneously by splitting the primary rf signal into two mixing stages with 90° phase shifted refernce frequencies. The Analog-Digital Converter (ADC) For storage and processing the audio frequency signal has to be digitized first. There are two critical parameters involved: 1. dynamic range describes how fine the amplitude resolution is that can be achieved; usually 12 bit or 16 bit. 16 bit corresponds to a resolution of 1:215 (since the FID amplitudes will go from -215 up to 215), meaning that features of the FID smaller than 1 /32768 of the maximum amplitude will be lost! 2. time resolution corresponds to the minimum dwell time that is needed to digitize a single data point (by loading the voltage into a capacitor and comparing it to voltages within the chosen dynamic range). This needs longer for higher dynamic range, limiting the range of offset frequencies that can be properly detected (the sweep width SW). High resolution spectrometers: dynamic range 16 bit, time resolution ca. 6 ms (= 133,333 Hz SW) Solid state spectrometers: dynamic range 9 bit, time resolution ca. 1 ms (= 1,000,000 Hz SW) NYQUIST frequency: the highest frequency that can be correctly detected from digitized data, corersponding to two (complex) data points per period. After FT, the spectral width will go from -NYQUIST freq. to +NYQUIST freq.. Signals with absolute offset frequencies w larger than the NYQUIST freq. will appear at wrong places in the spectrum (folding):
Usually electronic band pass filters are set automatically to suppress signals(and noise!) from far outside the chosen spectral range. Really sharp edges are only possible with digital signal processing Characteristic for folded signals out of phase(but: phase error varies!) due to the band pass filter, signal intensity decreases with offset(of the unfolded signal) beyond the spectral width Fourier Transformation All periodic functions(e.g, of time t) can be described as a sum of sine and cosine functions f(t)=a,/2+ a, cos(t)+ a,cos( 2t)+ a, cos( 3t)+ b,sin(t)+b sin(2t)+ basin(3t)+ The coefficients an and bn can be calculated by fourier transformation F(o)=f(exp(ior dr with expiot=cos(ot)+ i sin(ot) F(o)-the FOURIER transform of f(t)-is a complex function that can be divided into a real and maginary part Re(F(O)=∫ f(t)cose(otdt Im(F(o)=f(t)sin(ot)dt
16 Usually electronic band pass filters are set automatically to suppress signals (and noise!) from far outside the chosen spectral range. Really sharp edges are only possible with digital signal processing. Characteristic for folded signals: - out of phase (but: phase error varies!) - due to the band pass filter, signal intensity decreases with offset (of the unfolded signal) beyond the spectral width - Fourier Transformation All periodic functions (e.g., of time t) can be described as a sum of sine and cosine functions: f(t) = a0 / 2 + a1 cos(t) + a2 cos(2t) + a3 cos(3t) + ... + b1 sin(t) + b2 sin(2t) + b3 sin(3t) + ... The coefficients an and bn can be calculated by FOURIER transformation: F(w) = f (t)exp{iwt}dt -¥ +¥ ò with exp{iwt} = cos(wt) + i sin(wt) F(w) – the FOURIER transform of f(t) – is a complex function that can be divided into a real and an imaginary part: Re(F(w)) = õó -¥ +¥ f(t)cos(wt)dt Im(F(w)) = õó -¥ +¥ f(t)sin(wt)dt
