Basic Principles of NMR Bloch equations can be represented in the rotating frame,which is related to their form in the laboratory frame according to the following relationship: (xx)-MxyBa (1.20) in which Bem =B +/y and w is the angular frequency of the rotating fra same as orf ng山 ided the field Bis repa y the 0 rotatin It i app rth no ha to Equation(1.5).wh sB is the transformation from the lal boratory frame to the rotating ction1.3】 Since the hulk ma tization at equilibrium is independent of time based on the bloch equations it is not an observable NMR signal.The observable NMR signals are the time dependent transverse magnetization However at equilibrium the net oiections of the magnetization (nuclear angular moments)are zero due to precession of the nuclear spins.The simple solution to this is to bring the bulk magnetization to the xy plane by applying the B.electromagnetic field.The transverse magnetization generated by the B field (90 pulse) will not stay in the transverse plane indefinitely,instead it decays under the interaction of the static magnetic field Bo while precessing about the z axis and realigns along the magnetic field direction or the z axis of the laboratory frame (Figure 1.7).The decay of the transverse magnetization forms the observable NMR signals detected by the receiver in the xy plane in the rotating frame,which is called the free induction decay,or FID. The Bloch theory has its limitations in describing spin systems with nuclear interaction othe han chemi such as strong sca the Bloch re app to sy ms nteracting spin ss,it remains a ery useful tool to il simple NMR experiments a】 (b) Figure 1.7.Vector model repre tation of a one-pulse iment (a)The equilibrium bulk ma netiz ation shown by the shaded arrow is brought to the y axis by a 90 pulse along the x axis.(b)After the back toh state while about bserved by qu ure actecnon on
Basic Principles of NMR 11 Bloch equations can be represented in the rotating frame, which is related to their form in the laboratory frame according to the following relationship: dM dt rot = dM dt lab + M × ω = M × (γ B + ω) = M × γ Beff (1.20) in which Beff = B + ω/γ and ω is the angular frequency of the rotating frame and is the same as ωrf . The motion of magnetization in the rotating frame is the same as in the laboratory frame, provided the field B is replaced by the effective field Beff . When ω = −γ B = ω0, the effective field disappears, resulting in time-independent magnetization in the rotating frame. It is worth noting that −γ B is the Larmor frequency of the magnetization according to Equation (1.5), whereas γ B is the transformation from the laboratory frame to the rotating frame, −ωrf (section 1.3). Since the bulk magnetization at equilibrium is independent of time, based on the Bloch equations it is not an observable NMR signal. The observable NMR signals are the timedependent transverse magnetization. However, at equilibrium the net xy projections of the magnetization (nuclear angular moments) are zero due to precession of the nuclear spins. The simple solution to this is to bring the bulk magnetization to the xy plane by applying the B1 electromagnetic field. The transverse magnetization generated by the B1 field (90◦ pulse) will not stay in the transverse plane indefinitely, instead it decays under the interaction of the static magnetic field B0 while precessing about the z axis and realigns along the magnetic field direction or the z axis of the laboratory frame (Figure 1.7). The decay of the transverse magnetization forms the observable NMR signals detected by the receiver in the xy plane in the rotating frame, which is called the free induction decay, or FID. The Bloch theory has its limitations in describing spin systems with nuclear interactions other than chemical shift interactions, such as strong scalar coupling. In general, the Bloch equations are applied to systems of noninteracting spin 1 2 nuclei. Nevertheless, it remains a very useful tool to illustrate simple NMR experiments. x y B0 MZ (b) y x B0 M0 B1 (a) Figure 1.7. Vector model representation of a one-pulse experiment. (a) The equilibrium bulk magnetization shown by the shaded arrow is brought to the y axis by a 90◦ pulse along the x axis. (b) After the 90◦ pulse, the transverse magnetization decays back to the initial state while precessing about the z axis. An FID is observed by quadrature detection on the transverse plane
12 1.5.FOURIER TRANSFORMATION AND ITS APPLICATIONS IN NMR The FID is the sum of many tim ne domai signa with different frequ es,amplitude and phases d digitized ng the signal acqur ton peri do in d a (FID ned after its disc overy b by French mathe atician Jose The questions to be addressed in this section include: What are the properties of Fourier transformation us ful for NMR? ship between excitation bandwidth and pulse length in terms of 3 What adratur on and why is it necessary? 1.5.1.Fourier Transformation and Its Properties Useful for NMR el.1986: Fa=RuUo=Lroea 1.2) f()=FF=F)dv （022 Although there are many methods to perform Fourier transformation for the NMR data,the Cooley-lukey as max et o mear predi ina Ba 990.B rkhuuse 1985).F0 function (or s ,the Fourier pair is is the FI d spectrum witn f(t)=e(ioo-1/T)t (1.23) F(@)=Ft(f(t))= poefie-e dri(ox )+1/T) (a咖-ω)2+(1/T2） 120 which indicate that the spectrum is obtained by Fourier transformation of the FID and the fre SRowaecTSoag9gnaomia 1 The Fou rier transform of the sum of functions is the same as the sum of Fourier transforms of the functions: Ft(f(t)+g(t))=Ft(f(t))+Ftlg(t)) 129
12 Chapter 1 1.5. FOURIER TRANSFORMATION AND ITS APPLICATIONS IN NMR The FID is the sum of many time domain signals with different frequencies, amplitudes and phases. These time domain signals are detected and digitized during the signal acquisition period. In order to separate the individual signals and display them in terms of their frequencies (spectra), the time domain data (FID) are converted to frequency spectra by applying Fourier transformation, named after its discovery by French mathematician Joseph Fourier. The questions to be addressed in this section include: 1. What are the properties of Fourier transformation useful for NMR? 2. What is the relationship between excitation bandwidth and pulse length in terms of the Fourier transformation? 3. What is quadrature detection and why is it necessary? 1.5.1. Fourier Transformation and Its Properties Useful for NMR The Fourier transformation describes the connection between two functions with dependent variables such as time and frequency (ω = 2π/t), called a Fourier pair, by the relationship (Bracewell, 1986): F (ω) = Ft{f (t)} = ∞ −∞ f (t)e−iωt dt (1.21) f (t) = Ft{F (ω)} = ∞ −∞ F (ω)eiωt dν (1.22) Although there are many methods to perform Fourier transformation for the NMR data, the Cooley–Tukey fast Fourier Transformation algorithm is commonly used to obtain NMR spectra from FIDs combined with techniques such as maximum entropy (Sibisi et al., 1984; Mazzeo et al., 1989; Stern and Hoch, 1992) and linear prediction (Zhu and Bax, 1990; Barkhuusen et al., 1985). For NMR signals described as an exponential function (or sine and cosine pair) with a decay constant 1/T , the Fourier pair is the FID and spectrum with the forms of: f (t) = e(iω0−1/T )t (1.23) F (ω) = Ft{f (t)} = ∞ −∞ e(iω0−1/T )t e−iωtdt = i(ω0 − ω) + (1/T ) (ω0 − ω)2 + (1/T 2) (1.24) which indicate that the spectrum is obtained by Fourier transformation of the FID and the frequency signal has a Lorentzian line shape. Some important properties of Fourier transformation useful in NMR spectroscopy are discussed below (Harris and Stocker, 1998): 1. Linearity theorem. The Fourier transform of the sum of functions is the same as the sum of Fourier transforms of the functions: Ft{f (t) + g(t)} = Ft{f (t)} + Ft{g(t)} (1.25)
Basic Principles of NMR 名 This tells us that the sum of time domain data such as an FID will yield individual frequency signals after Fourier transformation. 2.Translation theorem.The Fourier transform of a function shifted by time t is equal to the product of the Fourier transform of the unshifted function and the factore: Fuf(+))=ei Ftif())=ei2Ftif()) (1.26 This states that a delay in the time function introduces a frequency-dependent phase shift in the frequency function.A delay in the acquisition of the FID will cause a first-order phase shift in the corresponding spectrum(frequency-dependent phase shift,see section 4.96).This also allows the phase of a spectrum to be adjusted after acquisition without altering the signal information contained in the time domain data f(r)(FID).The magnitude representation of the spectrum is unchanged because integration ofexp()over all possibleyields unity. Similarly, Ftlf(@-@n)=eo f(t) (1.2切 A frequency shift in a spectrum is equivalent to an oscillation in the time domain with the same frequency.This allows the spectral frequency to be calibrated after acquisition. 3. Convolution theorem.The Fourier transform of the convolution of functions f and f2 is equal to the product of the Fourier transforms of fi and f2: Ftn()*(t=FtIf(]*Ft[f(t) 1.28) in which the convolution of two functions is defined as the time integral over the product of one function and the other shifted function: 1.29) Based on this the to the a red F ctral peaks.known as 4.c na the orem.The Fourier transform of a function with which a scaling trans- formation is car ried out (t→t/c)is nual to the fourier tra nsform of the original function with the transformation cmultiplied by the absolute value of factor c: Ft(f(t/c))=IclF(co) (1.30) According to this theo m.the narrowing of the time domain function by a factor ofc causes 5 Parseval's theorem. IfOPdr-IF(wPdv-IFOPdo (131)
Basic Principles of NMR 13 This tells us that the sum of time domain data such as an FID will yield individual frequency signals after Fourier transformation. 2. Translation theorem. The Fourier transform of a function shifted by time τ is equal to the product of the Fourier transform of the unshifted function and the factor eiωτ : Ft{f (t + τ )} = eiωτ Ft{f (t)} = ei2πφFt{f (t)} (1.26) This states that a delay in the time function introduces a frequency-dependent phase shift in the frequency function. A delay in the acquisition of the FID will cause a first-order phase shift in the corresponding spectrum (frequency-dependent phase shift, see section 4.96). This also allows the phase of a spectrum to be adjusted after acquisition without altering the signal information contained in the time domain data f (t) (FID). The magnitude representation of the spectrum is unchanged because integration of | exp(iωτ )| over all possible ω yields unity. Similarly, Ft{f (ω − ω0)} = eiω0τ f (t) (1.27) A frequency shift in a spectrum is equivalent to an oscillation in the time domain with the same frequency. This allows the spectral frequency to be calibrated after acquisition. 3. Convolution theorem. The Fourier transform of the convolution of functions f1 and f2 is equal to the product of the Fourier transforms of f1 and f2: Ft{f1(t) ∗ f2(t)} = Ft[f1(t)] ∗ Ft[f2(t)] (1.28) in which the convolution of two functions is defined as the time integral over the product of one function and the other shifted function: f1(t) ∗ f2(t) = ∞ −∞ f1(τ ) f2(t − τ ) dτ (1.29) Based on this theorem, desirable line shapes of frequency signals can be obtained simply by applying a time function to the acquired FID prior to Fourier transformation to change the line shape of the spectral peaks, known as apodization of the FID. 4. Scaling theorem. The Fourier transform of a function with which a scaling transformation is carried out (t → t/c) is equal to the Fourier transform of the original function with the transformation ω → cω multiplied by the absolute value of factor c: Ft{f (t/c)}=|c|F (cω) (1.30) According to this theorem, the narrowing of the time domain function by a factor of c causes the broadening of its Fourier transformed function in frequency domain by the same factor, and vice versa. This theorem is also known as similarity theorem. 5. Parseval’s theorem. ∞ −∞ |f (t)| 2 dt = ∞ −∞ |F (ν)| 2 dν = ∞ −∞ |F (ω)| 2 dω (1.31)
14 cy domain is identical possessed by the signals in both time doma 1.5.2.Excitation Bandwidth ncies,the excitation band width is quired to be sufficien tly lar ent is achieved by oulses.In ertain other situatio ns。th gecxciaioabandidhisre uired ing short RE siderably arrow to excite a of re nce frequencies such as in selective excitation the following relationships of Fourier transform pairs are helpful in understanding the ADirac delta function()in the time domain at=gives rise to aspectrum with an infinitely wide frequency range and uniform intensity: P(S)s(-t)e-dr 0.32) which produces a frequency domain function with a perfectly flat magnitude at all frequencies because le=1.The function can be considered as an infinitely short pulse centered at. This infinitely short pulse excites an infinitely wide frequency range.Whenr equals zero,each Equation (1.32)means that in orde r to excite a wide frequency vee> n,a narroy range of frequen y is excited when a long RF pulse d.A 8 fur the domain n the time do Ftew]=ewe-iad山=2x8o-ow） 0.33 For a single resonance excitation.the RF pulse is required to be infinitely long.In practice the short pulses are a few microseconds,which are usually called hard pulses,whereas the long pulses may last a few seconds,and are called selective pulses.Shown in Figure 1.8 are the Fourier transforms of the short and long pulses.The bandwidth of the short pulse may cover several kilohertz and the selectivity of a long pulse can be as narrow as several hertz.A Gaussian function is the only function whose Fourier transformation gives another same-type (Gaussian)function [Figure 1.8(d)and(h)]: 1.34 A Gaussian shaped pulse will selectiv The value of eterm ils on se lective shaped pulsesv
14 Chapter 1 This theorem indicates that the information possessed by the signals in both time domain and frequency domain is identical. 1.5.2. Excitation Bandwidth In order to excite the transitions covering all possible frequencies, the excitation bandwidth is required to be sufficiently large. This requirement is achieved by applying short RF pulses. In certain other situations, the excitation bandwidth is required to be considerably narrow to excite a narrow range of resonance frequencies such as in selective excitation. The following relationships of Fourier transform pairs are helpful in understanding the process. A Dirac delta function δ(t − τ ) in the time domain at t = τ gives rise to a spectrum with an infinitely wide frequency range and uniform intensity: Ft{δ(t − τ )} = ∞ −∞ δ(t − τ )e−iωt dt = e−iωτ (1.32) which produces a frequency domain function with a perfectly flat magnitude at all frequencies because |e−iωτ | = 1. The δ function can be considered as an infinitely short pulse centered at τ . This infinitely short pulse excites an infinitely wide frequency range. When τ equals zero, each frequency has the same phase. Equation (1.32) means that in order to excite a wide frequency range, the RF pulse must be sufficiently short. Alternatively, for selective excitation, a narrow range of frequency is excited when a long RF pulse is used. A δ function in the frequency domain representing a resonance at frequency ω0 with a unit magnitude has a flat constant magnitude in the time domain lasting infinitely long in time: Ft{eiω0t } = ∞ −∞ eiω0t e−iωt dt = 2πδ(ω − ω0) (1.33) For a single resonance excitation, the RF pulse is required to be infinitely long. In practice, the short pulses are a few microseconds, which are usually called hard pulses, whereas the long pulses may last a few seconds, and are called selective pulses. Shown in Figure 1.8 are the Fourier transforms of the short and long pulses. The bandwidth of the short pulse may cover several kilohertz and the selectivity of a long pulse can be as narrow as several hertz. A Gaussian function is the only function whose Fourier transformation gives another same-type (Gaussian) function [Figure 1.8(d) and (h)]: Ft{e−t2/σ2 } = ∞ −∞ e−t2/σ2 e−iωt dt = −σ √πe−(ω2σ2)/4 (1.34) A Gaussian shaped pulse will selectively excite a narrow frequency range. The value of σ determines the selectivity of the pulse. The broadening of a Gaussian pulse results in narrowing in the frequency domain. More details on selective shaped pulses will be discussed in Chapter 4
Basic Principles of NMR 1.5.3.Quadrature Detection Two important time domain functions in NMR are the cosine and sine functions.The Fourier transformations of the two functions are as follows: Ft{cos(oot)月= e+e)ed=6w-w)++o】 135 Ft(sin(@ot))= ze-。ed=-w-+o】o The time domain signal may be considered as a cosine function.If an FID is detected by a single detector in the y plane in the rotating frame during acquisition after a 9 pulse,th Fourier transfo on of t as d d by the s in Equ 0n1.5 As a re e the inform n on gn or sine 90° mhined signal dete ted by the uadratu or is a lex idal function.which produces a resonance at Ft(eiao!)= e'e-dr=2π8(o-b）=(v-6） 137
Basic Principles of NMR 15 a b c d e f g h Figure 1.8. RF pulses and their Fourier transforms. Long and short rectangular pulses (a)–(c) and corresponding Fourier transforms (e)–(g), Gaussian shaped pulse (d) and its Fourier transform (h). 1.5.3. Quadrature Detection Two important time domain functions in NMR are the cosine and sine functions. The Fourier transformations of the two functions are as follows: Ft{cos(ω0t)} = ∞ −∞ 1 2 eiωt + e−iωt e−iωt dt = 1 2 [δ(ν − ν0) + δ(ν + ν0)] (1.35) Ft{sin(ω0t)} = ∞ −∞ 1 2i eiωt − e−iωt e−iωt dt = 1 2i[δ(ν − ν0) − δ(ν + ν0)] (1.36) The time domain signal may be considered as a cosine function. If an FID is detected by a single detector in the xy plane in the rotating frame during acquisition after a 90◦ pulse, the Fourier transformation of the cosine function gives rise to two frequency resonances located at ν0 and −ν0, as described by the δ functions in Equation (1.35). This indicates that the time domain signal detected by a single detector does not have information on the sign of the signal. As a result, each resonance will have a pair of peaks in the frequency domain. In order to preserve the information on the sign of the resonance, a second detector must be used, which is placed perpendicular to the first one in the xy plane. The signal detected by the second detector is a sine function (a time function which is 90◦ out of phase relative to the cosine function). The combined signal detected by the quadrature detector is a complex sinusoidal function, eiω0t , which produces a resonance at ω0 or ν0: Ft{eiω0t } = ∞ −∞ eiω0t e−iω0t dt = 2πδ(ω − ω0) = δ(ν − ν0) (1.37)