6 1.3.ROTATING FRAME The questions to be addressed in this section include 2 What is the rotating frame and why is it needed? lating electr netic feld? 3 the hulk t when a B field is lied to it? 4.What is the relationship between radio frequency (RF)pulse pow The larmor frequency ofa nuclear isotope is the resonance frequency ofthe isotope in the magnetic field.For example.H Larmor frequency will be 500 MHz for all protons of a sample in a magnetic field of 11.75 T.If the Larmor frequency were the only observed NMR signal. NMR spectroscopy would not be useful because there would be only one resonance signal for allH.In fact,chemical shifts are the NMR signals of interest(details in section 1.7),which have a frequency range of kilohertz,whereas the Larmor frequency of all nuclei is in the range of megahertz.For instance,the observed signals of protons are normally in the range of several kilohertz with a Larmor frequency of 600 MHz in a magnetic field of 14.1 T.How the Larmor frequency is removed before NMR data are acquired,what the rotating frame is,why we need it,and how the bulk magnetization changes upon applying an additional electromagnetic field are the topics of this section. not be present inany NMR spectrum,it is nec essary in the an e e xy pla the frequency with respect t tory frame of (a) I the ce simplifying on of the spi n the nuclei in the when the fre c of the of the le This Geld is turne ethe Larm edin NMR experiments,a new coordinate frame is introduced to eliminate the Larmor frequency from consideration.called the me In the otating frame.the xy plane of the laboratory frame is rotating at or near the Larmor frequency o with re ect to the z axis of the laboraton frame The transformation of the laboratory frame to the rotating frame can be illustrated by taking a merry-go-round as an example.The merry-go-round observed by one standing on the ground is rotating at a given speed.When one is riding on it,he is also rotating at the same speed.However,he is stationary relative to others on the merry-go-round.If the ground is considered as the laboratory frame,the merry-go-round is the rotating frame.When the person on the ground steps onto the merry-go-round,it is transformed from the laboratory frame to the rotating frame.The sole difference between the laboratory frame and the rotating frame is that the rotating frame is rotating in the xy plane about the z axis relative to the laboratory frame. By transforming from the laboratory frame to the rotating frame. e nuclear m ments ar The ter s that its abo rotating am a result. the bulk cy o rmor freq e quency e血he f thi rame results in a sta long an axis on the xy plan n the rotating frame fo
6 Chapter 1 1.3. ROTATING FRAME The questions to be addressed in this section include: 1. What is the rotating frame and why is it needed? 2. What is the B1 field and why must it be an oscillating electromagnetic field? 3. How does the bulk magnetization M0 react when a B1 field is applied to it? 4. What is the relationship between radio frequency (RF) pulse power and pulse length? The Larmor frequency of a nuclear isotope is the resonance frequency of the isotope in the magnetic field. For example, 1H Larmor frequency will be 500 MHz for all protons of a sample in a magnetic field of 11.75 T. If the Larmor frequency were the only observed NMR signal, NMR spectroscopy would not be useful because there would be only one resonance signal for all 1H. In fact, chemical shifts are the NMR signals of interest (details in section 1.7), which have a frequency range of kilohertz, whereas the Larmor frequency of all nuclei is in the range of megahertz. For instance, the observed signals of protons are normally in the range of several kilohertz with a Larmor frequency of 600 MHz in a magnetic field of 14.1 T. How the Larmor frequency is removed before NMR data are acquired, what the rotating frame is, why we need it, and how the bulk magnetization changes upon applying an additional electromagnetic field are the topics of this section. Since the Larmor frequency will not be present in any NMR spectrum, it is necessary to remove its effect when dealing with signals in the kilohertz frequency range. This can be done by applying an electromagnetic field B1 on the xy plane of the laboratory frame, which rotates at the Larmor frequency with respect to the z axis of the laboratory frame. This magnetic field is used for the purposes of (a) removing the effect of the Larmor frequency and hence simplifying the theoretical and practical consideration of the spin precession in NMR experiments and (b) inducing the nuclear transition between two energy states by its interaction with the nuclei in the sample according to the resonance condition that the transition occurs when the frequency of the field equals the resonance frequency of the nuclei. This magnetic field is turned on only when it is needed. Because the Larmor frequency is not observed in NMR experiments, a new coordinate frame is introduced to eliminate the Larmor frequency from consideration, called the rotating frame. In the rotating frame, the xy plane of the laboratory frame is rotating at or near the Larmor frequency ω0 with respect to the z axis of the laboratory frame. The transformation of the laboratory frame to the rotating frame can be illustrated by taking a merry-go-round as an example. The merry-go-round observed by one standing on the ground is rotating at a given speed. When one is riding on it, he is also rotating at the same speed. However, he is stationary relative to others on the merry-go-round. If the ground is considered as the laboratory frame, the merry-go-round is the rotating frame. When the person on the ground steps onto the merry-go-round, it is transformed from the laboratory frame to the rotating frame. The sole difference between the laboratory frame and the rotating frame is that the rotating frame is rotating in the xy plane about the z axis relative to the laboratory frame. By transforming from the laboratory frame to the rotating frame, the nuclear moments are no longer spinning about the z axis, that is, they are stationary in the rotating frame. The term “transforming” here means that everything in the laboratory frame will rotate at a frequency of −ω0 about the z axis in the rotating frame. As a result, the bulk magnetization does not have its Larmor frequency in the rotating frame. Since the applied B1 field is rotating at the Larmor frequency in the laboratory frame, the transformation of this magnetic field to the rotating frame results in a stationary B1 field along an axis on the xy plane in the rotating frame, for
Basic Principles of NMR 7 example the x axis.Therefore,when this B magnetic field is applied,its net effect on the bulk magnetization is to rotate the bulk magnetization away from the z axis clockwise about the axis of the applied field by the left-hand rule in the vector representation. In practice,the rotation of the B field with respect to the z axis of the laboratory frame is achieved by generating a linear oscillating electromagnetic field with the magnitude of 2B because it is easily produced by applying electric current through the probe coil (Figure 1.3). The oscillating magnetic field has a frequency equal to or near the Larmor frequency of the nuclei.As the current increases from zero to maximum,the field proportionally increases from zero to the maximum field along the coil axis(2BI in Figure 1.3).Reducing the current from the ma. 0 um creases the fiel Finally, the If the 2B13 人 one cycle. itude,B P9 two eq the ane at the se site dire o each other (thir When th field has the on the axis with an itude of B1.Th sum of the two is 2B[Fis e 1 4(a)l 八hen the current is ze which gives zero in the field magnitude.each c nt still has the same magnitude of B but aligns on the x and-x axes,respectively,which gives rise to a vector sum of zero [Figure 1.4(c)].As the current reduces,both components rotate into 000不 VVVV 2B Figure 1.3.The electromagnetic field generated by the current passing through the probe coil.The 000A000e000000①00000C a) (b) c) d Figure 1.4.Vector sum of the oscillating B field generated by passing current through aprobe coil.The vectors rotating in opp When R is said to he on resonance (a)When the cur reaches the ximum the twe vectors align on y axis.The sum of the two vectors is the same as the field produced in the coil.(b)As
Basic Principles of NMR 7 example the x axis. Therefore, when this B1 magnetic field is applied, its net effect on the bulk magnetization is to rotate the bulk magnetization away from the z axis clockwise about the axis of the applied field by the left-hand rule in the vector representation. In practice, the rotation of the B1 field with respect to the z axis of the laboratory frame is achieved by generating a linear oscillating electromagnetic field with the magnitude of 2B1 because it is easily produced by applying electric current through the probe coil (Figure 1.3). The oscillating magnetic field has a frequency equal to or near the Larmor frequency of the nuclei. As the current increases from zero to maximum, the field proportionally increases from zero to the maximum field along the coil axis (2B1 in Figure 1.3). Reducing the current from the maximum to zero and then to the minimum (negative maximum, −i) decreases the field from 2B1 to −2B1. Finally, the field is back to zero from −2B1 as the current is increased from the minimum to zero to finish one cycle. If the frequency of changing the current is νrf , we can describe the oscillating frequency as ωrf(ω = 2πν). Mathematically, this linear oscillating field (thick arrow in Figure 1.4) can be represented by two equal fields with half of the magnitude, B1, rotating in the xy plane at the same angular frequency in opposite directions to each other (thin arrows). When the field has the maximum strength at 2B1, each component aligns on the y axis with a magnitude of B1. The vector sum of the two is 2B1 [Figure 1.4(a)]. When the current is zero, which gives zero in the field magnitude, each component still has the same magnitude of B1 but aligns on the x and −x axes, respectively, which gives rise to a vector sum of zero [Figure 1.4(c)]. As the current reduces, both components rotate into i –i –2B1 2B1 Figure 1.3. The electromagnetic field generated by the current passing through the probe coil. The magnitude of the field is modulated by changing the current between −i and +i. The electromagnetic field is called the oscillating B1 field. i i -i x y -i (a) (b) –ω (c) rf ωrf (d) (e) Figure 1.4. Vector sum of the oscillating B1 field generated by passing current through a probe coil. The magnitude of the field can be represented by two equal amplitude vectors rotating in opposite directions. The angular frequency of the two vectors is the same as the oscillating frequency ωrf of the B1 field. When ωrf = ω0, B1 is said to be on resonance. (a) When the current reaches the maximum, the two vectors align on y axis. The sum of the two vectors is the same as the field produced in the coil. (b) As the B1 field reduces, its magnitude equals the sum of the projections of two vectors on the y axis. (c) When the two vectors are oppositely aligned on the x axis, the current in the coil is zero
8 Figure 1.5.B field in the laboratory frame.The bulk magnetization Mo is the vector sum of individual nuclear moments which are precessing about the static magnetic field Bo at the Larmor frequency Bfeld anguar lrequeney of ot he i hed is cquan to the Lamor rrequeney.hat s.ofh the-y region and the sum produces a negative magnitude [Figure 1.4(d)I.Finally.the two components meet at the-y axis,which represents a field magnitude of-2B [Figure 1.4(e)]. At any given time the two decomposed components have the same magnitude of B,the same frequency of f,and are mirror images of each other. If the frequency of the rotating frame is set to,which is close to the Larmor frequency @o,the component ofthe B field which has in the laboratory frame has null frequency in the rotating frame because of the transformation by.The other with in the laboratory frame now has an angular frequency of-2 after the transformation.Since the latter has a frequency far away from the Larmor frequency it will not interfere with the NMR signals which are心nhc8rof6c业therctor.心componnu freque8y2 ssion unless specifically mentioned.T ormer component wi rotating f ent the BI field en said to 15).Simce i t ncy o,the BI the r ting frame th ear spir T eld un whe crBstapliediedh will rotate about the ction u n its inter with the bulk m ag axis where B otating frame.The frequer ncy of the otati ined by: =-YB1 .1) This should not be misunderstood as f of the B field since f is the field oscillating frequency determined by changing the direction of the current passing through the coil,which is set to be the same as or the Larm frequency.Frequency f is often called the carrie mitter dete ed b f the B By modula the ng the am ere in t eld i when Mo moves from h eaxis to the xy plane,this is calleda9 pulse
8 Chapter 1 x y M0 B1 1 0 Figure 1.5. B1 field in the laboratory frame. The bulk magnetization M0 is the vector sum of individual nuclear moments which are precessing about the static magnetic field B0 at the Larmor frequency ω0. When the angular frequency ωrf of the B1 field is equal to the Larmor frequency, that is, ωrf = ω0, the B1 field is on resonance. the −y region and the sum produces a negative magnitude [Figure 1.4(d)]. Finally, the two components meet at the −y axis, which represents a field magnitude of −2B1 [Figure 1.4(e)]. At any given time the two decomposed components have the same magnitude of B1, the same frequency of ωrf , and are mirror images of each other. If the frequency of the rotating frame is set to ωrf , which is close to the Larmor frequency ω0, the component of theB1 field which has ω1 in the laboratory frame has null frequency in the rotating frame because of the transformation by −ωrf . The other with −ωrf in the laboratory frame now has an angular frequency of −2ωrf after the transformation. Since the latter has a frequency far away from the Larmor frequency it will not interfere with the NMR signals which are in the range of kilohertz. Therefore, this component will be ignored throughout the discussion unless specifically mentioned. The former component with null frequency in the rotating frame is used to represent the B1 field. If we regulate the frequency of the current oscillating into the coil as ω0, then setting ωrf equal to the Larmor frequency ω0, the B1 field is said to be on resonance (Figure 1.5). Since in the rotating frame the Larmor frequency is not present in the nuclei, the effect of B0 on nuclear spins is eliminated. The only field under consideration is the B1 field. From the earlier discussion we know that nuclear magnetization will rotate about the applied field direction upon its interaction with a magnetic field. Hence, whenever B1 is turned on, the bulk magnetization will be rotated about the axis where B1 is applied in the rotating frame. The frequency of the rotation is determined by: ω1 = −γ B1 (1.11) This should not be misunderstood as ωrf of the B1 field since ωrf is the field oscillating frequency determined by changing the direction of the current passing through the coil, which is set to be the same as or near the Larmor frequency. Frequency ωrf is often called the carrier frequency or the transmitter frequency. The frequency ω1 is determined by the amplitude of the B1 field, that is, the maximum strength of the B1 field. By modulating the amplitude and time during which B1 is turned on, the bulk magnetization can be rotated to anywhere in the plane perpendicular to the axis of the applied B1 field in the rotating frame. If B1 is turned on and then turned off when M0 moves from the z axis to the xy plane, this is called a 90◦ pulse
Basic Principles of NMR 9 receiver applyinga90°pulse anda180°pulsc pulse length (or the9 te puise power se a e nd .The puls sho 90 lei (all nuclei ptH.Be l ratios than r ons,they have longer 90 pulse lengths at a gi Thepulse length (w)is proportional to the Bfield stre gth: 1 1= 12 T 1 pw9o=2yB1-4v1 1.13) in which v is the field strength in the frequency unit of hertz.A higher B field produces a shorter 90 pulse.A90 pulse of 10 us corresponds to a 25 kHz B field.Nuclei with smaller gyromagnetic ratios will require a higher B to generate the same pwo as that with larger y.When a receiver is placed on the transverse plane of the rotating frame,NMR signals are observed from the transverse magnetization.The maximum signal is obtained when the bulk magnetization is in the xy plane of the rotating frame,which is done by applying a9 pulse. No signal is observed when a 180 pulse is applied(Figure 1.6). 1.4.BLOCH EQUATIONS As we now know,the nuclei inside the magnet produce nuclear moments which cause them to spin about the magnetic field.Inaddition,the interaction ofthe nuclei with the magnetic
Basic Principles of NMR 9 x y B0 M0 B1 90°x 180° x x y B0 M0 B1 x y B0 M0 B1 receiver receiver Figure 1.6. Vector representation of the bulk magnetization upon applying a 90◦ pulse and a 180◦ pulse by B1 along the x axis in the rotating frame. The corresponding time during which β1 is applied is called the 90◦ pulse length (or the 90◦ pulse width), and the field amplitude is called the pulse power. A 90◦ pulse length can be as short as a few microseconds and as long as a fraction of a second. The pulse power for a hard (short) 90◦ pulse is usually as high as half of a hundred watts for protons and several hundred watts for heteronuclei (all nuclei except 1H). Because heteronuclei have lower gyromagnetic ratios than protons, they have longer 90◦ pulse lengths at a given B1 field strength. The 90◦ pulse length (pw90) is proportional to the B1 field strength: ν1 = γ B1 2π = 1 4pw90 (1.12) pw90 = π 2γ B1 = 1 4ν1 (1.13) in which ν1 is the field strength in the frequency unit of hertz. A higher B1 field produces a shorter 90◦ pulse. A 90◦ pulse of 10 μs corresponds to a 25 kHz B1 field. Nuclei with smaller gyromagnetic ratios will require a higher B1 to generate the same pw90 as that with larger γ . When a receiver is placed on the transverse plane of the rotating frame, NMR signals are observed from the transverse magnetization. The maximum signal is obtained when the bulk magnetization is in the xy plane of the rotating frame, which is done by applying a 90◦ pulse. No signal is observed when a 180◦ pulse is applied (Figure 1.6). 1.4. BLOCH EQUATIONS As we now know, the nuclei inside the magnet produce nuclear moments which cause them to spin about the magnetic field. In addition, the interaction of the nuclei with the magnetic
10 Chapter 1 tra the the Bl ons to be add ed in the current section 1 What phenomena do the Bloch equations describe? 2 at is free induction de ay (FID)? 3.What are the limitations of the Bloch equations? In the presence of the magnetic field Bo.the torque produced by Bo on spins with the angular moment u causes precession of the nuclear spins.Felix Bloch derived simple semi- classical equations to describe the time-dependent phenomena of nuclear spins in the static magnetic field(Bloch,1946).The torque on the bulk magnetization,described by the change of the angular momentum as a function of time,is given by: T=dp -=M X B 1.14) dr n的 of magnetization with time is described by: dM =y(M x B) (1,15) dt When B is the static magnetic field Bo which is along the z axis of the laboratory frame the change of magnetization along the x,y,and z axes with time can be obtained from the determinant of the vector product: dM dt =业++k=yMM,M=iyM,-jM.a1 dt dt dt 00 Bo in whichi,j,k are the unit vectors along the,y,andaxes,respectively.Therefore, dM. dr =yMy Bo 17) dM dr =-yMx Bo (1.18 dM: =0 dt (1.19 The abo the n components agnetic field Boprodu n NMR sp ti pe and the rate of decay is depen nt on ield nuclear gyromagnetic ratio
10 Chapter 1 field will rotate the magnetization toward the transverse plane when the electromagnetic B1 field is applied along a transverse axis in the rotating frame. After the pulse is turned off, the magnetization is solely under the effect of the B0 field. How the magnetization changes with time can be described by the Bloch equations, which are based on a simple vector model. Questions to be addressed in the current section include: 1. What phenomena do the Bloch equations describe? 2. What is free induction decay (FID)? 3. What are the limitations of the Bloch equations? In the presence of the magnetic field B0, the torque produced by B0 on spins with the angular moment μ causes precession of the nuclear spins. Felix Bloch derived simple semiclassical equations to describe the time-dependent phenomena of nuclear spins in the static magnetic field (Bloch, 1946). The torque on the bulk magnetization, described by the change of the angular momentum as a function of time, is given by: T = dP dt = M × B (1.14) in which M × B is the vector product of the bulk magnetization M (the sum of μ) with the magnetic field B. Because M = γ P (or P = M/γ ) according to Equation (1.3), the change of magnetization with time is described by: dM dt = γ (M × B) (1.15) When B is the static magnetic field B0 which is along the z axis of the laboratory frame, the change of magnetization along the x, y, and z axes with time can be obtained from the determinant of the vector product: dM dt = i dMx dt + j dMy dt + k dMz dt = γ ijk Mx My Mz 0 0 B0 = iγMyB0 − jγMxB0 (1.16) in which i, j , k are the unit vectors along the x, y, and z axes, respectively. Therefore, dMx dt = γ MyB0 (1.17) dMy dt = −γ MxB0 (1.18) dMz dt = 0 (1.19) The above Bloch equations describe the time dependence of the magnetization components under the effect of the static magnetic fieldB0 produced by the magnet of an NMR spectrometer without considering any relaxation effects. The z component of the bulk magnetization Mz is independent of time, whereas the x and y components are decaying as a function of time and the rate of decay is dependent on the field strength and nuclear gyromagnetic ratio. The