1 Basic Principles of NMR 1.1.INTRODUCTION Energy states and population distribution are the fundamental subjects of any spectro scopic technique.The energy difference between energy states gives rise to the frequency ofthe spectra,whereas intensities of the spectral peaks are proportional to the population difference of the states.R her funda ntal phe n ne s es an ng these In principle,an NMR spectrometer is more or less like a radio.In a radio,audio signals in the frequency range of kilohertz are the signals of interest,which one can hear.However, the signals sent by broadcast stations are in the range of 100 MHz for FM and of up to 1GHz for AM broadcasting.The ansmis n requenci re they are t to spea ns hay me Ia h nals ofn are the chemical shifts generated by the electron density surrounding an individual proton which are in the kilohertz frequency range.Many of the protons in the molecule have different chemical environments which give different signals in the kilohertz range.One must find a egahertz Larmor frequency in order to observe the kilohertz chemical 1.2.NUCLEAR SPIN IN A STATIC MAGNETIC FIELD 1.2.1.Precession of Nuclear Spins in a Magnetic Field As mentioned above,energy and population associated with energy states are the basis of the frequency position and the intensity ofspectral signals.In order to understand the principles of NMR spectroscopy,it is necessary to know how the energy states of nuclei are generated What kind of nuclei will give N How do nuclear spins orent in the magnetic field?
1 Basic Principles of NMR 1.1. INTRODUCTION Energy states and population distribution are the fundamental subjects of any spectroscopic technique. The energy difference between energy states gives rise to the frequency of the spectra, whereas intensities of the spectral peaks are proportional to the population difference of the states. Relaxation is another fundamental phenomenon in nuclear magnetic resonance spectroscopy (NMR), which influences both line shapes and intensities of NMR signals. It provides information about structure and dynamics of molecules. Hence, understanding these aspects lays the foundation to understanding basic principles of NMR spectroscopy. In principle, an NMR spectrometer is more or less like a radio. In a radio, audio signals in the frequency range of kilohertz are the signals of interest, which one can hear. However, the signals sent by broadcast stations are in the range of 100 MHz for FM and of up to 1 GHz for AM broadcasting. The kilohertz audio signals must be separated from the megahertz transmission frequencies before they are sent to speakers. In NMR spectroscopy, nuclei have an intrinsic megahertz frequency which is known as the Larmor frequency. For instance, in a molecule, all protons have the same Larmor frequency. However, the signals of interest are the chemical shifts generated by the electron density surrounding an individual proton, which are in the kilohertz frequency range. Many of the protons in the molecule have different chemical environments which give different signals in the kilohertz range. One must find a way to eliminate the megahertz Larmor frequency in order to observe the kilohertz chemical shifts (more details to follow). 1.2. NUCLEAR SPIN IN A STATIC MAGNETIC FIELD 1.2.1. Precession of Nuclear Spins in a Magnetic Field As mentioned above, energy and population associated with energy states are the basis of the frequency position and the intensity of spectral signals. In order to understand the principles of NMR spectroscopy, it is necessary to know how the energy states of nuclei are generated and what are the energy and population associated with the energy states. Key questions to be addressed in this section include: 1. What causes nuclei to precess in the presence of a magnetic field? 2. What kind of nuclei will give NMR signals? 3. How do nuclear spins orient in the magnetic field? 1
2 Not any kind of nucleuswlve charge and mas ve a spin quantum nun f nucle will n to an N signa ctive th prith a no app eld Nue 1 nagnitud is determi angul P=hvI(I+1) er and his the Planck constant divided by2 ofl is or half as n integra er The z compone of the ntum P:is given by P:=hm 12) in which the magnetic quantum number m has possible values of./-1. -1-and a total of2/+1.This equation tells us that the pr oiection of nuclear angular momentum on the axis is quantized in space and has a total of2/+1possible values.The orientations ofnuclear angular momentum are defined by the allowed m values.For example,for spin nuclei,the an angle of 54 79 relative to the magnetic field (Figure 11) The nuclei with a nonzero spin quantum number will rotate about the magnetic field Bo due to the torque generated by the interaction of the nuclear angular momentum with the magnetic field.The magnetic moment(or nuclear moment),,is either parallel or antiparallel to their angular momentum: u=yP=yhvI(I.+万 .3) in which y is the nuclear gyromagnetic ratio,which has a specific value for a given isotope. Thus,y is a characteristic constant for a specific nucleus.The angular momentum P is the same for all nuclei with the same magnetic quantum number,whereas the angular moment m=- Figure 1.1.Orientation of lear angular mo and it comp the laboratory frame
2 Chapter 1 Not any kind of nucleus will give NMR signals. Nuclei with an even number of both charge and mass have a spin quantum number of zero, for example, 12C. These kinds of nuclei do not have nuclear angular momentum and will not give rise to an NMR signal; these are called NMR inactive nuclei. For nuclei with a nonzero spin quantum number, energy states are produced by the nuclear angular moment interacting with the applied magnetic field. Nuclei with a nonzero spin quantum number possess nuclear angular momentum whose magnitude is determined by: P = I (I + 1) (1.1) in which I is the nuclear spin quantum number and is the Planck constant divided by 2π. The value of I is dependent on the mass and charge of the nucleus and can be either an integral or half integral number. The z component of the angular momentum Pz is given by: Pz = m (1.2) in which the magnetic quantum number m has possible values of I , I −1, ... , −I +1, −I , and a total of 2I +1. This equation tells us that the projection of nuclear angular momentum on the z axis is quantized in space and has a total of 2I +1 possible values. The orientations of nuclear angular momentum are defined by the allowed m values. For example, for spin 1 2 nuclei, the allowed m are 1 2 and −1 2 . Thus, the angular momentum of spin 1 2 has two orientations; one is pointing up (pointing to the z axis) and the other pointing down (pointing to the −z axis) with an angle of 54.7◦ relative to the magnetic field (Figure 1.1). The nuclei with a nonzero spin quantum number will rotate about the magnetic field B0 due to the torque generated by the interaction of the nuclear angular momentum with the magnetic field. The magnetic moment (or nuclear moment), μ, is either parallel or antiparallel to their angular momentum: μ = γ P = γ I (I + 1) (1.3) in which γ is the nuclear gyromagnetic ratio, which has a specific value for a given isotope. Thus, γ is a characteristic constant for a specific nucleus. The angular momentum P is the same for all nuclei with the same magnetic quantum number, whereas the angular moment μ z x z 2 m 1 = 2 m 1 = − Figure 1.1. Orientation of nuclear angular moment μ with spin 1 2 and its z component, μz. The vectors represent the angular moment μ rotating about the magnetic field whose direction is along the z axis of the laboratory frame.
Basic Principles of NMR is different for different nuclei.For instance,3C and H have the same angular momentum P because they have same spin quantum number of,but have different angular momentsu because they are different isotopes with differenty.Therefore,the nuclear angular moment is used to characterize nuclear spins.The moment u is parallel to the angular momentum ify is positive or antiparallel ify is negative (e.g,5N).Similar to thecomponent of the angular momentum,P..the z component of angular moment u.is given by: H:=yP=yhm The equation indicates thathas a different value for diffe if they may have etic field.they the etic field rated hy the of the etic field Bo with the clear n nt The relativ to Bo is de Nuclei with called nucle spins because their ents make the in in the netic field in summar the nuelear ang lar mome tum is whateau ses the nucleus to rotate relative to the magnetic field.Different nuclei have a characteristic nuclear moment because the moment is dependent on the gyromagnetic ratio y.whereas nuclei with the same spin quantum number possess the same nuclear angular momentum.Nuclear moments have quantized orientations defined by the value of the magnetic quantum number,m.The interaction of nuclei with the magnetic field is utilized to generate an NMR signal.Because the energy and population of nuclei are proportional to the magnetic field strength(more details discussed below),the frequency and intensity of the NMR spectral signals are dependent on the field strength. 1.2.2.Energy States and Population It has been illustrated in the previous section that nuclei with nonzero spin quantum numbers orient along specific directions with respect to the magnetic field.They are rotating continuously about the field direction due to the nuclear moment u possessed by nuclei.For each orientation state,also known as the Zeeman state or spin state,there is energy associated with it,which is characterized by the frequency of the precession. Key questions to be addressed in this section include: 1.What is the energy and population distribution of the Zeeman states? 2.What are the nuclear precession frequencies of the Zeeman states and the frequency of the transition hetween the states and how are they different? 3.How are energy and population related to measurable spectral quantities? The intrinsic fred ency of the precession is the Lam the ze an state with magnetic quantum mhe m can be des armo frequency: E=-u:Bo =-mhy Bo=mhao 15 in which Bo is the magnetic field strength in the unit of tesla,T,and o=-yBo is the Larmor frequency.Therefore,the energy difference of the allowed transition (the selection rule is that
Basic Principles of NMR 3 is different for different nuclei. For instance, 13C and 1H have the same angular momentum P because they have same spin quantum number of 1 2 , but have different angular moments μ because they are different isotopes with different γ . Therefore, the nuclear angular moment μ is used to characterize nuclear spins. The moment μ is parallel to the angular momentum if γ is positive or antiparallel if γ is negative (e.g., 15N). Similar to the z component of the angular momentum, Pz, the z component of angular moment μz is given by: μz = γ Pz = γ m (1.4) The equation indicates that μz has a different value for different nuclei even if they may have the same magnetic quantum number m. When nuclei with a nonzero spin quantum number are placed in a magnetic field, they will precess about the magnetic field due to the torque generated by the interaction of the magnetic field B0 with the nuclear moment μ. The angle of μ relative to B0 is dependent on m. Nuclei with nonzero spin quantum numbers are also called nuclear spins because their angular moments make them spin in the magnetic field. In summary, the nuclear angular momentum is what causes the nucleus to rotate relative to the magnetic field. Different nuclei have a characteristic nuclear moment because the moment is dependent on the gyromagnetic ratio γ , whereas nuclei with the same spin quantum number possess the same nuclear angular momentum. Nuclear moments have quantized orientations defined by the value of the magnetic quantum number, m. The interaction of nuclei with the magnetic field is utilized to generate an NMR signal. Because the energy and population of nuclei are proportional to the magnetic field strength (more details discussed below), the frequency and intensity of the NMR spectral signals are dependent on the field strength. 1.2.2. Energy States and Population It has been illustrated in the previous section that nuclei with nonzero spin quantum numbers orient along specific directions with respect to the magnetic field. They are rotating continuously about the field direction due to the nuclear moment μ possessed by nuclei. For each orientation state, also known as the Zeeman state or spin state, there is energy associated with it, which is characterized by the frequency of the precession. Key questions to be addressed in this section include: 1. What is the energy and population distribution of the Zeeman states? 2. What are the nuclear precession frequencies of the Zeeman states and the frequency of the transition between the states, and how are they different? 3. How are energy and population related to measurable spectral quantities? The intrinsic frequency of the precession is the Larmor frequency ω0. The energy of the Zeeman state with magnetic quantum number m can be described in terms of the Larmor frequency: E = −μzB0 = −mγ B0 = mω0 (1.5) in which B0 is the magnetic field strength in the unit of tesla, T, and ω0 = −γ B0 is the Larmor frequency. Therefore, the energy difference of the allowed transition (the selection rule is that
4 △E=yB 1.6 Beca h,the frequency of the required electromagnetic radiation for the transition @=YBo 1) which ha on Larmo 23Twh amp agne 14 adian strength of ns per secc frequency can also be represent v-2 1.8 diffe ce be ween two transition states as doe ition The in lati on diff f th en d h the Rolt- such as H.or C.the lo rgy state (g round state)is defined as thestate form .whereas the higher energy state (excited state)is labeled a the B state for m .For I5 N,m =-is the lower energy a state because of its negativey The ratio ofthe populations in the states is quantitatively described by the Boltzmann equation: =e△ET=e-T= （1.9 N in which N and Ne pulation of the a and 8 states.respectively.T is the absolute stant Thee uation states that hath the e ergy difference of the transition states and the population difference of the states increase with the magnetic field strength.Furthermore,the opulation difference has a temperature dependence if the sample temperature reaches absolute zero,there is no population at the B state and all spins will lie in the a state,whereas both states will have equal population if the temperature is infinitely high.At T near room temperature,~300 K,hy BokT.As a consequence,a first-order Taylor expansion can be used to describe the population difference: ≈1、yB 1.10) Na 7 elver than that su small fra n will contribu sity due o the low energy diffe rence and ence NMR spectroscopy intrinsically isa very insensitive
4 Chapter 1 only a single-quantum transition, that is, m = ±1, is allowed), for instance, between the m = −1 2 and m = 1 2 Zeeman states, is given by: E = γ B0 (1.6) Because E = ω, the frequency of the required electromagnetic radiation for the transition has the form of: ω = γ B0 (1.7) which has a linear dependence on the magnetic field strength. Commonly, the magnetic field strength is described by the proton Larmor frequency at the specific field strength. A proton resonance frequency of 100 MHz corresponds to the field strength of 2.35 T. For example, a 600 MHz magnet has a field strength of 14.1 T. While the angular frequency ω has a unit of radians per second, the frequency can also be represented in hertz with the relationship of: ν = ω 2π (1.8) As the magnetic field strength increases, the energy difference between two transition states becomes larger, as does the frequency associated with the Zeeman transition. The intensity of the NMR signal comes from the population difference between two Zeeman states of the transition. The population of the energy state is governed by the Boltzmann distribution. For a spin 1 2 nucleus with a positive γ such as 1H, or 13C, the lower energy state (ground state) is defined as the α state for m = 1 2 , whereas the higher energy state (excited state) is labeled as the β state for m = −1 2 . For 15N, m = −1 2 is the lower energy α state because of its negative γ . The ratio of the populations in the states is quantitatively described by the Boltzmann equation: Nβ Nα = e−E/kT = e−γ B0/kT = 1 eγ B0/kT (1.9) in which Nα and Nβ are the population of the α and β states, respectively, T is the absolute temperature and k is the Boltzmann constant. The equation states that both the energy difference of the transition states and the population difference of the states increase with the magnetic field strength. Furthermore, the population difference has a temperature dependence. If the sample temperature reaches absolute zero, there is no population at the β state and all spins will lie in the α state, whereas both states will have equal population if the temperature is infinitely high. At T near room temperature, ∼300 K, γ B0 kT . As a consequence, a first-order Taylor expansion can be used to describe the population difference: Nβ Nα ≈ 1 − γ B0 kT (1.10) At room temperature, the population of the β state is slightly lower than that of the α state. For instance, the population ratio for protons at 800 MHz field strength is 0.99987. This indicates that only a small fraction of the spins will contribute to the signal intensity due to the low energy difference and hence NMR spectroscopy intrinsically is a very insensitive
Basic Principles of NMR spectroscopic technique.Therefore,a stronger magnetic field is necessary to obtain better sensitivity,in addition to other advantages such as higher resolution and the TROSY effect (transverse relaxation optimized spectroscopy). 1.2.3.Bulk Magnetization Questions to be addressed in this section include: re is it located? Why do no transverse components of bulk magnetization exist at equilibrium? The observable NMR sigr als ome from the sembly of nuclea spins in the presence the magne e (o mag gives in of the nlane onent of the bulk magnetization at the equilibrium state is av ged to ero and hen is not observable (Figure 1.2).The bulk magnetization Mo results from the small p lation difference hety een the a and B states.At equilibrium,this vector lies along h zaxis and is parallel to the magnetic field direction for nuclei with positive because the spin population in the a state is larger than that in the B state.Although the bulk magnetization is stationary along the zaxis,the individual spin moments rotate about the axis. B B Figure 1.2.Bulk magnetization of spin nuclei with positive v.r.y.and z are the axes of the labora frame.The thin arrows represent individual nuclear moments.The vector sum of the nuclear moments on the xy plane is zero because an individual nuclear moment has equal probability of being in any directionof th exy pla The bu arrow,is ge f they the smal
Basic Principles of NMR 5 spectroscopic technique. Therefore, a stronger magnetic field is necessary to obtain better sensitivity, in addition to other advantages such as higher resolution and the TROSY effect (transverse relaxation optimized spectroscopy). 1.2.3. Bulk Magnetization Questions to be addressed in this section include: 1. What is the bulk magnetization and where is it located? 2. Why do no transverse components of bulk magnetization exist at equilibrium? The observable NMR signals come from the assembly of nuclear spins in the presence of the magnetic field. It is the bulk magnetization of a sample (or macroscopic magnetization) that gives the observable magnetization, which is the vector sum of all spin moments (nuclear angular moments). Because nuclear spins precess about the magnetic field along the z axis of the laboratory frame, an individual nuclear moment has equal probability of being in any direction of the xy plane. Accordingly, the transverse component of the bulk magnetization at the equilibrium state is averaged to zero and hence is not observable (Figure 1.2). The bulk magnetization M0 results from the small population difference between the α and β states. At equilibrium, this vector lies along the z axis and is parallel to the magnetic field direction for nuclei with positive γ because the spin population in the α state is larger than that in the β state. Although the bulk magnetization is stationary along the z axis, the individual spin moments rotate about the axis. x y z M0 B0 Figure 1.2. Bulk magnetization of spin 1 2 nuclei with positive γ . x, y, and z are the axes of the laboratory frame. The thin arrows represent individual nuclear moments. The vector sum of the nuclear moments on the xy plane is zero because an individual nuclear moment has equal probability of being in any direction of the xy plane. The bulk magnetization M0, labeled as a thick arrow, is generated by the small population difference between the α and β states, and is parallel to the direction of the static magnetic field B0.