NUCLEAR MAGNETIC RESONANCE SPECTROSCOPY | Principles

NMR SPECTROSCOPY / Principles 211 Rae AIM (1992) Quantum Mechanics, 3rd edn. Bristol: Institute of Physics. Vanhamme L, van den Boogaart A, and Van Huffel S (1997) Improved method for accurate and efficient

quantification of MRS data with use of prior-knowledge. Journal of Magnetic Resonance 129: 35–43. Young IR (ed.) (2002) Methods in Biomedical Magnetic Resonance Imaging and Spectroscopy. Chichester: Wiley.

Principles J B Grutzner, Purdue University, West Lafayette, IN, USA & 2005, Elsevier Ltd. All Rights Reserved.

Introduction Nuclear magnetic resonance (NMR) spectroscopy is a rich source of molecular information. It is widely used for molecular structure analysis and for medical imaging under the name MRI (magnetic resonance imaging). NMR is a nondestructive quantitative method for the analysis of complex mixtures such as reactions, petroleum, materials, and foods. It finds major application in the determination of molecular structure and dynamics, especially of pharmaceuticals, polymers, and proteins. Diverse applications include measurement of internuclear distances, water content of food, characterization of oil deposits using the earth’s magnetic field, in vivo studies of metabolic pathways in cells and humans, diffusion of adsorbates into porous materials, reaction dynamics, fluid flow, generation and measurement of temperatures in the millikelvin range, and online monitoring of manufacturing processes. Samples may be gases, liquids, or solids. This article provides a description of the basic principles of NMR spectroscopy. The many applications of this technique, which make it indispensable as a tool for the analytical chemist, are described in subsequent articles, which also include discussion of advanced techniques such as the NMR of solids, multidimensional NMR, and MRI.

Principles of NMR NMR detects signals from nuclei in atoms and molecules when they are excited in a strong homogeneous magnetic field (B0 in tesla). NMR was invented independently by Bloch and Purcell to accurately measure nuclear magnetic moments (m). It is now used to measure magnetic field strength with high precision based on known m values.

The nuclear magnets are ordered into 2I þ 1 states determined by the nuclear spin quantum number I. The spin angular momentum is quantized in units of _ and the change in the z component is detected in the NMR experiment. For many common NMR nuclei I ¼ 1/2 (1H, 13C, 19F, 29Si, 31P) and there are two allowed states labeled spin up and spin down. Irradiation with radiofrequency (RF) electromagnetic radiation (n in Hz) causes spin flip transitions provided the resonance condition is satisfied: hn ¼ _gB0 ¼ mz bB0

½1


where h is Planck’s constant (6.62608  10  34 J s), _ ¼ h=ð2_Þ, b is the nuclear Bohr magneton (eh/ 4pmp ¼ 5.0508  10  27 J T  1) and mz is the z-component of the total moment ¼ O[I(I þ 1)]mz. The nuclear magnetogyric ratio (g in rad s  1 T  1) is an exquisitely sensitive discriminator between elements and also of the immediate electronic environment of each nucleus. Tables list values of m in units of b according to different conventions. For protons, the values are 5.5854 with m ¼ g_; 2.7927 with mz ¼ g_I for the z-component of spin I; or 4.8372 with m ¼ g_O [I(I þ 1)] for the total magnetic moment. The common information obtainable from an NMR spectrum is summarized in Table 1 and illustrated with the proton NMR spectrum of ethanol (CH3CH2OH) (Figure 1). The horizontal axis, which gives the ‘chemical shift’ of the signal, is a linear frequency scale in hertz, which is almost always converted to the dimensionless d scale (ppm) with tetramethylsilane (TMS) as reference at d ¼ 0.000. The d scale is independent of magnetic field strength and so is transferable from spectrometer to spectrometer. It is defined as di ¼ Dni =nref

½2


where nref is the absolute resonance frequency of TMS (e.g., 200.010078 MHz in a field of 4.70 T) and Dni ¼ ni  nref is the difference between the frequency of the nucleus of interest (ni) and nref. The vertical axis is a linear relative intensity scale that is generally proportional to the number of nuclei in the resonant

212 NMR SPECTROSCOPY / Principles Table 1 Basic information in an NMR spectrum Observable

NMR parameter

Symbol (units)

Chemical information

Peak position Peak multiplicity

Chemical shift Coupling constant

d (ppm) (absolute Hz) J (Hz)

Peak intensity Lineshape and linewidth

Integrated area Spin–spin relaxation time

Time dependence of intensity a

Spin–lattice relaxation time and nuclear Overhauser effect (nOe)

Functional group identification Number and type of nuclei within three bonds of the nucleus being detected Count of nuclei of given type Lifetime of spin state determined by dynamic processes and local magnetic environment Molecular dynamics and magnetic interactions

a

T2 ¼ (1/pDn) (s) Peak width at half-maximum intensity (Dn, Hz) T1 (s), nOe

Not shown in Figure 1. Requires repetitive experiments.

1200

600 J = 5 triplet (CH2)

0

J = 7 quartet (CH3)

J = 7 triplet (CH2)

J = 5 doublet (OH) TMS

CH3CH2OH − CH3 6.0

5.5

5.0

4.5

3.5 ppm

4.0 −CH2

H2O −OH

29 Si Satellites

25 Hz SSB SSB 6

5 1.27

4 0.44

SSB SSB 3

2.25

2 ppm

1 3.00

SSB SSB 0 1.08

Figure 1 200 MHz proton NMR spectrum of ethanol containing water (13 mol.%) and tetramethylsilane (TMS) as reference. The inset shows dry ethanol with OH coupling. Splitting diagrams are shown over each multiplet. The upper scale is in Hz and lower scale in d/ppm. SSB indicates spinning side-bands, which are separated from the main peaks by the spinner frequency. The 29Si satellites on the TMS resonance arise from 1H–29Si coupling. 29Si (1 ¼ 1/2) is present at 4% natural abundance. (Spectrum courtesy of Dr J. Hopkins.)

chemical group. More precisely, it is the relative magnitude of the magnetization precessing at the frequency of interest. The spectrum of ethanol shown in Figure 1 consists of five signals – a single sharp peak at d ¼ 0.0 from the TMS added as reference, a triplet at d ¼ 1.16 for the methyl group, a quartet at d ¼ 3.61 for the methylene group, and broad singlets at d ¼ 5.41 and 4.62, which are the signals from the alcohol OH and the water present as a ‘contaminant’. An integral curve is shown above the spectrum giving the integrated area under each peak. Each

integral is normalized to the methyl peak intensity of 3.0 and gives the proton count in each group. The fine structure within each multiplet is generated by the n protons within three bonds of the observed proton. The multiplet intensity pattern is governed by the statistical probability of the 2nI þ 1 possible spin arrangements of the neighboring nuclei. As I ¼ 1/2 for protons, n neighboring protons cause splitting into n þ 1 peaks with binomial intensity. The spacing within each multiplet is called the coupling constant, J (Hz), and the same spacing occurs in each multiplet for the pair of coupled

NMR SPECTROSCOPY / Principles 213 H3a H3e H1 CH3 H7

H5 H4e CH3

8

DO

CH3 H 9 2

H4a

H6e

8

10

10 9

H6a

3a 1

7

6e

4e3e

2 6a 4a

5

40 µg in 5 s

4 µg in 10 min

400 ng in 10 h 3.4

3.2

3.0

2.8

2.6

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

0.8

ppm

Figure 2 500 MHz 1H NMR spectrum of menthol (OD) in CD2Cl2. The spectra show the effects of sample size and signal averaging. Note the growing importance of solvent impurities as the sample size decreases. (Spectrum courtesy of Varian Associates.)

groups. For ethanol, the methyl group is split into a 1:2:1 triplet with J ¼ 7 Hz by the neighboring CH3 group. Similarly, the methylene peak is split with the same spacing into a 1:3:3:1 quartet by the adjacent CH3 group. In the normal spectrum there is no coupling observed between the CH2 and OH groups because the OH bond is being broken several times per second by exchange with the water. In ultrapure ethanol (inset in Figure 1) when the OH exchange is slow, the CH2 group appears as a doublet (5 Hz) of quartets (7 Hz) and the OH signal is a 1:2:1 triplet with J ¼ 5 Hz. The methyl triplet is unchanged as it is separated by four bonds from the OH. Peak linewidths may be used to measure the spin life time. In Figure 1, the linewidths of the water and alcohol OH peaks are substantially broader than those from protons bound to carbon. The broadness shows that the water protons and alcohol OH are exchanging on the order of 10 times per second. This example illustrates the power of NMR. About 0.5 ml of colorless liquid, contained in a glass tube, has been identified as ethanol containing B15% water. Less than 0.2% of other protonated impurities are present and the kinetics of proton

exchange was estimated. The whole experiment took a few seconds. Sensitivity is always an issue for NMR analysis. Millimolar solutions and milligram quantities are required for most applications. The effect of sample size on spectral quality is shown in Figure 2. Spectral analysis has been achieved on 10 mg of a compound in 0.3 ml of solution and spectra of 100 ng samples have been recorded and interpreted.

NMR and the Periodic Table The NMR periodic table (Table 2, Bruker Almanac (2003) Karlsruhe: Bruker Biospin) provides the nuclear properties for the main NMR active isotopes. Nuclear Spin Quantum Number (I )

In a magnetic field, a nucleus exists in 2I þ 1 possible spin states. Protons and neutrons have I ¼ 1/2. The vector sum of coupled proton and neutron spins gives the nuclear spin quantum number for each nucleus. Isotopes with an even number of both protons and neutrons may have zero spin and are NMR inactive (e.g., 12C, 16O).

Table 2 List of elements showing the nuclear properties of the main NMR-active isotopes

NMR SPECTROSCOPY / Principles 215 Quadrupole Moment (Q, 10  24 cm2)

w ¼ e2 Qqzz =h ðHzÞ

½3


where Q is the intrinsic electric quadrupole moment and eqzz measures the effective electric field gradient at the nucleus induced by local bonding. In a symmetrical environment, e.g., 14N in the NH4 cation, q and hence w may be zero. For monovalent elements, e.g., 2H and 79Br, the electric field gradient is directed along the bond axis and is nonzero. The spectra of quadrupolar nuclei with large Q values are usually very broad since the electric field gradient is modulated by molecular motion and the nuclear relaxation time becomes very short. The linewidth is inversely proportional to the spin lifetime, and linewidths of the order of kilohertz to megahertz are common. The spectra of spin 1/2 nuclei are influenced by coupling to quadrupolar nuclei. The spectra of 14N–H protons are often broad because the rate of interchange of the 14 N spin states is of the same order of magnitude as the 14N–H spin–spin coupling. This contrasts with O–H protons, where chemical bond breaking is responsible for line broadening. For CDCl3, D has a small Q and the carbon is split into a 1:1:1 triplet, but 35Cl has a large Q and relaxes very rapidly, and so no coupling between 13C and 35Cl is observed. In solid samples, the magnitude of w depends on the orientation of the electric field gradient with respect to B0. As a consequence, deuterium NMR has been developed as a sensitive probe of molecular motion in materials over the 0–125 kHz range. A novel probe capable of rotating a sample about two different axes simultaneously has been developed to obtain narrow peaks for quadrupolar nuclei in solids. Quadrupolar nuclei give narrower signals at higher fields, and pulse sequences have been developed to further narrow lines.

200

−1/2(H)

Benzene TMS

100 Energy,  (MHz)

Isotopes with I41/2 are called quadrupolar nuclei. The nucleus responds to both magnetic and electric fields. Quadrupolar nuclei respond to the electric field gradient at the nucleus through the quadrupole coupling, w:

200.0 0

−1(D) −1/2(C)

200.0

30.7 50.3 30.7

0(D) +1/2(C) +1(D)

−100 TMS −200

Benzene

+1/2(H)

4.7 Field strength,
0 (T) Figure 3 The spin energy diagram for 1H, 13C, and 2H (I ¼ 1) as a function of field strength against energy (E ¼ hn) showing allowed transitions at 4.698 T. The energy difference between TMS and benzene protons has been exaggerated to show that, at fixed field, TMS resonates at lower frequency, while at fixed frequency, TMS resonates at higher field. The orientation of the nuclear moments and spin populations (eqn [4]) are shown on the right of the diagram.

Natural Isotopic Abundance and Relative Sensitivity

The product of isotopic abundance and relative sensitivity determines the overall sensitivity to detection of the isotope relative to hydrogen with its 99.985% natural abundance and unit relative sensitivity. Low g nuclei may be quite difficult to detect as sensitivity is proportional to g3 (e.g., 57Fe which is 3  10  5 times less sensitive than protons for equal numbers of nuclei). Sensitivity may be enhanced by repetitive spectral accumulation as the signal-to-noise (S/N) ratio is proportional to the square root of the number of accumulations. It requires 109 accumulations to give a 57 Fe signal 1/50 the S/N of protons – a daunting task.

Nuclear Resonance Frequency (m, MHz)

The NMR Experiment

Table 2 lists the resonance frequencies of each nucleus in a magnetic field of 4.698 T. Protons resonate at 200.0 MHz at this field. The frequency values are equivalent to g as g ¼ 2pn/B0. For protons g ¼ 26.7519  107 rad T  1 s  1. The resonance frequency in other fields is obtained by multiplying by the ratio of the fields. The proton resonance frequency is often used synonymously with magnetic field strength, as illustrated in Figure 3.

The essential components of an NMR spectrometer are an exceptionally stable magnet, a tunable source of RF, a high-power broadband amplifier, a highsensitivity resonantly tuned coil surrounding the sample, and a computer for rapid data acquisition and analysis. Samples may be solids, liquids, or gases. Solutions (millimoles to moles per liter) are most widely used. There are four basic steps in an NMR experiment, as described below.

216 NMR SPECTROSCOPY / Principles Generation of Magnetization

When a sample is placed in a magnetic field the nuclei immediately experience a torque and begin to precess at their Larmor frequencies (n) in the plane perpendicular to the magnetic field direction. The nuclear spins are oriented according to their magnetic moments. The energies of each of the 2I þ 1 energy states are established very rapidly (the timescale is 1/n). Initially, each spin state will be equally populated as the levels were degenerate without an applied field. Energy exchange with the surroundings occurs as spins flip. The spin temperature of the sample drops from infinity (equally populated states) until a Boltzmann distribution of spin populations is established at equilibrium: ni =n0 ¼ expðDE=kTÞ ¼ expð_gB0 =kTÞ

½4


The approach to equilibrium is a first-order rate process and is exponential in time (cf. Newton’s law of cooling). The time constant is the spin lattice relaxation time, T1 (s). The T1 of water at 200 MHz is B15 s. The resultant Z magnetization – Mz ¼ gh(n0  ni) arising from the excess population in the lower spin state – is then available for manipulation in the experiment (see Figure 3). As DE for NMR is of the order of millijoules (B2 mJ mol  1 for protons at 4.7 T), the ni/n0 ratio is very close to 1. However, it is this excess population in the lower state (n0  niB3 per 105 for protons) that generates the NMR signal. High g nuclei, high fixed field, and low T all favor a larger population excess and hence higher sensitivity.

small to influence sample temperature. A high-energy RF pulse does induce substantial heating, especially if the solvent has a high dielectric constant or the sample is highly conducting. However, the heating is electrical, not magnetic. Most importantly, a new coherent spin motion is created in the x–y plane perpendicular to B0. At equilibrium there is no x–y field and so no x–y magnetization. When the B1x field is applied, a new coherent precessional motion occurs about it and x–y magnetization is detectable (see Figure 4). The spin magnetization rotates about the combined B0 static field and the B1 RF field oscillating at n0. The description of the spin motion is simplified by transformation to the rotating frame where B1x0 is static since the rotating frame frequency is chosen to be the RF oscillator frequency. When the RF pulse is applied the magnetization rotates about B1eff for the pulse duration (tp, s). The effective B1 field is the vector sum of B1x0 and Bz0 ¼ ðni  n0 Þ=g. Pulses are usually described in terms of the angle y through which a resonant nucleus would rotate in time tp: y ¼ gB1 tp

½5


There are two important pulses: a 901 pulse, which creates maximum magnetization in the x–y plane and hence maximum signal; and a 1801 pulse, which inverts the spin population present immediately prior to the pulse (see Figure 5). The 1801 pulse is particularly important for the accurate determination of T1 and T2. The 1801 pulse initiates time reversal for the x–y magnetization resulting in a Hahn spin echo.

Resonant Energy Absorption and Coherence Creation

Detection

The spins resonantly exchange energy with an oscillating RF electromagnetic field applied through a tuned coil. In pulse Fourier transform (FT) spectroscopy the RF field oscillating at n0 is applied as a large-amplitude, short pulse with magnetic field strength 2gB1. For a typical spectrometer, gB1 is of the order of 25 kHz and the pulse duration is 10 ms. This establishes the width of the spectral range which can be excited by such a pulse at B712.5 kHz (7gB1/2). Radiofrequency radiation induces up or down spin flips with equal probability. The number of quanta exchanged between the spins and the RF coil is enormous. Consequently, the overall behavior of the system is statistical and can be described classically. Energy is absorbed by the spins provided there is an excess population in the lower state at the time the RF is applied. This spin energy change is far too

The x–y magnetization created in the excitation step is minute in absolute terms. Thus it is essential that detection is separated in either time or space from excitation. The magnetization precessing at each nuclear resonance frequency induces an alternating current in the receiver coil, which is closely wound around the sample. In Bloch’s original work separate transmitter and receiver coils were mounted orthogonally to one another to provide spatial isolation of the detector from the exciting RF field, which was applied continuously. The magnetic field was swept to bring signals into resonance with the fixed frequency detector in this continuous-wave (CW) NMR experiment. In a modern pulse spectrometer, the same coil is used for excitation and detection. With pulse excitation, the detected signals are separated in time. A few microseconds after the high-power excitation has been turned off, the detector is gated on.

NMR SPECTROSCOPY / Principles 217

Mz


0

X

Y

(A)

X MX−Y Y

(B) Figure 4 The precessing spins and net magnetization (A) at equilibrium and (B) following a 901 pulse. Note the equalization of populations and generation of x–y coherence. (Reproduced with permission from Campbell ID and Dwek RA (1984) Biological Spectroscopy. Menlo Park, CA: Benjamin/Cummings.)

The multiple frequencies present are resolved by signal processing. Signal Processing

The oscillating output current from the receiver coil is amplified and mixed with the original excitation

signal. This phase-sensitive detection of the signal greatly improves the S/N ratio. The resulting signal oscillates at the difference between the transmitter frequency and the NMR resonance and is now at audiofrequencies, usually o20 kHz. The multicomponent time-decaying analog signal (free induction decay, FID, or Bloch decay) is digitized into amplitude and time values in a computer (Figure 5). The process of Fourier transformation converts the FID into its frequency-equivalent spectrum. The spectrum is computed with NP points as a sum of cosine waves (real) and a sum of sine waves (imaginary) in the frequency domain, each containing NP/2 points. The Nyquist requirement for a minimum of two data points to define a wave sets the frequency range. The best digital resolution – smallest resolvable frequency difference between adjacent peaks – is set by the total acquisition time, AT (Dnmin ¼ 1/AT). This corresponds to resolving two waves whose frequencies differ by a single data point during the acquisition. There is a constant sampling time for each data point in the FID called the dwell time (DT). The finite time for signal sampling sets the highest frequency (fmax), which can be accurately represented such that fmax ¼ 1/2DT. The amplitudes of any higher frequencies (including noise!) are digitized identically to frequencies in the window and appear at n ¼ ni  nmax (fold back). Ubiquitous white noise contains all possible frequencies. For realistic S/N, electronic filters with defined frequency bandwidth are essential to limit fold-back of noise from frequencies outside the spectral window. Prior to Fourier transformation, a variety of digital weighting functions may be applied to improve the spectral appearance (Figure 5). For example, a matched filter – an exponentially decaying function with a time constant equal to the T2 decay time of the NMR signal – provides S/N optimization. Resolution enhancement is achieved at the expense of S/N by intensifying the long time tail of the FID. In addition to the standard fast FT algorithm, other data processing methods may be used for the time-tofrequency conversion. These include linear prediction – especially useful for truncated FIDs with few data points; Bayesian statistics; and maximum entropy, which are advantageous in special circumstances but with the cost of increased calculation time and memory. More elaborate acquisition schemes, which employ unequal time step functions, may be used to shorten spectral acquisition times for multidimensional experiments but at the expense of specialty data processing routines. The absolute phase of the real and imaginary halves of the FT output cannot be established a priori. The experimental phase depends on the phase lags that occur inevitably in electronic circuitry. The real

218 NMR SPECTROSCOPY / Principles AT FID

Signal intensity

exp (−t /T 2*) Digitize and store data

Plot spectrum

Intensity Intensity vs. vs. time frequency

Time (t )

90°

Computer 2356 101 −716 211 −612 FT 1126 1014 ⇒ 888 310 91

RF pulse response

Signal intensity

tp RF pulse

W1/2 =1/πT 2*

Frequency

FID

Spectrum intensity

FT

360°

180°

Multiply by exp (− at )

Weighted FID FT

Weighted FID 270°

Multiply by 2 exp (at − bt )

FT

Increasing pulse time (tp)

Figure 5 The basic pulsed NMR experiment showing essential processes and times. The spectrum is excited with a short resonant RF pulse of length tp. The FID oscillates at n1 and decays with time constant T*2, which determines the linewidth and frequency following digital data acquisition and Fourier transformation. The influence of increasing pulse angle (pulse duration tp) is shown at lower left. For steady-state spectral acquisition, the pulse length is set for the Ernst angle (cos aE ¼ exp(  AT/T1)). The effect of digital filtering functions applied to the FID prior to FT is shown in the lower right. The exponential filter, exp(  at), discriminates against highfrequency noise at the expense of line broadening; the shifted Gaussian, exp ( þ at  bt2, generates sharp lines at the expense of the signal-to-noise ratio. (Reproduced with permission from Field LD and Sternhell S (1989) Analytical NMR. Chichester: Wiley. 1989; & John Wiley & Sons Ltd.)

and imaginary components must be empirically phase shifted so that the real series is in pure absorption (cosine) and the imaginary series is in pure dispersion (sine). This phase correction is optimized by maximizing the peak integral that corresponds to pure absorption. The integral for pure dispersion is zero. Furthermore, if data acquisition is delayed for a set of waves that begin with a common phase, they will dephase at a rate that is proportional to their frequency difference. Because acquisition delays appear regularly in FT-NMR experiments, a second frequency-dependent phase correction is usually required. The frequency independent phase correction is always in the range 0–3601. Much larger values may be found for the frequency-dependent correction, which is directly proportional to the acquisition time delay and the frequency offset. Finally, all the intensity information in an FID is contained in the first data point as no new magnetization is created

during acquisition. This intensity is resolved into its individual frequency contributions during the acquisition time.

Chemical Shift The chemical shift of a nucleus measures its immediate electromagnetic environment. The local circulation of electrons around the nucleus, induced when the sample is placed in the external magnetic field, modifies the local magnetic environment of the nucleus, resulting in the chemical shift. Its absolute magnitude is directly proportional to B0 but is only a tiny fraction of it. The range of chemical shifts varies enormously from B1 ppm for 3He through 20 ppm for 1H and 2H (isotopes have the same shifts) and 300–1500 ppm for main group elements 13C, 11B, 15 N, and 19F to over 10 000 ppm for transition metals such as 59Co and 119Sn and the rare-earth elements.

NMR SPECTROSCOPY / Principles 219

Detailed correlations of chemical shifts with functional groups may be found in articles dealing with particular nuclei. The chemical shift literature represents an interesting blend of philosophical approaches. Since frequencies are measured with great accuracy and linewidths are narrow in solution, chemical shifts are exquisitely sensitive reporters of local molecular environment. For example, the relative stereochemistry of the methyl groups, separated by up to five monomer units (13 bonds) along the chain, in polypropylene can be distinguished by carbon NMR (see Figure 6). Empirical group additivity rules have proved to be effective predictors of chemical shifts for many classes of compounds. Extensive spectral tables are available in many texts and are very useful for structure identification provided representative model compounds are available (e.g., tripeptides for proteins, additivity rules for 13C in organics). The ad hoc explanations that accompany such assignments often have little basis in reality. For accurate interpretation of chemical shift trends, the individual components of the chemical shift tensor must be considered. Molecular orbital calculations on systems of up to

rrrr mmmm mmrm + mmrr rmrr rmrrrm

mmmr

rmrrrr mmrrrm rmmr

22

rmrrmr rmrrrmm mmrrmm

rmrm

21 ppm versus TMS

20

Figure 6 The 90 MHz methyl 13C spectrum of atactic polypropylene in 1,2,4-trichlorobenzene at 1001. Empirical shift predictions for different stereoisomers reflecting the meso (m) or racemic (r) relative orientation of neighboring methyl groups. (Reprinted with permission from Schilling FC and Tonelli AE (1980) Carbon-13 nuclear magnetic resonance of atactic polypropylene. Macromolecules 13: 270; & American Chemical Society.)

50 first-row atoms may now be done with close to experimental accuracy. Small solvent and related effects may be useful for resolving spectral overlap, particularly when magnetically anisotropic solvents are used such as pyridine and acetone. Common NMR solvents are listed in Table 3. Anisotropic circulation of electrons in molecular groups, e.g., aromatic rings, ring currents, and shift reagents, may induce chemical shifts. The relative orientation and distance to the anisotropic group control the magnitude. This anisotropy induced shift is most important for protons as its magnitude is independent of the nucleus observed. Chemical shift is measured on a ppm scale relative to a standard of known frequency. TMS is the practical frequency standard for 1H, 13C, and 29Si. In principle, the absolute frequency reference is the electron-free nucleus. For theoretical work, the shift of the neutral diamagnetic atom is used as it may be calculated accurately. Alternatively, the absolute scale may be established independently of the NMR experiment from the rotational splitting constant measured by microwave spectroscopy on a reference gas, e.g., CO for 13C and 17O. In Figure 7, the chemical shift/shielding scale for 13C is shown together with the terms used to describe changes in shift. Chemical shift is a tensor quantity whose three principal elements can be determined for molecules with a small number of unique resonances from the turning points of the shielding pattern in solid samples. Multidimensional methods are required for resolution in more elaborate spectra. The isotropic chemical shift observed in solution is the simple average of the principal components measured in the solid. Thus, siso ¼ ðsxx þ syy þ szz Þ=3. Chemical shift is a highly localized interaction. Shielding is induced by electron circulation in the plane perpendicular to the applied magnetic field direction and so changes with molecular orientation. The original Ramsey perturbation theory of chemical shift defines the essential chemical shift components:
 
2
x þ y2
e2

0
r 3
0 2mc2      eh 2 X h0jLz jni nj2Lz =r3 j0 þ 2mc En  E0 n   0j2Lz =r3 jn hnjLz j0i þ En  E0

szz ¼ sdzz þ spzz ¼

½6


The first term is the diamagnetic contribution (sd, upfield, more shielded), which arises predominantly from the induced circulation of the inner-shell electrons generating a magnetic field opposed to the applied field. It increases monotonically with atomic

Table 3 Useful properties of NMR solvents Solvent

Acetic acid-d4

1

H chemical shift a (ppm from TMS) (multiplicity)

JHD (Hz)

11.65 (1) 2.04 (5)

2.2

2.05 (5)

2.2

1.94 7.16 7.27 1.38 4.80 4.81 8.03 2.92 2.75 2.50 3.53 5.29 3.56 1.11 4.87 3.31 5.32 8.74 7.58 7.22 3.58 1.73

2.5

Acetone-d6 Acetonitrile-d3 Benzene-d6 Chloroform-d Cyclohexane-d12 Deuterium oxide N,N-Dimethylformamide-d2

Dimethyl sulfoxide-d6 p-Dioxane-d8 Ethanol-d6

Methanol-d4 Methylene chloride-d2 Pyridine-d5

Tetrahydrofuran-d8

(5) (1) (1) (1) (DSS) (TSP) (1) (5) (5) (5) (m) (1) (1) (m) (1) (5) (3) (1) (1) (1) (1) (1)

1.9 1.9 1.9

Trifluoroacetic acid-d Trifluoroethanol-d3 a

(m) (1) (m) (5) (1)

5.02 (1) 3.88 (4  3)

C chemical shift a (ppm from TMS) (multiplicity)

178.99 20.00 206.68 29.92 118.69 1.39 128.39 77.23 26.43

163.15 34.89 29.76 39.51 66.66

(1) (7) (13) (7) (1) (7) (3) (3) (5)

(3) (7) (7) (7) (5)

56.96 (5) 17.31 (7) 1.7 1.1

Toluene-d8 7.09 7.00 6.98 2.09 11.50

13

2.3

2 (9)

49.15 (7) 54.00 (5) 150.35 (3) 135.91 (3) 123.87 (5) 67.57 (5) 25.37 (1) 137.86 (1) 129.24 (3) 128.33 (3) 125.49 (3) 20.4 (7) 164.2 (4) 116.6 (4) 126.3 (4) 61.5 (4  5)

JCD (Hz)

20 0.9 19.4 21 24.3 32.0 19

29.4 21.0 21.1 21.0 21.9

1

H chemical shift of HOD b (ppm from TMS)

Physical properties Density at 20 1C

Melting point ( 1C) c

Boiling point ( 1C) c

Dielectric constant

Molecular weight

11.5

1.12

17

118

6.1

64.08

2

0.87

 94

57

20.7

64.12

2.1

0.84

 45

82

37.5

44.07

0.4 1.5 4.8

0.95 1.50 0.89 1.11

5  64 6 3.8

80 62 81 101.4

2.3 4.8 2.0 78.5

84.15 120.38 96.24 20.03

3.5

1.04

 61

153

36.7

80.14

3.3 2.4 5.3

1.18 1.13 0.91

18 12 o  130

189 101 79

46.7 2.2 24.5

84.17 96.16 52.11

4.9

0.89

 98

65

32.7

36.07

1.5 5

1.35 1.05

 95  42

40 116

12.4

86.95 84.13

2.4–2.5

0.99

 109

66

7.6

80.16

0.4

0.94

 95

111

2.4

100.19

11.5

1.50

 15

72

115.03

5

1.45

 44

75

103.06

22 19 21.4 27.2 27.5 24.5 25 22.2 20.2 23 24 24 19

22

The 1H spectra of the residual protons and 13C spectra were obtained on a Varian Gemini 200 spectrometer at 295 K. The samples for the 1H and 13C spectra contain a maximum of 0.05 and 1.0% TMS (v/v), respectively. Since deuterium has a spin of 1, triplets arising from coupling to deuterium have the intensity ratio of 1:1:1. (m) denotes a broad peak with some fine structure. It should be noted that the chemical shifts, in particular, can be dependent on solute, concentration, and temperature. b Approximate values only: may vary with pH, concentration, and temperature. c Melting and boiling points are those of the corresponding compound (except for D2O). These temperature limits can be used as a guide to determine the useful liquid range of the solvents. Courtesy of Cambridge Isotope Labs, Andover, MA, USA, used by permission.

NMR SPECTROSCOPY / Principles 221 C6+CO

C

TMS CH4 s g

(A)

190 181 0 4 C1s22s22p2 1

D2

(B)

−71 261

0−6−13 194 203 C5+

TMS C1s2

C1s22s2

 (Experimental)  (Theoretical) C1s22s22p2

~600 km

2000 000 000

1

D0

0

Paramagnetic Deshielded Low field High frequency

190 201 

230

257 261

Diamagnetic Shielded High field Low frequency

Figure 7 The carbon chemical shift scale. (A) The experimental shift (dTMS ¼ 0.0) and theoretical shielding (s ¼ 0) scales for standard reference compounds. (B) The theoretical shielding scale for atomic carbon together with common shielding synonyms.

number. The second, paramagnetic, term is dominant for atoms containing electrons with orbital angular momentum (p, d, f, etc., but not s). It determines the induced paramagnetic circulation of electrons as the orbital angular momentum in the atom is quenched on bond formation. Linear molecules have particularly large chemical shift anisotropy, as only diamagnetic shielding is possible when the field and molecular axes are collinear. In the perpendicular orientation, large paramagnetic contributions are the rule. There are three essential components of eqn [6]. The r  3 distance term accounts for the common and generally valid correlation of chemical shift with charge density (e.g., 160 ppm/e for 13C in p systems). However, other terms can dramatically alter this simple correlation. For example, cations are known that appear upfield of anions, contradicting the simple charge prediction of cations at low field. The /0jLz jnS orbital angular momentum terms select for orbital angular momentum contributions such as px -py . In symmetric situations, this term can go to zero as it does in linear molecules leaving only the diamagnetic shielding. Finally, there is the excitation energy for electron promotion (En  E0), a consequence of the Ramsey perturbation treatment. This is best illustrated with transition metal shifts where a linear relationship is found between the ‘forbidden’ t2g–eg optical transition energy and the chemical shift. Note that this energy term is generally not the HOMO–LUMO gap (HOMO is the acronym for highest occupied molecular orbital, and LUMO the acronym for lowest unoccupied molecular orbital). Induced electron circulation generates the chemical shift and so the perturbation terms must represent an electronic excitation that changes the electron angular momentum in direct proportion to the applied magnetic field.

Proton chemical shifts of O–H and N–H groups are strongly dependent on solvent and temperature because of changes in hydrogen bonding. The shift difference between alkyl and hydroxyl protons is used as a thermometer (methanol below room temperature, ethylene glycol above). The 59Co shift in K3Co(CN)6 is an even more sensitive thermometer. For solids, the 207Pb shift in lead nitrate is the standard thermometer. In systems with unpaired electrons (e.g., metals, radicals, paramagnetic transition metal complexes), a much larger range of chemical shifts is possible. Now the major magnetic interaction is between the nucleus of interest and the unpaired electron(s). The observed shift depends on the excess electron spin population and the coupling constant to the nucleus. The induced contact shifts or Knight shifts (in metals and conductors) often exceed 1000 ppm.

Spin–Spin Coupling The presence of coupling between a pair of nuclei establishes molecular connectivity usually over one, two, or three bonds and is a very powerful tool for molecular structure determination. The internuclear coupling constant J is transmitted by the bonding electrons. Coupling through as many as 10 bonds has been detected in delocalized p systems. By tracing coupling paths (e.g., in COSY spectra), molecular connectivity can be established uniquely and unequivocally. In a sophisticated experiment (INADEQUATE), one-bond 13C–13C coupling in natural abundance is used to trace carbon connectivity in organic molecules. The magnitude of the three-bond coupling constant often provides stereochemical information through the Karplus relationship relating J to the internuclear dihedral angle (see Figure 8).

222 NMR SPECTROSCOPY / Principles

1

14

2

1

J HH (Hz)

12  H

10

C

C

1

10

H

1 1

8

8 6

6

4

4

2

2

1 1

0

0

40

80 120 Dihedral angle, 

160

1 1

0

F

J >0





J =0

J/4

J<0 J/4

3

4

3

4

3

4

J/4


J/4




J/4 2

1



J/4

2

1

2

1

J/4 J

J

32

41

J/4

2 3

1 4

10

6

8

45

4

20 35

56

5

70

1

15 35

126

120 210

1

10

15

84

1

6 10

21 28

36

1 3

4 5

7

9

3

6

1

21

7

56 126

28 84

252 210

1 8

36

120

1 9

45

1 10

1

Figure 10 Pascal’s triangle.

Figure 8 The Karplus relationship for three-bond proton–proton coupling relating dihedral angle and J. (Adapted from Campbell ID and Dwek RA (1984) Biological Spectroscopy. Menlo Park, CA: Benjamin/Cummings.)

H

1

1

16

J

J

23

14

Figure 9 Energy diagram and spectrum for a two-spin coupled system showing allowed transitions and effect of J coupling. The energy scale is severely distorted as the energy gaps are in the megahertz range and the coupling energies are in hertz.

The coupling constant measures the energy of interaction between two nuclei and is independent of the applied magnetic field. When the spin dipoles are opposed, a positive coupling constant denotes that the energy is lowered, and vice versa. Positive coupling is the norm, but negative coupling is routinely found between protons in CH2 groups. For two spins, such as molecular HF, there are four possible spin combinations each with a specified energy – both up (aa), both down (bb), and two with one up and one down (ab and ba) (see Figure 9).

The four-line spectrum arises from the four allowed transitions in which there is a single spin flip. They have equal probability and so equal intensity. In the absence of coupling the pair of transitions for a given nucleus is of equal energy and appears as single lines. When the nuclei are coupled, the transitions differ in energy by J Hz. The splitting is the same for each nucleus. If a nucleus is coupled to n other spin 1/2 nuclei, there are 2n possible lines for the observed nucleus. For n nuclei coupled with equal J (e.g., the three hydrogens of a methyl group) to the observed nucleus, the peak multiplicity is 2nI þ 1. They have binomial intensities corresponding to the statistical probability of the spins having a given total spin quantum number – the familiar Pascal’s triangle shown in Figure 10. Splitting patterns are multiplicative as shown by the CH2 group of dry ethanol which is a doublet of quartets (see Figure 1). Spectral splitting patterns reveal the quantum character of the nuclear coupling interactions. Two energy levels of a coupled spin system may not be closer in energy than the J coupling between the nuclei. The spectral consequences of these quantum effects include distorted line intensities (haystacking), line spacings that no longer equal J, and the appearance of a finite number of extra lines and in some cases fewer lines than predicted by first-order rules. The single line spectrum observed for benzene is one such example. Even though each proton is coupled to the five other protons on the ring, no splitting is observed. Equivalent nuclei do not split each other. The proton spectra of 1,1-difluoromethane and 1,1-difluoroethene are dramatically different (see Figure 11). In CH2F2, the pairs of protons are indistinguishable and the fluorines are split into a simple triplet. For CH2CF2 the protons are magnetically nonequivalent. They are distinguished by their different couplings to the cis and trans orientated fluorines. Despite the complexity of the pattern, the spectral symmetry reflects the symmetry in the

NMR SPECTROSCOPY / Principles 223

Some notable exceptions to the generalization have been observed in systems with three hydrogen ligands on a common metal center where tunneling dominates the coupling surface. In its full form J is also a tensor quantity and additional orientation dependent contributions involving orbital and spin–dipolar coupling interactions have been established, especially with multielectron atoms. In orientated samples (e.g., liquid crystals, solids), the through-space dipole–dipole coupling is combined with the J coupling. Because the dipolar coupling is often in the kilohertz range, J coupling is usually too small to detect in anything except fluid media. Residual dipole couplings measured in slightly oriented fluid medium find use in structure determination.

CH2F2

(A)

CH2 = CF2

Relaxation and Molecular Rotational Dynamics (B) Figure 11 60 MHz 1H NMR spectra of (A) CH2F2 and (B) CH2QCF2 showing the effect of magnetic nonequivalence. The pattern is identical in the 19F spectrum and is unchanged at higher B0. These spectra were obtained by sweeping the field at constant frequency as shown by the ringing distortion. (Reproduced with permission from Becker ED (1969) High Resolution NMR. Orlando, FL: Academic Press.)

coupled groups. Related effects are commonly observed in equivalently ortho and para disubstituted benzenes, and in X–CH2CH2 systems. All these spectra may be computed with high accuracy in programs such as LAOCOON. The input information is the chemical shifts and coupling constants. The line positions and intensities are then computed in a quantum mechanical calculation without approximation. The Fermi contact interaction is the major mechanism for J coupling. It is governed by the electron density at the nucleus (Si) and the nuclear gyromagnetic ratio according to the expression JðA; XÞ ¼

m 2 0

3p

b2 hgA gX jSA j2 jSX j2 P2AX =ðET  ES Þ ½7


PAX is the A–X bond order and ET  ES the electronic excitation energy for triplet formation. The magnitude of the 13C–1H one-bond J value increases with S character at carbon ( JC–H ethane ¼ 125 Hz, acetylene ¼ 210 Hz). Isotopic substitution is generally a minor perturbation on the electronic coupling and the observed J coupling is proportional to g. For example, JXH ¼

gH JXD gD

½8


Without relaxation, NMR spectra would be unobservable! NMR relaxation is a major tool for the determination of molecular dynamics on the microsecond to picosecond timescale. The nucleus is the ultimate molecular level sensor and is sensitive to motion on the scale of molecular fragments and small groups. The nuclear relaxation probe is most effective on timescales comparable to the inverse of the resonance frequency (1/o0), i.e., in the nanosecond to picosecond regime, which occurs most frequently in fluid solution. The energy absorbed by the spins from the RF excitation must be dissipated to the surroundings prior to repetition of the experiment. If not, the populations of each level will become equal (saturation) and there is no net magnetization to detect. Nuclei with I ¼ 1/2 are unaffected by electric fields and are only magnetically coupled to their surroundings. Thus, the broadening observed in electronic spectra from collisional processes is absent. As molecular magnetism is weak, spin relaxation times are usually long, ranging from milliseconds to kiloseconds. Relaxation is assumed to be a random dissipative process and follows an exponential decay with time constant T1. Just as for spin excitation, the dissipative relaxation process requires an oscillatory magnetic field at the resonance frequency (i.e., an energy match). A change in nuclear spin state is limited to unit change in resonant spin quantum energy and requires a change in angular momentum by one unit. The nuclear spin quantum is the smallest quantum of energy. The relaxation rate R1 ¼ 1/T1 is specified by the expression R1 ¼ CB2J(o) where C is a constant, B is the mean local magnetic field at the nucleus, and J(o) is the spectral density

224 NMR SPECTROSCOPY / Principles Table 4 Equations for spin–lattice relaxation rates, R1, of nucleus A, for various mechanisms in the extreme narrowing approximation Mechanism

R1

DD(intra) (homo)

m 2 3 o gA _2 tc =r 6 4p 2 4

DD(intra) (hetero)

m 2 o

4p DD(inter) (hetero)

Unpaired electron (intra) (dipolar)

g2A g2x _2 tc =r 6

m 2 2 o Nx g2A g2x _2 =Da 4p 15

m 2 4 o

4p

3

g2A g2e _2 SðS þ 1Þtc =r

Spin rotation

2Ir kTC 2 tsr =_2

CSA

2 2 2 2 g B Ds tc 15 A 0

Scalar coupling

8 p2 J 2Ax lx ðlx þ 1Þtsc 3 1 þ ðvx  vA Þ2 t2sc

Quadrupole

ðeqQ=_Þ2 tc

Raman scattering

 2 81p T ðF2 Þ2 10oD yD

Notes

Diagnostic

Examples

For a single pair of spin1/2 nuclei of separation r For a single pair of spin1/2 nuclei AX of separation r For paramagnetic relaxation of a spin1/2 nucleus by particle X, when X has unpaired spins For relaxation by unpaired electrons of total spin S at distance r For isotropic molecular moment of inertia (Ir) For nuclei with large CSA

Overhauser effect

Protons at low conc. In deuterated solvent 13 C in cholesterol, most protonated carbons Relaxation of water protons by dissolved O2 or Cr(AcAc)3

Relaxation by coupling to spin, X, of quantum number Ix. Requires gAEgX Relaxation of quadrupole by electric field gradient Relaxation of high Z spin 1/2 nuclei in solids

Overhauser effect

Proportional to Nx

EPR

Most radicals: relaxation agent Gd (DPM)3

Proportional to T

Gaseous CS2: most gases Quaternary 13C and 31 P in phosphates at high field 195Pt in square planar complexes 13 C in cyclohexylbromide ‘Unique’ to 13 C–79Br Almost all quadrupolar nuclei 17 O in compounds; halides except 19F 207 Pb in lead salts

Proportional to B02

Inversely proportional to B02

I41/2

Proportional to T 2 Independent of B0

AcAc, acetylacetonate; EPR, electron paramagnetic resonance; DPM, dipivaloylmethane; tc, Correlation time for molecular tumbling; NX, concentration of spins X (per unit volume); D, mutual translational self-diffusion coefficient of the molecules containing A and X; a, distance of closest approach of A and X; ge, magnetogyric ratio for the electron; C, spin–rotation interaction constant (assumed to be isotropic); Ds, shielding anisotropy (s8  s>); oD, Debye frequency; yD, the corresponding Debye temperature; F2, spin–phonon coupling constant. Adapted from Harris RK (1983) Nuclear Magnetic Resonance Spectroscopy. A Physicochemical View. London: Pitman Press.

function. The major local fields contributing to relaxation are shown in Table 4, together with representative examples and a method for confirming each contribution. For molecules in solution with I ¼ 1/2, T1 values are usually in the range 0.1–10 s. In the presence of unpaired electrons, and for quadrupolar nuclei, values of milliseconds to microseconds are found. For immobilized systems, such as solids, and spins isolated from other magnets, times of minutes to days have been measured. The probability of finding the molecular fragment oscillating at the resonance frequency o is determined by J(o). The oscillatory motion arises from random Brownian motion. The frequency distribution of Brownian motion is Gaussian and may cover

12 orders of magnitude for a mobile liquid: Z JðoÞ ¼

N

GðtÞeiot dt

½9


GðtÞ ¼ /f ðt þ tÞ f ðtÞS

½10


N

with

being the autocorrelation function. It measures the persistence of local fluctuations, f(t), and is assumed to decay exponentially with a time constant tc known as the correlation time. This gives JðoÞ ¼

2tc 1 þ o2 t2c

½11


NMR SPECTROSCOPY / Principles 225

The correlation time can be interpreted as the mean time for the molecular fragment to rotate through 1 radian. Alternatively, it is the inflexion point on the Gaussian distribution function describing the motion. For moderate molecular weight, approximately spherical molecules of volume V, in a solvent of viscosity Z, the Stokes–Einstein relationship tc ¼ VZ/kT applies surprisingly well. For small molecules, a semiempirical correction factor may also be needed when dealing with solvents whose volume is comparable to the solute. For anisotropically rotating groups (e.g., methyl groups, side chains in polymers, and substituted phenyl groups rotating about their C2 axis) a more detailed motional analysis is required. The relationship between the correlation time and relaxation time is shown in the log–log plot of Figure 12. On the short correlation time, fast motion side of the diagram where otc {1, an increase in fluidity makes relaxation less efficient, resulting in a longer T1. This is known as the ‘extreme narrowing’ regime and T1 is inversely proportional to tc. In this regime, fast molecular motion averages dipole– dipole coupling and chemical shift anisotropy to zero, giving narrow spectral lines. On the slow motion side of the relaxation maximum, otc c1, increased motion generates faster relaxation and T1 is proportional to tc. On the slow motion side molecular systems are usually solids or high molecular mass polymers and broad lines are the rule. A treatment in terms of random Brownian motion is often too simplistic for such materials. A broad distribution of correlation times and differential segmental motion descriptions are usually needed to account for the relaxation behavior. In solids it is not uncommon to find that nuclei in all molecular fragments relax with a single correlation time, which corresponds to the motion of the mobile portion of the molecule such as a rotating methyl group. The maximum relaxation rate occurs when otc ¼ 1 and R1 ¼ CB2 =2o0 . Relaxation efficiency decreases with increasing resonance frequency as B2 is usually independent of field. Chemical shift anisotropy relaxation is an exception as the magnetic interaction increases as the square of the magnetic field. NMR relaxation may be used to probe molecular motion in the kilohertz range, using the relaxation time in the rotating frame, T1r. Here the relaxation is determined in the presence of the RF magnetic field. In this case there is a nonzero x–y field given by the amplitude of the RF field (gB1). The relaxation rate maximum (1/T1r) occurs at a frequency o ¼ gB1/p. The T1 relaxation time is best measured in an inversion-recovery pulse experiment. A 1801 pulse is applied to the signal of interest that inverts the spin

4

100 MHz

2 Fast motion

Slow motion T1

25 MHz

0 log (T1/s)

Extreme narrowing

log (T2 /s) −2

−4

−6 −12

T2 −10

−8 −6 log (τc /s)

−4

−1

4

3 2.988

nOe (n +1)

25 MHz

2

100 MHz

1.17

Fast motion 0

−2

Slow motion T1 min −1

0

1

log ( c /s)

Figure 12 The relationship between carbon T1 and T2, relaxation times, and the correlation time for rotational diffusion, tc, for 13 C spin being dipolar relaxed by a bonded proton. Relaxation is less efficient at higher B0 field (indicated by carbon resonance frequencies). Curves for other relaxation mechanisms may be generated by appropriate scaling. The lower curve shows the nOe in the region of the T1 minimum where otcB1 and differential cross-relaxation is effective. Curves for other dipolar pairs may be constructed by scaling the limiting nOes according to eqn [15].

population. z-Magnetization is perturbed as far as possible from equilibrium. A composite pulse is recommended to ensure full inversion and that there is no residual x–y magnetization. A variable time delay

226 NMR SPECTROSCOPY / Principles

is introduced during which the z magnetization relaxes back toward equilibrium. A 901 pulse is then used to sample the residual z magnetization. The T1 is determined by fitting the recovery curve MðtÞ ¼ MðNÞ½1  2 expðt=T1 Þ


½12


where t is the variable time delay and M(t) is the magnetization at time t. It is necessary to wait at least 5  T1 between 1801 pulses to ensure that spin equilibrium is restored and M(N) is at its maximum value. This is a general requirement for repetition of experiments. It applies to any experiment that requires equilibrium spin populations initially.

common, while Gaussian and other more complex shapes are also found in poorly shimmed (inhomogeneous) magnets. Deviation from magnet ideality contributes to the shape of all peaks in the spectrum. Magnetic interactions also shorten spin lifetimes and contribute to line broadening. Paramagnetic impurities such as ferromagnetic particulates, dissolved oxygen, and transition metals are a common source of broadening. For large molecules – greater than 10 kDa – and in highly viscous media dipolar interactions between neighboring nuclei cause broadening when the slow tumbling in solution is insufficient to average the dipolar interaction. Chemical Exchange and NMR Timescales

Dynamics, Lineshapes, and Homogeneity Frequency and time are inversely correlated in any NMR experiment. In order to observe an interaction in the frequency spectrum, the nuclei must experience that interaction for a time that is long with respect to the inverse of the interaction frequency. Fast processes result in frequency averaging. Examples include isotropic molecular tumbling in solution which averages dipolar coupling to zero with the resulting narrow lines; magic angle spinning in solidstate NMR to remove chemical shift anisotropy; and molecular dynamics which provides conformationally averaged coupling constants. A solution NMR peak commonly has a Lorentzian shape. A Lorentzian line in the frequency domain results from a first-order exponential decay process with a time constant, T2 It is specified by the lineshape function, f(n), where f ðnÞ ¼

2T2 1þ

4p2 T22 ðni

 nÞ2

½13


Here T2 (s) is the spin–spin relaxation time. T2 can be estimated from the full linewidth at half-height Dn ¼ 1/pT2. For accurate work T2 is determined in a spin echo experiment, which cancels linewidth contributions from magnetic field inhomogeneities. In a highly homogeneous magnet the linewidths of small molecules, such as TMS, may be as narrow as 0.1 Hz. The linewidth specification for commercial solution spectrometers is commonly p0.3 Hz across a 5 mm sample, independent of absolute magnetic field. Note that this requires a magnetic field that is stable (see eqn [1]), and uniform (homogeneous) to better than 1 part in 109 across the sample – a remarkable engineering feat. The sharpest line in a spectrum provides a measure of the actual instrument homogeneity. In practice, ideal Lorentzian lineshapes are

The intrinsic linewidth of a peak measures the lifetime of a spin in a given configuration. Lifetimes may depend on chemical processes such as the proton exchange between water and the ethanol OH shown in Figure 1. In such cases, an order of magnitude estimate of the kinetics of the exchange process can be made using the uncertainty principle DEDtBh=2p or

DnDtB1

½14


For the OH exchange, Dn1/2 for water was measured to be 25 Hz and the exchange rate is B75 s  1 for this OH group. The hydroxyl proton line of ethanol is significantly narrower. This is because the lifetime of the proton bound to alcohol is longer than its lifetime when bound to water, as required for chemical equilibrium. The exchange lifetimes are directly proportional to the population at each site. If the rate is very much faster than the chemical shift difference in hertz, then a single sharp line is observed at the population-weighted average shift. Accurate kinetics may be determined from exact lineshape calculations. NMR has a unique ability to reveal ‘invisible’ chemically degenerate processes. Examples include proton exchange rates in water, bond rotations, and ligand interchanges in complexes. Intramolecular processes may be distinguished from intermolecular processes by the presence of averaged coupling constants. Chemical kinetics may also be measured by saturation transfer. Spin population information is transferred from site to site by chemical exchange. Saturation at one site will appear at a new site provided exchange is faster than T1 relaxation. Diffusion

Molecular diffusion (and flow) may be measured in the presence of magnetic field gradients using the Stejskal–Tanner spin-echo technique. Diffusion in a gradient dephases the magnetization between pulses. The technique may be incorporated into

NMR SPECTROSCOPY / Principles 227

multidimensional pulse sequences, e.g., DOSY, to discriminate signals in mixtures on the basis of diffusion coefficient/molecular size. Combined nOe and diffusion experiments provide information on binding small molecules to large molecules, which finds application in drug discovery. Sample Spinning, Field/Frequency Lock, and Field Gradients

Sample spinning about the z-axis averages the static x–y magnet inhomogeneities. In poorly shimmed magnets, spinning side-bands are found separated from the main peak at multiples of the spinning frequency. Ideal lineshapes may be recovered by computer deconvolution to remove the effects of magnetic inhomogeneity if an ideal reference line is available. Most spectrometers also employ a field/ frequency lock to overcome effects of drifting fields. A signal from a reference – commonly deuterium in a deuterated solvent – is detected in dispersion mode. Magnet drift produces a signal which is used in a feedback loop to compensate for the drifting field. Pulsed field gradients are used to shorten experimental times in multidimensional NMR by removing the T2 limit on pulse repetition times. As spin frequency is proportional to the applied field, field gradients are used to discriminate between multiple quantum spin orders in multipulse experiments, e.g., DQ filtered COSY experiments that discriminate against the strong diagonal signals arising from single quantum excitation.

Double Resonance and Decoupling Coupled spectra contain a wealth of bonding information. However, in many situations, the information content is more readily interpreted if this spectral complexity is removed by decoupling. Decoupling is achieved by applying a second RF field at the resonance of the nucleus whose coupling is to be removed. The decoupling may be selective, involving a single group or even a single line within a group, or may be broadbanded and cover the full spectrum. The amplitude of the decoupler field must be larger than the coupling constant to the nucleus to be decoupled. If lower fields are used, or the frequency is offset, additional lines and line broadening may be observed (see Figure 13). A low-power RF pulse applied selectively may be used for specific inversion of populations. Broadband proton decoupling is used routinely when carbon spectra are obtained. For complete decoupling, the observed spin and the decoupled spin must be orthogonal. J coupling is a scalar, and the

coupling contribution to the energy is determined by the dot product of the spin angular momenta J.I.S. Alternatively, but not equivalently, spins may be decoupled by randomizing their orientation relative to one another in a time which is fast with respect to 1/J. Many processes randomize spin orientation: rapid T1 relaxation which decouples quadrupolar nuclei, chemical exchange which breaks bonds, and by noise decoupling in which the frequency of the irradiation is rapidly and randomly varied around the resonance frequencies to average J.I.S. to zero. More sophisticated multiple pulse versions for broadband decoupling exist, which minimize the power levels required (e.g., WALTZ, GARP, MLEV 17, etc.). Since an additional oscillating magnetic field is present while the decoupler is on, the irradiated spins have new coherent motions and new energy states. The decoupler is also an energy source and so spin populations are perturbed. While the new energy states and line positions are generated immediately, spin population changes still require times comparable to T1 and T1r. The most important population changes occur in dipolar coupled systems and give rise to the nuclear Overhauser effect (nOe). Pulse sequences have been developed to exploit the time differentials to obtain decoupled spectra without nOe for spin counting or coupled spectra with nOe to enhance sensitivity. Nuclear Overhauser Effect

The dipolar relaxation mechanism is unique because the dipolar coupling interaction contains two spin terms involving mutual spin flips – zero quantum mk$km and double quantum mm$kk transitions. All other relaxation processes are limited to single spin interchange. While the decoupler is on, the spin population differences between the levels irradiated are equalized as the rate of energy input from the decoupler greatly exceeds the outflow by relaxation. In dipolar coupled systems, the availability of double and zero quantum relaxation pathways (cross-relaxation) produces non-Boltzmann spin populations in the energy levels of the observed nucleus. These perturbed populations are measured as the nuclear Overhauser effect (nOe) effect (Z). The relationship is Z¼

Rc gS Mirr ¼ 1 R1 þ Rx gI M0

½15


where Rc is the cross-relaxation rate between spins I and S. I is the observed and S the irradiated nucleus. R1 is the dipolar relaxation rate and Rx is the relaxation rate for all other relaxations (Table 4). Mirr and M0 are the integrated intensity of I in the presence

228 NMR SPECTROSCOPY / Principles EFFECT

TECHNIQUE Low power−spin tickling (homo or hetero)

Actual appearance depends on sample

INDOR (home or hetero) 1

H−13C

Observe frequency fixed on top of a proton line. Decoupler swept through the

13

Decoupler frequencies correspond to 13C transitions

C region

Medium power−selective decoupling (home or hetero) 13

1

H

13

C

C

High power−proton noise decoupling (hetero)

HA

HB

HC

CA

CB

CC

CA

CB

CC

CC

CA

CB

CC

Off−resonance decoupling (hetero) 2000 Hz

HA

HB

HC

CA

CB

Figure 13 Summary of double resonance techniques and their influence on the spectrum. (Reproduced with permission from Abraham RJ, Fisher J, and Loftus P (1988) Introduction to NMR Spectroscopy. Chichester: Wiley; & Wiley.)

and absence of irradiation of S. For extreme narrowing when zero, single, and double quantum dipolar relaxations contribute, Rc/R1 ¼ 1/2. In the slow motion limit when only zero quantum terms apply, Rc/R1 ¼  1. The nOe dependence on correlation time was illustrated in Figure 12. The detection of an nOe is diagnostic of dipolar relaxation and usually implies that nuclei are within 0.5 nm of each other. The relative rate of alternative relaxation processes sets the distance cutoff. If differential nOe values or

the rate of change of nOe can be measured, relative distance information may be obtained. A full relaxation analysis is necessary for accurate distance measurement. Polarization Transfer

The ability to transfer population information via dipolar coupling and double resonance is particularly useful for enhancing the detectability of insensitive

NMR SPECTROSCOPY / Principles 229

nuclei. The low-sensitivity nuclear resonance is irradiated, the information transferred via the coupling, and the sensitive nucleus is detected. Over 10% of the total spins in gaseous helium and xenon can be polarized through dipole coupling with rubidium atoms that have been optically pumped with a laser. The spin relaxation time for rare gases at low pressure can be of the order of hours. This enormous signal enhancement and the great sensitivity of xenon chemical shifts to local environment has been used to probe surfaces and also for MRI. Remarkable polarization transfers approaching 50% have been achieved by microwave irradiation of organic radicals. This technique allowed detection of 13C labeled urea by MRI.

Quantitative Analysis At first sight, quantitative analysis by NMR might be expected to be highly accurate and broadly applicable since the ‘extinction coefficient’ for NMR is unity. In practice, it is found that the best accuracy achievable in extensive studies of proton NMR of known mixtures is 71%. Results at this level of accuracy require very careful attention to the selection of the experimental parameters such as pulse width, pulse repetition time, and nOe effects and the availability of calibration mixtures. For 13C analysis the accuracy is commonly poorer, B1–5%, because of the reduced S/N ratio and the variation of nOe contributions with molecular structure and field strength. Errors of 710% or more are not uncommon in quantitative analyses especially when conducted by inexperienced operators. In proton NMR, linewidth variation is an underappreciated issue. Careful control of experimental parameters, and the use of calibration mixtures are minimum requirements for accurate analyses and accuracies of 70.5% have been achieved by experts. A fundamental characteristic of Lorentzian lines is their slow return to baseline as a function of frequency offset from resonance. Thus, broad lines such as the OH peak in ethanol, and especially intense solvent peaks, contribute intensity throughout a spectrum. The decay of the signal to the baseline is linearly dependent on the linewidth at half-height. For example, to obtain an integral corresponding to 99.0% of the total peak intensity of a Lorentzian line, one must integrate over 63.6 linewidths. This value is obtained from f(n) – the Lorentzian function – eqn [13]: Z Int ¼ K

n

n

f ðnÞ dn

½16


where 7n are the integration limits and K is the instrument response factor. If the frequency scale is normalized in units of the full-peak linewidth at halfheight, the integral takes the form Z Int ¼ 2K

xL

f 0 ðxÞ dx ¼ 2ðK=pÞ arctanðxL Þ

½17


0

where xL is the integration cutoff in units of linewidths. Precise integrals require that the cutoff limit be equal for every peak. Further, because of the variation of long range 13 C–H couplings and multiplicities, the extent of incorporation of satellite contributions within the linewidth cutoff ranges will vary from group to group. Solvents often contain trace amounts of impurities, especially water, and can cause overlap problems, as can their isotopic satellites.

Magnetic Resonance Imaging, Field Gradients, and Diffusion Application of magnetic field gradients adds spatial dimensions to the standard NMR experiment. The spatial location of spins – often the protons of water – is determined by making B0 spatially dependent across the sample with additional defined field gradients. The local field, which varies from point to point, determines the resonance frequency. This is the basis of MRI. Field gradients may be applied in three dimensions and generate sliced 3D images of objects such as the brain (Figure 14) and physical processes such as tablet dissolution (Figure 15). Data processing allows slicing through any desired plane. In early experiments static field gradients were used. In the simplest experiment the object was orientated at specific angles with respect to the field and the image intensity was projected onto the gradient axis. The image was reconstructed by a back projection computation. More recently pulsed field gradients are used and the image encoded in a frequency- and phase-modulated 2D array with intensity providing the third dimension. It is now possible to obtain useable images in a few seconds with such methods. The resolution in an image is determined by the number of data points per dimension. The volume elements are called voxels. Their number is usually limited by the total acquisition time and the S/N ratio desired. The intrinsic limit for imaging of living systems is the water diffusion distance B50 mm. In special situations 5 mm resolution has been achieved. Practical images of flow, including blood flow, have been obtained. Polarized 3He provides images of breathing function in lungs.

230 NMR SPECTROSCOPY / Principles

Figure 14 (A) Planning images for magnetic resonance spectroscopy (MRS) of the brain: top and middle: spin-echo image TE ¼ 96 ms, TR ¼ 3 s. Bottom gradient echo image TE ¼ 1.9 ms, TR ¼ 116 ms. (B) Chemical shift metabolic image (CSI) of the brain overlaid on the anatomic image. Spectra are displayed for the selected regions. Chemical shift range is from 4.3 to 0.5 ppm. (Spectra were obtained by Dr JA Hopkins, GE Medical Systems and reproduced by permission of GE Medical Systems, Milwaukee, WI.)

Figure 15 NMR microscopy: dissolving of a tablet in 0.1 mol l  1 HCl solution. (Spectra were reproduced by permission of Bruker Biospin.)

NMR SPECTROSCOPY / Principles 231

Magnetic field gradients may also be generated with RF in homogeneous static fields. Surface coils are tuned to resonate spins at specified depths within a material. As the spins are in a homogeneous field, it is possible to resolve chemical shift information as a function of distance. Metabolic studies in vivo, especially of phosphate materials, provide unique information. Clinical MRI studies of disease states are becoming practical. Functional MRI is used to identify sites of neurological activity in the brain. See also: Nuclear Magnetic Resonance Spectroscopy: Overview; Instrumentation. Nuclear Magnetic Resonance Spectroscopy-Applicable Elements: Hydrogen Isotopes; Carbon-13; Fluorine-19; Nitrogen-15; Phosphorus-31. Nuclear Magnetic Resonance Spectroscopy Applications: Proton NMR in Biological Objects Subjected to Magic Angle Spinning. Nuclear Magnetic Resonance Spectroscopy Techniques: Nuclear Overhauser Effect; Multidimensional Proton; Solid-State.

Further Reading Abragam A (1963) The Principles of Nuclear Magnetism. Oxford: Clarendon. Akitt JW (1992) NMR and Chemistry. London: Chapman and Hall. Braun S, Kalinowski H-O, and Berger S (1998) 150 and More Basic NMR Experiments. Weinheim: Wiley-VCH. Brevard C and Granger P (1981) Handbook of High Resolution Multinuclear NMR. New York: Wiley. Callaghan PT (1991) Principles of Nuclear Magnetic Resonance Microscopy. Oxford: Clarendon Press. Carrington A and McLachlan AD (1967) Introduction to Magnetic Resonance. New York: Harper & Row. Chujo R, Hatada K, Kitamuru R, et al. (1987) NMR measurement of identical polymer samples by roundRobin mehtod. 1. Reliability of Chemical-shift and signal intensity measurements. Polymer Journal 19: 413–424. Derome AE (1987) Modern NMR Techniques for Chemistry Research. Oxford: Pergamon. Freeman R (1988) A Handbook of NMR. New York: Wiley. Frydman L, Lupulescu A, and Scherf T (2003) Principles and features of single-scan two-dimensional NMR spectroscopy. Journal of the American Chemical Society 125: 9204–9217. Fukushima E (1989). NMR in Biomedicine. American Institute of Physics, New York. Fukushima E and Roeder SBW (1981) Experimental Pulse NMR. Reading, MA: Addison-Wesley. Fyfe CA (1983) Solid State NMR for Chemists. Guelph, Ontario: CFC Press. Grant DM and Harris RK (eds.) (1996–2002) Encyclopedia of Nuclear Magnetic Resonance. New York: Wiley. Hore PJ (1995) Nuclear Magnetic Resonance. Oxford: Oxford University Press. Jackman LM and Sternhell S (1969) Applications of Nuclear Magnetic Resonance Spectroscopy in Organic Chemistry. Oxford: Pergamon.

Johnson CS Jr. (1999) Diffusion ordered nuclear magnetic resonance spectroscopy: Principles and applications. Progress in NMR Spectroscopy 34: 203. Mason J (1987) Multinuclear NMR. New York: Plenum Press. MacNaughtan MA, Hou T, Xu J, and Raftery D (2003) High-throughput nuclear magnetic resonance analysis using a multiple coil flow probe. Analytical Chemistry 75: 5116–5123. Oslon D, Norcross J, O’Neil-Jounson M, et al. (2004) Microflow NMR: Concepts and capabilities. Anal. Chem. 76: 2966–2974. Popov AI and Hallenga K (1991) Modern NMR Techniques and their Application in Chemistry. New York: Dekker. Randall JC (1977) Polymer Sequence Determination. New York: Academic Press. Singleton D and Thomas A (1995) High-precision simultaneous determination of multiple small kinetic isotope effects at natural abundance. Journal of the American Chemical Society 117: 9357–9358. Slichter CP (1990) Principles of Magnetic Resonance, 3rd edn. Berlin: Springer. Van de Ven FJM (1995) Multidimensional NMR in Liquids: Basic Principles and Experimental Methods. New York: VCH.

Glossary Anisotropy

Unequally distributed in space, a common situation for electrons in bonds where symmetry is rare. Anisotropy may be spatial, magnetic, or molecular in origin. Magnetic anisotropy is created by anisotropic electron motion induced by the applied magnetic field in groups such as phenyl. The anisotropic magnetic effects generate differential chemical shifts parallel and perpendicular to B0 for groups in fixed locations close to the anisotropic group.

B0, B1, B2

The magnetic fields used in the nuclear magnetic resonance experiment quantified by their strengths. B0 is the large static field generated by the instrument magnet. Its direction defines the z-axis. B1 is the RF electromagnetic field oscillating close to the resonance frequency of the observed nucleus. The strength of B1 is determined by its amplitude. Its direction is usually perpendicular to B0. B1 is applied as an oscillating current in a tuned coil. B2 is similar to B1 but is tuned to the nucleus to be decoupled and is often of higher amplitude.

Bloch decay

This is also known as the free-induction decay (FID). The summed nuclear response detected by the receiver as the magnetization induced by an excitation

232 NMR SPECTROSCOPY / Principles pulse decays to steady state with time constant T2 . The response is digitized and Fourier transformed to produce the observed spectrum. Chemical shift

Chemical shift anisotropy (CSA)

Coalescence

The position of a nuclear resonance in the spectrum. The chemical shift is specified on the d scale relative to a reference resonance. The chemical shift is characteristic of the local electronic environment of the nucleus and is used to identify functional groups. Chemical shift is a tensor quantity and depends on the molecular orientation with respect to the applied magnetic field. Different electron circulations induced by the applied field generate magnetic fields that differentially shield the nucleus depending on molecular orientation. The CSA is the difference between the extreme values of the chemical shift observable in solids but averaged to zero in solution (see Figure 16). The point at which two separate peaks merge to a single unresolved peak. Coalescence usually refers to a system undergoing chemical exchange and exhibiting exchange-line broadening.

Composite pulse A sequence of RF pulses designed to generate a desired spectral response. The individual pulses within a composite pulse may vary in time, frequency, and orientation to achieve the desired response. Composite pulses are commonly used to invert spins over a broad frequency range where a single pulse gives imperfect inversion far from resonance. Composite pulses are also used for excitation of selected frequencies. Coherence

Continuous wave (CW)

=90°

Processes are said to be coherent when they maintain a fixed-time or phase relationship with respect to each other. The Larmor precession frequencies are brought to coherence by an RF pulse, for example, generating x–y magnetization. Phase coherence is used in pulsed field gradient experiments to selectively detect different quantum spin states. NMR spectroscopy performed with time-independent fixed-frequency excitation and/or detection. From 1950 to 1975, most spectrometers operated in CW mode with nuclei being brought to the fixed resonance frequency by sweeping the B0 magnetic field. Fourier

=0°

=0°

(A) 22

11

33

(B)

(C) Figure 16 Spectral lineshapes for powdered solids. (A) Peak doublet produced by dipole coupling between two spin 1/2 nuclei. The doublet is composed of two parts (shown dotted). They correspond to the observed proton flip occurring when its neighbor is spin up (left) or spin down (right). The indicated turning points correspond to the angle between the internuclear vector and B0. (B) Chemical shift anisotropy pattern with shielding tensor components s11, s22, and s33. (C) Combined DD and CSA spectrum. Note that this is not simply (A) þ (B). (Reproduced with permission from Power WP and Wasylishen RE (1991) In: Webb GA (ed.) Annual Reports in NMR Spectroscopy, vol. 23, p. 17. London: Academic Press.)

transform spectrometers by contrast operate at fixed field and use broadband frequency excitation and detection. Continuous wave may also refer to fixed-frequency continuous decoupling experiments. Correlation time The mean time for a molecule or molecular fragment to rotationally diffuse (tc)

NMR SPECTROSCOPY / Principles 233 magnetic field. Internuclear interactions are treated accurately with the point-dipole approximation.

through one radian. May also refer to the inverse of the most probable angular frequency for rotational motion. Coupling constant

Nuclei are coupled when their energies depend on their mutual spin orientations. The magnitude of the interaction is the coupling constant, which is independent of magnetic field. Coupling may be through bond ( J coupling), or through space (dipole coupling). Dipole coupling is inversely proportional to the third power of the internuclear distance and depends on the orientation of the spin pair to the magnetic-field direction. J coupling is transmitted by the bonding electrons and attenuates with the number of intervening bonds. J coupling confirms nuclear connectivity.

Cross polarization with magic angle spinning (CPMAS)

A combination of techniques to narrow signals and facilitate observation of high-resolution spectra of solids. Crosspolarization enhances the sensitivity of low abundance and low g-nuclei by transferring polarization from high abundance, high g-nuclei such as protons. See also magic angle spinning.

Cross polarization

The exchange of magnetic polarization from one nuclear type to another, e.g., 1 H to 13C. Exchange requires an energy match and a coupling interaction for polarization to be transferred. Usually employed to increase the sensitivity of low-frequency low-abundance nuclei using the polarization generated by high-frequency, high-abundance nuclei.

Decoupling and double resonance (DD)

Diamagnetic

Dipole

The use of high-power resonant RF to collapse coupling interactions to zero. Versions include homonuclear (observed and decoupled nuclei have the same g), heteronuclear (observed and decoupled nuclei with different g), and noise (for wide frequency coverage). With lowpower selective irradiation it is possible to map coupled-energy diagrams and transfer polarization. Repelled from a magnetic field – opposite of paramagnetic; raised in energy in the presence of a magnetic field. Most molecules with all electrons paired are diamagnetic. Diamagnetic shielding contributions reduce the static magnetic field at the nucleus causing it to resonate at lower frequency. A nucleus with I ¼ 1/2 acts as a magnetic dipole having a north and south pole. Each dipole is associated with a local

Dipole–dipole coupling

The dipole–dipole coupling between two nuclei is given by: DAB ¼ ðm0 =4pÞgA gB _ð3 cos2 yAB  1Þ=r3AB  2p where A is the observed nucleus, B is neighboring nuclear dipole, yAB is the angle the A–B internuclear vector makes with the applied field direction, and rAB is the through-space internuclear distance. The two possible spin states of one nucleus differ in energy by the coupling to the magnetic field of the second spin dipole (see Figure 16).

Extreme narrowing

The fast motion regime where molecular motion is sufficiently rapid that o0 tc {1. Under these conditions, chemical shift anisotropy and dipole–dipole interactions are averaged to zero.

Exchange broadening

When two resonance frequencies interchange on a timescale that is of the order of the inverse of the frequency difference between them, exchange broadening occurs. When the exchange rate matches the frequency difference, the linewidth broadens to a maximum comparable to the frequency difference. Exchange may involve change in chemical shift and/or coupling constant, bond breaking, conformational change, etc.

Free induction decay

See Bloch decay.

Fermi contact

The major magnetic interaction between the electron and the nucleus. It is the major contribution to hyperfine coupling and is maximized for S electrons because of their high spin density at the nucleus.

Field gradient

A magnetic field applied to make the resonance frequencies depend – usually linearly – on the location of the spins in the sample. Specifically oriented coils in the probe generate fields (B100 mT m  1) in the x, y, and z directions. More complex gradients can be created by appropriate coil design (e.g., shim coils). Pulsed field gradients are used to measure diffusion and to select multiple quantum coherences. See also MRI.

Fourier transform (FT)

The mathematical operation for interconverting two related variables such as time and frequency. Fourier showed that any time-dependent function could be represented equivalently by the sum of a

234 NMR SPECTROSCOPY / Principles series of frequency functions such as sin and cos waves. The process of Fourier transformation converts the digitized Bloch decay into a frequency spectrum. Hahn spin echo

Hartmann– Hahn match

The recovery of signal following a 901– t–1801–t– pulse sequence. For example, a FID decays rapidly in an inhomogeneous field because different parts of the sample resonate at different frequencies. When a 1801 pulse is applied, at time t after the excitation pulse, the signal reappears after an additional time t – the Hahn echo. Alternatively, if the FID decays because of spin exchange processes (T2) no echo will occur. A method for allowing energy exchange between dipolar coupled nuclei with different g. No exchange is possible in the static field alone as the spin quanta differ in energy at common B0 fields. However, the precession frequencies can be made equal about the B1 RF field. The match requires that amplitudes be adjusted so that gAB1A ¼ gXB1X  Population and energy transfer is now possible.

Homospoil

The application of a brief field gradient pulse to the shim coils to vary B0 across the sample. This destroys any residual x–y magnetization because the magnet homogeneity has been spoiled and frequencies dispersed proportionally.

Knight shift

The large chemical shifts observed in the nuclear magnetic resonance of molecules with unpaired electrons – especially metals. It arises because the magnetic environment of the nucleus is dominated by the electron paramagnetism since ge cgN . Shifts to high or low frequency are observed depending whether the a or b electron spin state is low energy and hence more populated.

Larmor frequency

The precession frequency of the nuclear spin dipole when torqued by a magnetic field. The Larmor frequency is equal to the nuclear resonance frequency.

Lineshape, linewidth

A nuclear magnetic resonance line is usually found to have one of two ideal lineshapes – Gaussian, or more often, Lorentzian. A Gaussian line is found when there is a random distribution of static fields within the sample. A Lorentzian line by contrast arises because the spin lifetime follows a first-order decay law. Weighting functions can be applied to a free-induction decay to generate

any output lineshape. The linewidth is reported as the full-linewidth at halfpeak maximum. Lock or field/ frequency lock

A method for accurately maintaining (locking) the static magnetic field at a fixed value. The frequency of a nuclear magnetic resonance signal – commonly deuterium from a deuterated solvent – is held at constant value by a field/frequency feedback loop to compensate for any magnetic field drift during an experiment.

Magic angle spinning (MAS)

The process of spinning a sample, usually solid, rapidly about an axis set at 54.71 (arccos 1/O3) with respect to the static field. The magic angle corresponds to the body diagonal of a cube. Spinning about this axis averages over the three spatial dimensions equally. MAS is used to average CSA.

Magnetic moment (m, M)

Each nuclear spin has a magnetic moment, m quantified in units of its spin angular momentum. Values of m in units of the Bohr magneton, b, are recorded according to different conventions. For protons, the values are 5.5854 with m ¼ g_: 2.7927 with mz ¼ g_I for the z component pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiof ffi spin I, or 4.8372 with m ¼ g_ IðI þ 1Þ where the total magnetic moment is the scaling factor. The observable component p is given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi mI ¼ m= IðI þ 1Þ where mI, is the magnetic quantum number of an individual spin which has allowed values  I,  I þ 1,y, I  1, I. The vector sum of the individual spin moments gives the bulk observable magnetic moment which is denoted by M.

Magnetogyric ratio (g)

The proportionality constant, g, between the resonance frequency of a particular nucleus and the magnetic field strength.

Magnetic resonance imaging (MRI)

The use of nuclear magnetic resonance to measure the spatial distribution of nuclei, particularly water protons. Magnetic field gradients generate spatial dependence of the resonance frequency. Three-dimensional images are constructed from the signals.

Nuclear magnetic resonance (NMR) timescale

The NMR timescale most commonly refers to spin lifetimes of milliseconds to seconds and chemical exchange kinetic processes. The actual time is in the range 0.1/Do  10/Do where Do is the frequency difference in s  1 between exchanging signals. The NMR timescale

NMR SPECTROSCOPY / Principles 235 may also refer to the nanosecond regime when relaxation processes are being discussed. This timescale relates to the correlation time, tc, for molecular motion involved in relaxation. Nuclear Overha- The change in signal intensity of a nuuser effect (nOe) cleus when a transition of an adjacent spin is irradiated. The observed and irradiated spins must be dipole coupled to one another. The observation of an nOe implies the spins are within 0.1–1 nm of each other. Nyquist criteria/ A set of fundamental criteria establishfrequency ing the limits for digital data acquisition. For example, a minimum of two points taken in a specified time are required to define a wave and its frequency. The Nyquist frequency is the maximum frequency represented accurately in a spectrum and is set by the dwell time in the digital acquisition. Paramagnetic

Attracted into a magnetic field. Molecular paramagnetism is generated by unpaired electron spins. A paramagnetic interaction is one where energy is lowered by increasing the magnetic field. Electron orbital angular momentum may also generate paramagnetism. A paramagnetic chemical shift is to higher frequency.

Polarization

The total net magnetization of spins created by unequal spin populations. When nuclear spins are placed in a magnetic field their random motion becomes orientated with respect to the field direction. For I ¼ 1/2 nuclei, there are two spin states orientated with or against the field. At equilibrium the spin states are populated according to the Boltzmann distribution and so there are more spins in the lower energy state. This population imbalance creates an overall magnetization of the sample – the spins are polarized. Pulses generate nonequilibrium polarizations and polarization may be transferred between spins.

Pulse

A short, usually microsecond, burst of RF electromagnetic radiation. The fixed RF is chosen to be close to the nuclear resonance frequency to be excited. The shorter the pulse, the broader the frequency range that may be excited. The pulse is applied through a conducting coil wound closely around the sample. Pulse nuclear magnetic resonance involves spectral excitation with a short

RF pulse followed by detection of a freeinduction decay and subsequent Fourier transformation. Quadrupole

A nucleus with IX1. The nucleus responds to both magnetic fields and electric-field gradients caused by the nonspherical distribution of charge induced by bonding.

Radiofrequency (RF)

The energy difference between nuclear spin states in the magnetic field used for nuclear magnetic resonance falls in the RF region (kHz–GHz) of the electromagnetic spectrum. Thus, a source of RF electromagnetic radiation is used to excite nuclear spins.

Relaxation

The process where nuclei dissipate excess spin energy into other forms of molecular energy especially rotation and translation. T1 (spin–lattice, longitudinal) and T2 (spin–spin, latitudinal) relaxation are the independent time constants for the dissipation of z and x–y magnetization, respectively. Spin–lattice relaxation – time constant T1 – involves a change in the total z magnetization of the sample as quanta of spin energy are exchanged with the surroundings. Spin– spin relaxation, T2, is the time constant for decay of spin order and coherence commonly represented as x–y magnetization. There is no energy exchange with the surroundings and spin order is randomized. Spin order is commonly randomized by mutual spin flips between dipolar coupled pairs. The x–y magnetization decays with time constant T2 to its equilibrium value of zero. As there is no magnetic field in the x–y plane there is no x–y magnetization at equilibrium. When a B1 or B2 RF field is present, the T2 relaxation time constant is replaced with T1r – relaxation in the rotating frame.

Residual dipolar In samples containing large anisotropic coupling molecules such as phages and other additives, the motion of the small and large molecules becomes anisotropic. As a result the (1  3 cos2 y) term in the dipole coupling expression does not average to zero. Residual dipole couplings between closely spaced nuclei may then be detected and correlated with molecular geometry. Resolution

Two peaks may only be resolved if their linewidth is smaller than the frequency difference between them. In nuclear magnetic resonance, resolution

236 NMR SPECTROSCOPY / Principles is usually governed by experimental factors such as magnetic field homogeneity and digital resolution. Digital resolution is set by the number of data points in a spectral window which in turn depends on total acquisition time. Absolute resolution is set by the spin lifetime which establishes the linewidth at half height. See also T2. Resonance

When the RF equals the energy gap between spin states that differ by one in spin quantum number, the spins are said to be in resonance.

Rigid lattice

The slow motion regime where o0 tc c1 and chemical shift anisotropy and dipole– dipole effects broaden spectra.

Ring current

A magnetic dipole may be represented equivalently as a current loop. A magnetically anisotropic group such as a benzene ring acts as a magnetic dipole when placed in a magnetic field. The magnetic influence of such rings is often modeled as a current loop with electrons circulating around the p system at a rate proportional to B0 which is termed a ring current. Groups that are at fixed orientations relative to the magnetic dipole, ring current, have their resonances shifted by the induced ring current.

Rotating frame

Saturation

The simplest Cartesian frame of reference to envisage for the nuclear magnetic resonance experiment is the laboratory frame with the magnetic-field direction defining the z-axis. In the presence of an oscillating electromagnetic field (RF) the description of the motion of the nuclear spins and the net magnetization is complex in the laboratory frame. A mathematical transformation from the laboratory frame to a frame rotating at the RF oscillation frequency greatly simplifies the description of the magnetic field and the motion. (An analogy may be useful. The description of the motion of the moon in a solar coordinate system is greatly simplified by transforming to a coordinate system that rotates with the earth.) In the laboratory frame, the x and y coordinates are time independent, in the rotating frame the B1 field is static in the x0 –y0 plane which is rotating at the RF in the laboratory frame. The spins are said to be saturated when the spin populations are equalized by the absorption of RF energy. There is no residual z magnetization and no signal can be excited.

Second-order coupling

This refers to the patterns observed for J-coupled multiplets. In a first-order spectrum, the lines for spins coupled to groups containing n equivalent atoms of spin I are split into 2nI þ 1 lines. The spacing on both partners are equal, and equal to J. This requires that the chemical shift difference between coupled spins is at least ten times J. Secondorder effects occur when Dd BJ, so called strong coupling. As Dd approaches J, at first line intensities become unequal within the multiplet and are canted towards the mean chemical shift (haystacking). Next, line spacings become unequal and differ from J. When Dd BJ extra lines appear in the spectrum, intensities vary widely, and line spacings do not match J.

Shielding tensor

The electron motion around the nucleus induced by the applied magnetic field generates an internal magnetic field. The induced field is said to shield or screen the nucleus from the applied field and is observed as the chemical shift. The shielding tensor defines the principal values and orientation of the chemical shift in the molecular frame. The maximum difference between the principal values defines the CSA.

Shift reagent

A reagent which binds rapidly and reversibly to a molecule and induces a chemical shift. The common shift reagents are organometallic lanthanide complexes such as Eu(DPM)3 that act as Lewis acids in nonpolar solvents. The reagents have a large magnetic anisotropy and induce substantial changes in chemical shift. The changes in shift depend on the binding constant, proximity, and orientation with respect to the binding site. The magnetic anisotropy effect is independent of the nucleus observed, so shift reagents are most effective in proton spectra.

Shim

The adjustment of the magnet to achieve maximum homogeneity. In the early days of nuclear magnetic resonance, shimming involved physically moving the magnet pole faces relative to one another with pieces of wood (shims). On a modern spectrometer shimming is done by varying the current in carefully designed shim coils placed in specific orientations within the probe.

Spectral window The limited frequency range observed in a pulse nuclear magnetic resonance

NMR SPECTROSCOPY / Principles 237 n times. During each acquisition time a free-induction decay is acquired in time t2. Fourier transformation with respect to t2 will generate n spectral sets with a standard frequency axis (F2) containing peaks modulated according to t1. A new time series is obtained by selecting each of the common frequency points in F2 and following its magnitude as t1 is incremented. A second Fourier transform with respect to t1 and orthogonal to t2 will generate new spectra with a new frequency dimension (F1). A 2D plot can be made with intensity providing the third dimension. New peaks are found if the t1 and t2 processes are correlated and occur at n12 and n21 in addition to the 1D frequencies at n11 and n22. Pulse sequences have been developed to provide chemical shift information on one axis and coupling constant information on the other, i.e., 2D-J spectroscopy. COSY (HETCOR, COLOC, HOMCOR, HMBC) spectra correlate chemical shifts of J coupled nuclei, a very useful method for structure analysis. NOESY, ROESY, and TROESY spectra correlate chemical shifts of nuclei which are dipolar coupled. The higher dimension examples (3D, etc.) involve adding new timing sequences to separate variables of interest for a particular problem. Multidimensional spectra also provide enhanced peak separation in highly overlapped spectra of large proteins. 4D spectra may be created which show data with 1H, 13C, and 15N frequency axes with intensity given by the NOESY spatial coupling information.

experiment. The spectral window is set by the dwell time for data point acquisition in the computer (see Nyquist). The strength and duration of the radiofrequency pulse also limit the frequency window that may be excited. Spin lock

The application of an on-resonance RF field to maintain the x–y magnetization in the direction of the B1 field. The spin-lock pulse is usually applied immediately following a 901 pulse in the x–y plane and perpendicular to the original 901 pulse. It has the effect of fixing (locking) the x–y magnetization created by the initial pulse for a time of the order of T1r rather than T2.

Spinning sidebands

Peaks in a spectrum that are separated from the true peaks by integer multiples of the sample spinning frequency. Spinning sidebands occur as sample spinning modulates the effective field at the nucleus when the sample rotates through different field strengths caused by magnetic inhomogeneities (solution) or spatially dependent fields (e.g., chemical shift anisotropy, dipole coupling, quadrupole coupling) in solids.

Spin temperature

For a spin system at equilibrium with its surroundings the spin energy levels are populated according to the Boltzmann distribution: ni =n0 ¼ expðDE=kTÞ ¼ expð_gB0 =kTÞ (4). The energy gap is fixed for a given spectrometer and nucleus and so the spin population ratio is inversely proportional to absolute temperature with a calculable proportionality constant. The spin temperature of a sample is then defined by this spin population ratio. For example, following a 901 pulse when the spin populations are equal, the temperature is said to be infinite. It is possible to create a negative spin temperature by inverting the spin population with a 1801 pulse.

T1 and T2

The spin–lattice and spin–spin relaxation times. See Relaxation.

2D, 3D, 4D, etc., NMR

Nuclear spins may be perturbed by an enormous variety of pulse sequences. If the response of the spin system is detected as a function of independent time variables, subsequent Fourier transformation with respect to each time set will provide an independent frequency dimension for each time sequence. For example, a set of n free-induction decays could be obtained with the sequence Pulse–t1–Pulse–t1–Acquire with the time delay t1 incremented

x–y Magnetization (Mxy)

The instantaneous value of the magnetic moment of the sample along the axis perpendicular to the B0 field. In the laboratory frame, the x–y magnetization is rotating with the Larmor frequency. It varies with time constant T2 and has an equilibrium value of zero.

Zeeman energy

The energy of the nuclear spin states with different spin quantum numbers in a magnetic field. Often used to describe phenomena where the energy is proportional to the strength of the magnetic field.

Zeugmatography

The original name given to MRI by its inventor, Paul Lauterbur.

z-Magnetization The instantaneous value of the magnetic (Mz) moment of the sample along the axis of the applied field (z-axis). It varies with time constant T1 and has an equilibrium value denoted M0.