Nuclear magnetic resonance (NMR) spectroscopy is an analytical technique based on the magnetic properties of certain atomic nuclei. NMR is similar to other types of spectroscopy in that absorption of electromagnetic energy at characteristic frequencies provides analytical information. NMR differs from other types of spectroscopy because the discrete energy levels between which the transitions take place are created by placing the nuclei in a magnetic field of strength H0. Although the initial field strength of the applied field is H0, when the sample is inserted into the magnet, the field strength throughout the sample becomes B0, defined as follows:

in which µS is the magnetic susceptibility of the sample.

Atomic nuclei behave as if they were spinning on the nuclear axis. The angular momentum, 0, of the nucleus is characterized by a spin quantum number (I). The maximum observable component of the angular vector, , is

in which h is Planck’s constant and h is modified Planck’s constant.

Table 1 shows the values of I as a function of the mass number and the atomic number.

The angular momentum creates a magnetic moment, µ, which is parallel to and directly proportional to .

where is the magnetogyric ratio and is a constant for all nuclei of a given isotope, regardless of their position in a molecule.

Nuclei that have a spin quantum number I 0, when placed in an external uniform static magnetic field, align with respect to the field in (2I + 1) possible orientations. Thus, for nuclei with I = 1/2, which includes most isotopes of analytical significance (Table 2), there are two possible orientations, corresponding to two different energy states. The energies of these two states are ± µB0, and their separation is

with more nuclei populating the lower energy state (µB0) than the higher energy state (+µB0). The populations are in accordance with the Boltzmann distribution:

where N+ and N are the populations of the high and low energy states, respectively; k is the Boltzmann constant; and T is the temperature in K.

A nuclear resonance is the transition between these states, and upward as well as downward transitions are possible. In a static magnetic field the nuclear magnetic axis precesses (Larmor precession) about the B0 axis. The precessional angular velocity is often referred to as the Larmor frequency, 0, and is related to B0:

If energy from an oscillating radio-frequency (rf) field is introduced, then resonance is achieved when the rf frequency is the same as the precessional angular velocity. Although the probability of an upward transition is equal to that of a downward transition, more upward transitions take place than downward transitions because N is greater than N+. Hence, an overall absorption of energy takes place. As shown in Table 2, the resonance frequency of a nucleus increases in direct proportion with the increase of the magnetic field strength.

NMR is a technique of high specificity but relatively low sensitivity. The basic reason for the low sensitivity is the comparatively small difference in energy between the upper and lower energy states (0.08 Joules at 1.5 to 2.0 T field strength), which results in a population difference between the two levels of only a few parts per million. Another important aspect of the NMR phenomenon, with negative effects on sensitivity, is the long lifetime of most nuclei in the excited state. Long lifetimes affect the design of the NMR analytical test, especially in repetitive pulsed experiments. Simultaneous acquisition of the entire range of resonant frequencies instead of frequency-swept spectra can give increased sensitivity per unit time.

All characteristics of the signal–chemical shift, multiplicity, linewidth, coupling constants, relative intensity, and relaxation time contribute analytical information. The analytical usefulness of NMR arises from the observation that the same types of nuclei, when located in different molecular environments, exhibit different resonance frequencies. The reason for this difference is that the effective field associated with a particular nucleus is a composite of the external field provided by the instrument and the field generated by the circulation of the surrounding electrons. The latter field is generally opposed to the external field and lowers the overall field strength at the nuclear site. The phenomenon is termed shielding. Hence, the more shielded nuclei have lower Larmor frequencies.

It is not convenient to accurately measure the absolute values of transition frequencies, as is done with other spectroscopic procedures. However, it is convenient to accurately measure the difference in frequencies between two resonance signals. The position of a signal in an NMR spectrum is described by its separation from another resonance signal arbitrarily taken as standard. This separation is called chemical shift, which may be expressed in units of magnetic field or in frequency units that are readily interconverted by the equation for the resonance condition, Equation 6. This equation shows that when the separation is expressed in frequency units, it is directly proportional to the field strength. It is more convenient, therefore, to express the chemical shift in terms of the dimensional unit , which is independent of the field strength, and is defined by

where SS is the test substance resonance frequency, in Hz; RS is the reference resonance frequency in Hz; and 0 is the instrument frequency, in MHz. When 0 is expressed in units of MHz, then Equation 7 is expressed in units as parts per million (ppm). Hence, it is common to use the unit ppm to express the chemical shift difference between the resonance peak of the test substance and that of the reference.

By using this equation, one can use (with appropriate caution) the chemical shift of any known species (such as the residual 1H-containing species in a deuterated solvent) as a chemical shift reference. This equation, now in common use, is applicable to nearly all nuclei.

Tetramethylsilane (TMS) is the most widely used chemical shift reference for proton and carbon spectra. It is chemically inert, exhibits only one peak (which is more shielded than most signals), and is volatile, which allows ready specimen recovery. Either 2,2,3,3-d4 sodium 3-(trimethylsilyl)propionate (TMSP) or sodium 2,2-dimethyl-2-silapentane-5-sulfonate (DSS) is used as an NMR reference for aqueous solutions. The resonance frequency of the TMSP or DSS methyl groups closely approximates that of the TMS signal. Where the use of an internal NMR reference material is not desirable, an external reference may be used, such as a reference standard in a separate NMR tube.

Conventional NMR spectra are shown with shielding increasing and chemical shift decreasing from left to right because less shielded nuclei have higher Larmor frequencies than do more shielded nuclei. Resonances on the left are said to be deshielded (i.e., they show lower electron density). Resonance peaks appearing at the right are termed more shielded (i.e., they show greater electron density) than those appearing at the left. Resonances from the more shielded and the less shielded nuclei often are inappropriately called the high-field or upfield peaks and the low-field or downfield peaks, respectively, as a result of the outdated method of acquiring data by sweeping the magnetic field. Today, the overwhelming majority of spectra are acquired on a pulsed Fourier transform (FT) spectrometer, which sweeps neither the magnetic field nor the transmitter frequency. Therefore, it is more appropriate to refer to resonances at the left side of the spectrum as high-frequency or deshielded resonances and those on the right as low-frequency or shielded resonances.

The coupling between two nuclei can be described in terms of the spin-spin coupling constant, J, which is the separation (in Hertz) between the individual peaks of the multiplet. When two nuclei interact and cause reciprocal splitting, the measured coupling constants in the two resulting mutiplets are equal. Furthermore, J is independent of magnetic field strength.

Coupled spin systems are usually referred to as weak or strong. These terms depend on the separation of the Larmor frequencies of the coupled nuclei compared to the coupling constant between them. Both of these values are easily measured from the spectrum. For a weakly coupled system, the separation expressed in Hz (D) is large compared to J, which is always expressed in Hz. Thus, the ratio of the two is dimensionless. For a weakly coupled system, the ratio is large. Typically, spectroscopists consider a ratio above 10 to be weak. Only weakly coupled spin systems produce first-order spectra, which are comparatively easy to analyze. The number of individual peaks that are expected to be present in a multiplet and the relative peak intensities are predictable. The number of peaks is determined by 2 nI + 1, where n is the number of identical nuclei on adjacent groups that are active in splitting and I is the spin of those nuclei causing the splitting. For protons this becomes (n + 1) peaks. In general, the relative intensity of each peak in the multiplet follows the coefficient of the binomial expansion (a + b)n. These coefficients can conveniently be found by use of Pascal’s triangle, which produces the following relative areas for the specified multiplets: doublet, 1:1; triplet, 1:2:1; quartet, 1:3:3:1; quintet, 1:4:6:4:1; sextet, 1:5:10:10:5:1; and septet, 1:6:15:20:15:6:1. Two examples of first-order spectra arising from weak coupling are shown in Figure 1.

Coupling may occur between 1H and other nuclei, such as 19F, 13C, and 31P. This type of coupling can frequently be observed between nuclei separated by 1–5 bonds.

Magnetically active nuclei with I 1, such as 14N, possess a nuclear quadrupole moment, which produces line broadening of the signals from neighboring nuclei.

Another characteristic of an NMR signal is its relative intensity, which has wide analytical applications. In carefully designed experiments, the area or intensity of a signal is directly proportional to the number of protons that give rise to the signal. As a result, NMR can be used for quantitation (see Quantitative Analysis in this chapter and in Nuclear Magnetic Resonance, Qualitative and Quantitative NMR Analysis 761. NMR spectra may contain spinning side bands, peaks that appear symmetrically located around each signal. These signals are due to the failure to optimize the off-axis (x and y) shims. The homogeneity of modern superconducting magnets, coupled with computer shimming techniques, reduce the need for sample spinning and completely eliminate these sidebands.

NMR includes two types of relaxation: Spin-Spin Relaxation, sometimes referred to as Transverse Relaxation, or T2 Relaxation, and Spin-Lattice Relaxation, sometimes referred to as Longitudinal Relaxation, or T1 Relaxation. At least two mechanisms contribute to Transverse Relaxation: loss of signal due to B0 inhomogeneity and the natural relaxation that would take place even in a perfectly homogeneous field. The combined effects of these two mechanisms produce a new time constant for the relaxation, which is referred to as T2*.

During an rf pulse, a magnetic field (B1) is applied to the sample. The magnetization vector M precesses about B1 according to

where 1 is the precessional frequency and B1 is the strength of the magnetic field applied to the sample. During the time that the pulse is applied, M precesses to an angle () given by the precessional rate multiplied by the width of the pulse, in time, (PW). Then,

Tip angles typically are expressed in units of degrees or in radians. For example, a 90 pulse is sometimes referred to as a /2 pulse.

Surprisingly, the optimum S/N ratio per unit time is obtained when no relaxation delay is used for a given T1. That is when PR equals to AT, the acquisition time. However, because the nuclei in most molecules do not have the same T1 values, the relative intensity relationship will be lost.

For typical quantitation experiments, a relaxation delay is used and should be at least five times the longest T1 expected for any of the nuclei in the molecule; and the pulse width should be set to 90. Further details are provided in the section Quantitative Applications.

In NMR spectroscopy the typical definition of resolution is the ability to distinguish between two closely spaced resonance peaks in a spectrum. The industry standard for measuring resolution is to measure the width of a single peak, in units of Hz, at the half-height of the peak.

The uncertainty principle determines the best resolution that can be achieved in an NMR spectrum. The maximum resolution possible, or the minimum separation that can be observed between two frequencies, D, in a spectrum is equal to the reciprocal of the acquisition time, AT, of the FID.

The time set by the spectrometer operator should not exceed the required AT, because this would result in the collection of only noise after the signal has decayed to near zero. Collecting this noise does not improve the resolution.

The final appearance of the spectrum can usually be improved by applying a variety of mathematical procedures to the FID before the Fourier transform is performed. The two most common procedures are multiplying the FID by a mathematical function, generally known as a window function; or appending zeros to the end of the FID, generally known as zero filling.

NMR spectroscopy is a powerful technique for structure identification because of its specificity of detecting certain nuclei such as 1H, 13C, 31P, and 19F. Typically, a routine identification test can be performed by 1H NMR spectroscopy in a short period of time for simple molecules. The basis for identification is provided by a comparison of the signals from the test sample with the expected signals from a qualified reference standard. A positive identification can be concluded when the chemical shifts, multiplicities, and coupling constants of the spectrum of the test sample match those of the reference standard or, in the case of a USP monograph, the values listed in the monograph.

Data may be made unacceptable for analysis if incorrect sample preparation or poor adjustment of spectrometer parameters leads to poor resolution, decreased sensitivity, and spectral artifacts. It is preferable that the operator be familiar with the basic theory of NMR and operation of the spectrometer. Frequent checks of instrument performance are essential.

The procedures discussed here refer specifically to 1H and 13C NMR, but they are applicable, with modification, to other nuclei. The discussion assumes that the NMR spectra are obtained from solutions in suitable solvents.

NMR is one of the most useful techniques for quantitative analysis in chemistry. If appropriate experimental conditions are chosen, the relative intensities of resonances are proportional to the population of the nuclei causing those resonances. NMR experiments can be designed for relative or absolute quantitation, either with an internal standard or without one.

The analytical usefulness of solid-state NMR spectroscopy for studying solid materials lies in the fact that the same types of nuclei in different solid-state environments exhibit different resonance frequencies. Applications of this technique in pharmaceutical analysis range from solid form (polymorph, solvate) identification and quantitation in bulk drug substances to physical/chemical profiling of dosage forms. The technique has the unique ability to probe electronic environments of specific nuclei in the solid state over a large timescale without the requirement of single-crystal substrates or even homogeneous samples. Methods and procedures presented herein are directed at observing 13C, the most popular NMR nucleus for solids. The concepts may be equally applied to other relevant spin-1/2 nuclei such as low-natural-abundance 15N as well as high-natural-abundance 31P.

Low field NMR (LF-NMR), sometimes referred to as time domain NMR (TD-NMR), experiments are performed by measuring relaxation, relaxometry, or diffusion. Instrumentation for these applications is based on low-field permanent magnet technologies that operate in the 2–25 MHz frequency range. Inexpensive stationary bench-top and portable TD-NMR spectrometers are commercially available. Typical bore sizes are 10–50 mm in diameter. A recent development in spectrometer design uses a mobile mouse probe as an alternative to a stationary magnet. This design allows analysis of samples of unrestricted size.

Most TD-NMR applications are based on simple pulsing sequences, including FID, Hahn–echo, Carr–Purcell–Meiboom–Gill, and solid–echo acquisition. The choice of pulse sequence depends on the physical and chemical properties of the sample as well as the information that is desired from the experiment. These systems can be used to measure longitudinal (spin-lattice, T1) and transverse (spin-spin, T2) relaxation times. Diffusion properties of compounds can be exploited using pulsed field gradient (PFG-NMR) experiments.

The classical application of relaxometry is for the determination of food product components based on differences in longitudinal and transverse relaxation times of water, fats, and proteins.