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# Nuclear magnetic resonance

Nuclear magnetic resonance (NMR) is a physical phenomenon based upon the magnetic property of an atom's nucleus. NMR studies a magnetic nucleus, like that of a hydrogen atom, by aligning it with an external magnetic field and perturbing this alignment using an electromagnetic field. The response to the field (the perturbing), is what is exploited in NMR spectroscopy and magnetic resonance imaging.

NMR spectroscopy is one of the principal techniques used to obtain physical, chemical, electronic and structural information about a molecule. Also, NMR is one of the techniques that has been used to build quantum computers.

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## History

NMR was first described independently by Felix Bloch and Edward Mills Purcell in 1946 both of whom shared the Nobel Prize in physics in 1952 for their discovery. Purcell had worked on the development and application of RADAR during World War II at MIT's Radiation Lab. His work during that project on the production and detection of radiofrequency energy, and on the absorption of such energy by matter, preceded his discovery of NMR and probably contributed to his understanding of it and related phenomena.

It was noticed that magnetic nuclei, like 1H and 31P, could absorb RF energy when placed in a magnetic field of a specific strength. When this absorption occurs, the nucleus is described as on resonance. Interestingly for analytical scientists, different atoms within a molecule resonate at different frequencies and field strength. The observation of all the resonance frequencies of a molecule allows a user to discover structural information about the molecule.

The development of NMR as a technique of analytical chemistry and biochemistry parallels the development of electromagnetic technology and its introduction into civilian use.

Throughout its first few decades, NMR practice utilized a technique known as continuous-wave (CW) spectroscopy, in which either the magnetic field was kept constant and the oscillating field was swept in frequency to chart the on-resonance portions of the spectrum, or more frequently, the oscillating field was held at a fixed frequency, and the magnetic field was swept through the transitions.

The CW technique is limited in that it probes each frequency individually, in succession, which has unfortunate consequences due to the insensitivity of NMR--that is to say, NMR suffers from poor signal-to-noise ratio. Fortunately for NMR in general, signal-to-noise ratio (S/N) can be improved by signal averaging. Signal averaging increases S/N by the square-root of the number of signals taken.

The technique known as Fourier transform NMR spectroscopy (FT-NMR) can speed the time it takes to acquire a scan by allowing a range of frequencies to be probed at once. This technique has been made more practical with the development of computers capable of performing the computationally-intensive mathematical transformation of the data from the time domain to the frequency domain, to produce a spectrum.

Pioneered by Richard R. Ernst, FT-NMR works by irradiating the sample (still held in a static, external magnetic field) with a short pulse of radiofrequency energy (RF) containing all the frequencies required to excite every nucleus under study. Detectors record the decay of this excitation as a time-dependent pattern, known as the free induction decay (FID). This time-dependent pattern can be converted into a frequency-dependent pattern of nuclear resonances using a mathematical function known as a Fourier transformation, revealing the NMR spectrum. (A similar technique used for optical rather than NMR spectroscopy is simply called Fourier transform spectroscopy)

The use of pulses of different shapes, frequencies, and durations, in specifically-designed patterns or pulse sequences allows the spectroscopist to extract many different types of information about the molecule.

Multi-dimensional nuclear magnetic resonance spectroscopy is a kind of FT-NMR in which there are at least two pulses, and as the experiment is repeated, the delay between a pair of pulses is varied. The first dimension is the frequency of the excitation, and the second dimension is based on the time differential between the pair of pulses (because of the properties of the Fourier transform, this second dimension is eventually expressed as a frequency as well). In multidimensional nuclear magnetic resonance, there will be a sequence of pulses, and at least one variable time period (in 3D, two time sequences will be varied. In 4D, three will be varied).

There are many such experiments. In one, these time intervals allow for, among other things, magnetization transfer between nuclei and therefore the detection of the kinds of nuclear-nuclear interactions that allowed for the magnetization transfer. The kinds of interactions that can be detected are classed into two kinds, usually. There are through-bond interactions and through-space interactions, the latter usually being a consequence of the nuclear Overhauser effect. Experiments of the nuclear Overhauser variety may establish distances between atoms.

Kurt Wüthrich, Ad Bax , Vladimir Sklenar and many others, developed 2D and multidimensional FT-NMR into a powerful technique for studying biochemistry, in particular for the determination of the structure of biopolymers such as proteins or even small nucleic acids. Wüthrich shared the 2002 Nobel Prize in Chemistry for this work. This technique complements biopolymer X-ray crystallography in that it is most frequently applicable to biomolecules in a liquid or liquid crystal phase, whereas crystallography (as the name implies) is performed on molecules in a solid phase. Though NMR is used to study solids, extensive atomic-level biomolecular structural detail is especially difficult to obtain in the solid state.

Because the intensity of NMR signals, and hence the sensitivity of the technique, depend on the strength of the magnetic field, the technique has also advanced over the decades with the development of more powerful magnets. Advances made in the audio-visual technology sector have also improved the signal generation and processing capabilities of newer machines.

The sensitivity of NMR signals is also dependent, as noted above, on the presence of a magnetically-susceptible isotope, and therefore either on the natural abundance of such isotopes, or on the ability of the experimentalist to artificially enrich the molecules under study with such isotopes. The most abundant naturally occurring isotopes of hydrogen and phosphorus, for instance, are both magnetically susceptible and readily useful for NMR spectroscopy. In contrast, carbon and nitrogen have useful nuclei, but which occur only in very low natural abundance.

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## Uses of NMR

Nuclei are surrounded by orbiting electrons, which are also spinning charged particles [i.e. magnets] and so will partially shield the nuclei. The amount of shielding depends on the exact local environment. For example, a hydrogen bonded to an oxygen will be shielded differently than a hydrogen bonded to a carbon atom. In addition, two hydrogen nuclei can interact via a process known as spin-spin coupling if they are on the same molecule, which will split the lines of the spectra in a recognisable way. By studying the peaks of NMR spectra, skilled chemists can determine the structure of many compounds. It can be a very selective technique, distinguishing among many atoms within a molecule or collection of molecules of the same type, but which differ only in terms of their local chemical environment.

By studying T2* information, a chemist may determine the identity of a compound by comparing the observed nuclear precession frequencies to known frequencies. Further structural data can be elucidated by observing spin-spin coupling, a process by which the precession frequency of a nucleus can be influenced by the magnetization transfer from nearby nuclei.

T2 information can give information about dynamics and molecular motion.

Because the NMR timescale is rather slow (compared to other spectroscopic methods), changing the temperature of an T2* experiment can also give information about fast reactions, such as the Cope reaction or about structural dynamics, such as ring-flipping in cyclohexane.

A relatively recent example of NMR being used in the determination of a structure is that of buckminsterfullerene. This now famous form of carbon has 60 carbon atoms forming a sphere. The carbon atoms are all in identical environments and so should see the same internal H field. Unfortunately, buckminsterfullerene contains no hydrogen and so 13C NMR has to be used (a more difficult form of NMR to do). However in 1985 the spectrum was obtained by R. Curl and R. Smalley of Rice University and sure enough it did contain just the one single spike, confirming the unusual structure of C60.

NMR is extremely useful for analyzing samples non-destructively. Radio waves and static magnetic fields easily penetrate many types of matter (in practice, anything that is not inherently ferromagnetic). For example, if one wanted to decisively know whether or not a bottle of wine was 'off', NMR could be used to analyze the wine without ever opening the bottle. This also makes NMR a good choice for analyzing dangerous samples.

## Theory of Nuclear Magnetic Resonance

### Nuclear Spin and Magnets

Electrons, neutrons and protons, the three particles which constitute an atom, have an intrinsic property called spin. This spin is defined by the fourth quantum number for any given wave function obtained by solving relativistic form of the Schrödinger equation (SE). It represents a general property of particles which we can describe using the properties of electrons. Electrons flowing around a coil generate a magnetic field in a given direction; this property is what makes electric motors work. In much the same way electrons in atoms circulate around the nucleus, generating a magnetic field. This generated field has an angular momentum associated with it. It so turns out that there is also an angular momentum with the electron particle itself, denoted the spin, and this gives rise to the spin quantum number, ms.

Spin angular momentum is quantized and can take different integer or half-integer values depending on what system is under study. If we solve the relativistic SE for the electron we get the values +½ and -½. Since the Pauli principle states that no two species can have the same quantum number, it is why only two electrons, paired antiparallel (one positive one negative), can appear in a single atomic orbital.

Like the electron, protons and neutrons also have a spin angular momentum which can take values of + ½ and –½. In the atomic nucleus, protons can pair with other antiparallel protons, much in the same way electrons pair in a chemical bond. Neutrons do the same. Paired particles, with one positive and one negative spin, thus have a net spin of zero "0". We can see that a nucleus with unpaired protons and neutrons will have an overall spin, with the number unpaired contributing ½ to the overall nuclear spin quantum number, I. When this is larger than zero, a nucleus will have a spin angular momentum and an associated magnetic moment, μ, dependent on the direction of the spin. It is this magnetic moment that we manipulate in modern NMR experiments.

It is worth noting here that nuclei can have more than one unpaired proton and one unpaired neutron, much in the same way electronic structure in transition metals can have many unpaired spins. For example 27Al has an overall spin I=5/2.

NB: A technique related to NMR is electron spin resonance that exploits the spin of electrons instead of nuclei. The principles are otherwise similar.

#### Values of spin angular momentum

The spin angular momentum of a nucleus can take ranges from +I to –I in integral steps. This value is known as the magnetic quantum number, m. For any given nucleus, there is a total (2I+1) angular momentum states. Spin angular momentum is a vector quantity. The z component of which, denoted Iz, is quantised:
Iz = mh/2π
where h is Planck’s constant.

The resultant magnetic moment of this nucleus is intrinsically connected with its spin angular momentum. In the absence of any external effects the magnetic moment of a spin ½ nuclei lays approximately 52.3 from the angular momentum axis or 127.7 for the opposing spin. This magnetic moment is intrinsically related to I with a proportionality constant γ, called the gyromagnetic ratio:
μ=γI

#### Spin behaviour in a magnetic field

If we take the case of nuclei which have a spin of a half like 1H, 13C or 19F. The nucleus thus has two possible magnetic moments it could take, often referred to as up or down, +0.5 -0.5, or to be more in tune with physicists... α and β. The energies of each state are degenerate - that is to say that they are the same. The effect is that the number of atoms (population)in the up or α state is the same as the number of atoms in the β state.

If we place a nucleus in a magnetic field the angular momentum axis coincides with the field direction. The resultant magnetic momenta, space quantised from the angular momentum axis, no longer have the same energy since one states has a z-component aligned with an external field and are lower in energy (positive I values) and the other opposes the external field and is higher in energy. This causes a population bias toward the lower energy states.

The energy of a magnetic moment μ when in a magnetic field B0 (the zero subscript is used to distinguish this magnetic field from any other applied field) is the negative scalar product of the vectors:
E= zB0

We've already defined μz=γIz. So placing this in the above equation we get:
E = -mhγB0 / 2π

#### Resonance

The energy gap between our α and β states is (hγB0)/2π. We get resonance between the states, there for equalising populations, if we apply a radiofrequency with the same energy as the energy difference ΔE between the spin states. The energy of a photon is E=, where ν is its frequency.

ΔE = hγB0 / 2π

I.e. the frequency of electromagnetic radiation required to produce resonance of an specific nucleus in a field B is:

ν = γB0 / 2π

It is this frequency that we are concerned with, and detect in NMR. And it is this frequency which describes the sample we are observing. But importantly: it is this resonance that gives rise to the NMR spectrum

#### Nuclear Shielding

It would appear from the above equation that all nuclei of the same isotope, which take the same the gyromagnetic ratio (μ), resonate at the same frequency. This of course is not the case. Since the gyromagnetic ratio of a given isotope does not change we can only rationalise this by stating that the effect of the external magnetic field is different for different nuclei. Local effects of other nuclei, especially spin-active nuclei, and local electron effects shield each nucleus differently from the main external field.

It was stated that the energy of a spin state is defined by E= zB0. We can see that by shielding the strength of the magnetic field, the experienced effect, or effective magnetic field at the nucleus is lower: Beffective < B0. Thus the energy gap is different, and hence the frequence required to achieve resonance deviates from the expected value.

These differences due to nuclear shielding give rise to many peaks (frequencies) in an NMR spectrum. We can now see why NMR is a direct probe of chemical structure.

An astute reader will notice that differences in shielding would occur between two identical molecules orientated differently! However, these differences are averaged out due to molecular motion.

### Relaxation

The process called population relaxation refers to nuclei that return to the thermodynamic state in the magnet. This process is also called T1 relaxation, where T1 refers to the mean time for an individual nucleus to returns to its equilibrium state. Once the population is relaxed, it can be probed again, since it is in the initial state.

The precessing nuclei can also fall out of alignment with each other (returning the net magnetization vector to a nonprecessing field) and stop producing a signal. This is called T2 relaxation. In this state the population difference required to give a net magnetization vector is not at its thermodynamic state. Some of the spins were flipped by the pulse and will remain so until they have undergone population relaxation.

$\frac{1}{T_2} = \frac{1}{T_2^*} + \frac{1}{2T_1}$.

It is seen that T1 is larger (slower) than T2*.

## Correlation spectroscopy; a form of two-dimensional nuclear magnetic resonance

Correlation spectroscopy is one of several types of two-dimensional nuclear magnetic resonance (NMR) spectroscopy. Other types of two-dimensional NMR include J-spectroscopy, exchange spectroscopy (EXSY), and Nuclear Overhauser effect spectroscopy (NOESY.) Two-dimensional NMR spectra provide more information about a molecule than one-dimensional NMR spectra and are especially useful in determining the structure of a molecule, particularly for molecules that are too complicated to work with using one-dimensional NMR. The first two-dimensional experiment, COSY, was proposed by Jean Jeener, a professor at Université Libre de Bruxelles, in 1971. This experiment was later implemented by Walter P. Aue, Enrico Bartholdi and Richard R. Ernst, who published their work in 19761.

A two-dimensional NMR experiment involves a series of one-dimensional experiments. Each experiment consists of a sequence of radio frequency pulses with delay periods in between them. It is the timing, frequencies, and intensities of these pulses that distinguish different NMR experiments from one another. During some of the delays, the nuclear spins are allowed to freely precess (rotate) for a determined length of time known as the evolution time. The frequencies of the nuclei are detected after the final pulse. By incrementing the evolution time in successive experiments, a two-dimensional data set is generated from a series of one-dimensional experiments.2

An example of a two-dimension NMR experiment is the homonuclear correlation spectroscopy (COSY) sequence, which consists of a pulse (p1) followed by an evolution time (t1) followed by a second pulse (p2) followed by a measurement time (t2). A computer is used to compile the spectra as a function of the evolution time (t1). Finally, the Fourier transform is used to convert the time-dependent signals into a two-dimensional spectrum.

The two-dimensional spectrum that results from the COSY experiment shows the frequencies for a single isotope (usually hydrogen, 1H) along both axes. (Techniques have also been devised for generating heteronuclear correlation spectra, in which the two axes correspond to different isotopes, such as 13C and 1H.) The intensities of the peaks in the spectrum can be represented using a third dimension. More commonly, intensity is indicated using contours or different colors. The spectrum is interpreted starting from the diagonal, which consists of a series of peaks. The peaks that appear off of the diagonal are called cross-peaks. The cross-peaks are symmetrical (both above and below) the diagonal and indicate which hydrogen atoms are spin-spin coupled to each other. One can determine which atoms are connected to one another by only a few chemical bonds by matching the center of a cross-peak with the center of each of two corresponding diagonal peaks. The peaks on the diagonal when matched with cross-peaks are coupled to each other.

For example: a CH3CH2COCH3 molecule (ethanone) would show three peaks on the diagonal, due to the three distinct hydrogen groups. By drawing a line straight down from a cross-peak to the point on the diagonal directly above or below it, and then drawing a line from the cross-peak directly across to another peak on the diagonal, one can determine which peaks are coupled. This is done in such a way that the lines from the cross-peak form a 90° angle between the two peaks on the diagonal. The matching peaks, as determined by using the cross-peaks, indicate which hydrogen are coupled, giving a clearer understanding of the structure of the molecule under examination.

Above is an example of a COSY NMR spectrum of progesterone in DMSO-d6. The spectrum that appears along both the x- and y-axes is a regular one dimensional 1H NMR spectrum. The COSY is read along the diagonal - where the bulk of the peaks appear. Cross-peaks appear symmetrically above and below the diagonal.

### How COSY NMR works

COSY-90 is the most common COSY experiment. In COSY-90, the sample is irradiated with a radio frequency pulse, p1, which tilts the nuclear spin by 90°. After p1, the sample is allowed to freely precess during an evolution period (t1). A second 90° pulse, p2, is then applied, after which the experimental data are acquired. This is done repeatedly using a series of different evolution periods (t1). At the conclusion of data acquisition the data is Fourier transformed in each dimension to generate the two dimensional spectrum. It is only because the evolution period is varied that cross-peaks appear in the spectrum.

Cross-peaks result from a phenomenon called magnetization transfer. Depending on the experiment, this transfer can be achieved through space or bonds, or even through chemical or physical means. In COSY, magnetization transfer occurs through the bonds.

Another member of the COSY family is COSY-45. In COSY-45 a 45° pulse is used instead of a 90° pulse for the first pulse, p1. The advantage of a COSY-45 is that the diagonal-peaks are less pronounced, making it simpler to match cross-peaks near the diagonal in a large molecule. Additionally, the relative signs of the coupling constants can be elucidated from a COSY-45 spectrum. This is not possible using COSY-903. Overall, the COSY-45 offers a cleaner spectrum while the COSY-90 is more sensitive. Related COSY techniques include double quantum filtered COSY and multiple quantum filtered COSY.

COSY NMR has useful applications. Organic chemists often use COSY to elucidate structural data on molecules that are not satisfactorily represented in a one-dimensional NMR spectrum. Using cross-peaks, along with the diagonal spectrum, one can often discover much about the structure of an unknown molecule.

### Notes

1. Martin, G.E; Zekter, A.S., ‘’Two-Dimensional NMR Methods for Establishing Molecular Connectivity’’; VCH Pusblishers, Inc: New York, 1988 (p.59)
2. Akitt, J.W.; Mann, B.E., ‘’NMR and Chemistry’’; Stanley Thornes: Cheltenham, UK, 2000. (p273)
3. Akitt, J.W.; Mann, B.E., ‘’NMR and Chemistry’’; Stanley Thornes: Cheltenham, UK, 2000. (p287)

## References

Hornak, Joseph P. The Basics of NMR

J. Keeler, Understanding NMR Spectroscopy

Wuthrich, Kurt NMR of Proteins and Nucleic Acids Wiley-Interscience, New York, NY USA 1986.