NMR spectroscopy

NMR spectroscopy (MR spectroscopy, NMR or Nuclear Magnetic Resonance Spectroscopy) is a variant of spectroscopy that employs the principle of nuclear magnetic resonance to investigate chemical properties of samples, typically ranging from the molecular to the tissue scale but also occasionally beyond. As such, it measures the energy differences between spin states of nuclei in the presence of a magnetic field by using radio frequency electromagnetic radiation. The energy differences between the spin states of the nuclei depend upon the nature of the atom and are influenced by its environment. However, NMR spectroscopy is not limited to measurement of the energy differences between the spin states. NMR signals are also influenced by the motion of the nucleus and the rotational motion of the molecule within which the observed nucleus resides. Therefore, NMR spectroscopy provides static (structure and composition) as well as dynamic information regarding the system of interest, e.g., proteins, DNA and other natural products.

Pulses of radio-frequency electromagnetic radiation can be used to perturb the nuclear spin systems in a variety of ways; the time dependent response of the system of interest can be recorded and analyzed:

• to correlate different spectral properties of nuclei and/or
• to extract information regarding interactions between nuclear spins within the same molecule and/or
• to obtain information regarding intermolecular interactions.

The amplitude, frequency, phase and duration of pulses of radio-frequency electromagnetic radiation, as well as static and dynamic position dependent magnetic fields, can be varied in many combinations giving rise to a wide variety of NMR spectroscopic experiments.

NMR spectroscopic techniques are used extensively for structural elucidation of natural products and for quantitative analysis of components of complex mixtures such as body fluids. However, its applications are not limited to these systems; and NMR spectroscopy has been used for the study of matter in disordered, ordered and partially ordered systems such as gases, liquids, quantum fluids , superconductors, solutions, amorphous solids, crystalline solids, liquid crystals, membranes and living organisms. NMR spectroscopic methods have also found use in quantum computing.

NMR spectroscopy: Observable parameters

The Chemical shift

The nuclei in molecules are surrounded by electrons. The applied magnetic field induces a circulation of electrons, which in turn produces an additional magnetic field. This effect is called 'shielding', because the magnetic field induced by the circulation of charge is in opposite direction to the field (external) that was responsible for the circulation of charge. Therefore, the nuclei in a given molecule are subjected to a net magnetic field which is the sum of the applied magnetic field and the induced magnetic field.

$B_{induced}=-\sigma B_{applied}\quad \quad \quad B_{net}=B_{applied}+B_{induced}$

where $\scriptstyle \sigma$ is a tensor; however for molecules rotating rapidly in all directions, such as molecules in a gas, liquid or solution, $\sigma$ can be approximated by a scalar and is known as the shielding constant.

$\scriptstyle \sigma$ depends upon the electron density as well as polarizability which vary substantially depending upon the nature of the molecule and the location within the molecule. As a consequence, for a given value of the applied field, nuclei in different molecules and even nuclei in different parts of the same molecule may encounter different net values of the magnetic field - this results in a variation in the resonance frequency.

$\nu _{i}={\frac {\gamma B_{net,i}}{2\pi }}$

where $\scriptstyle \nu _{i}$ is the resonance frequency for the nucleus labeled 'i', $\scriptstyle B_{net,i}$ is the net magnetic field at nucleus 'i' and $\scriptstyle \gamma$ is the magnetogyric ratio.

The relative change in the resonance frequency is called the chemical shift.

$\delta _{i}(ppm)={\frac {(\nu _{i}-\nu _{ref})*10^{6}}{\nu _{ref}}}$

where $\scriptstyle \nu _{ref}$ is the resonance frequency of a standard. The chemical shift $\delta$ is generally quoted in ppm. For proton NMR spectroscopy of organic compounds, tetramethylsilane was chosen as the standard.

The chemical shifts of functional groups in many molecules have been measured and provide a basis for identification of functional groups in novel molecules.

Signal area:

The area of a signal is proportional to the total number of contributing nuclei. e.g. In the 1H NMR spectrum of a pure compound containing a single methyl group and a single methylene group, the relative areas of the corresponding signals will be in the ratio 3:2, because there are 3 protons in the methyl group and 2 in the methylene group.

If the pure compound can exist in multiple conformations, and if the transitions between the different conformational states occur on a time scale that is slow with reference to the relevant NMR time scale (typically msec) then it may be possible to observe separate signals corresponding to the different conformations in the NMR spectrum. In such a case, the relative areas of the signals corresponding to different conformations will be proportional to the fractions of the molecules that exist in each conformational state during the time that the NMR spectrum was acquired.

Many small molecules can exist in multiple conformations due to rotation about single bonds, however, such rotation is generally rapid at room temperature and hence separate signals are usually not observed for the different rotational isomers that exist at equilibrium. For large molecules, e.g. proteins the close packing of the constituent amino acids results in restricted rotation about many single bonds and complete loss of conformational freedom for many other single bonds. Such restricted rotation about single bonds results in separate NMR signal for each conformer and the areas of the NMR signals are again proportional to the populations of the conformers.

In the NMR spectra of mixtures, the relative areas of signals can be used to deduce information regarding the relative concentration of each compound. The very high resolution of NMR spectra enables us to resolve a very large number of compounds, and hence determine the relative concentrations of a large number of species. This is of particular use in vivo spectroscopy. The ability to resolve the signals of a large number of chemicals/biochemicals can be enhanced significantly by using 2D NMR.

Systems undergoing physical and chemical changes can also be monitored by NMR spectroscopy as the interconverting species usually have different chemical shifts. The changes in the relative areas of the reactants and products can be used to obtain information regarding the time course of the reaction. This is possible if the time scale of the conversion is slow with respect to the NMR time scale.

J-coupling and multiplicity

J-coupling, also known as scalar coupling, is due to the interaction between different nuclei in the same molecule that is mediated through electrons in chemical bonds. Usually, the J-coupling interaction is observable between nuclei that are separated from each other by three or fewer bonds. The effect of this interaction on the observable spectrum is that the signals of a given nuclei are 'split' if the nucleus has a J-coupling interaction with neighboring nuclei. If this coupling is weak, the resulting pattern can be used to deduce information regarding the number of neighboring nuclei. This information plays a critical role in structural elucidation of small organic molecules. In addition, the magnitude of the coupling constant provides information regarding the conformation of a molecule.

When weak coupling is applicable, the observed multiplicity of a signal provides useful structural information. The multiplicity of a signal is expected to be 2nI+1 where n is the number of nuclei with spin angular momentum quantum number (spin) I that have J coupling with the nucleus responsible for the signal being observed. If all magnetically active nuclei are protons, then the multiplicity is equal to n+1, because I=1/2 for protons. In addition, the relative intensities of the peaks within a multiplet is given by the coefficients of the binomial expansion. This is known as the n+1 rule. In using the n+1 rule, couplings between magnetically equivalent nuclei are not considered because the effects of J-coupling are not observable in a normal 1D NMR spectrum if the nuclei involved in the interaction are magnetically equivalent. e.g. in the fragment CH3-CH2-O- , the signal corresponding to the methyl (CH3) protons will have a multiplicity of three, because each of the methyl hydrogen has a J-coupling to each of the two methylene (CH2) protons, hence n=2, I=1/2 and 2nI+1 = 3. The signal corresponding to the methylene group of protons will be a quartet, as each methylene proton is expected to have J coupling to each of the three methyl (CH3) protons, hence n=3, I=1/2 and 2nI+1 = 4. In this case the pairs of nuclei that have observable J-coupling are separated by three bonds. Note that interactions between magnetically equivalent nuclei, such as interactions between the different protons of the same methyl group (separated by only two bonds), need not be considered when predicting the multiplicity. In this example the three protons in the methyl group are treated as magnetically equivalent and the two protons in the methylene group are treated as magnetically equivalent by assuming that free rotation occurs about the single bond separating the methyl and methylene groups. If this is not the case, e.g. if restricted rotation occurs about the single bond and if the fragment is placed in an asymmetric environment (such as part of a rigid macromolecule) then the methylene protons may no longer be assumed to be magnetically equivalent.

The 2nI+1 rule assumes that all the neighboring nuclei (i.e. those with observable J-coupling to nucleus being observed) are equivalent. If this is not the case, then the multiplet structure can still be predicted by considering first the coupling to one set of equivalent J-coupled nuclei and predicting that the effect of the remaining nuclei will cause additional splitting of the multiplet generated by interaction with the first set.

The magnitude of the J-coupling depends on the number of bonds separating the interacting nuclei as well as on the geometry. For ³J coupling, i.e., coupling between nuclei separated by three bonds, the magnitude of the coupling between the interacting nuclei is given by the Karplus equation

$^{3}J=A+B\cos {\theta }+C\cos ^{2}{\theta }\qquad or\qquad ^{3}J=A+B\cos {\theta }+C\cos {2\theta }$

where θ is the dihedral angle involving the nuclei of interest. The values of A,B and C depend on the chemical nature of the fragment. The Karplus equation is useful for conformational analysis of molecules, particurly biopolymers such as polypeptides and nucleic acids. The parameters A,B and C of the Karplus equation have been determined for most combinations of nuclei that occur in organic compounds, by comparing experimental NMR and X-ray crystallographic data. If relevant experiment data is not available the magnitude of the J coupling may be calculated by using quantum chemical calculations; however the prediction accuracy of ab initio calculations is often lower than that obtained from empirical parametrization.

The net Magnetization vector

In the presence of an external magnetic field B, the distribution of the number of nuclei in the different allowed nuclear spin states follows the Boltzmann distribution at equilibrium. i.e. for a two state system with states $\alpha$ and $\beta$

${\frac {n_{\beta }}{n_{\alpha }}}=e^{\frac {E_{\alpha }-E_{\beta }}{kT}}$

where

$n_{\beta }=$ number of nuclei in state $\beta$; $n_{\alpha }=$ number of nuclei in state $\alpha$

$E_{\beta }=$ Energy of state $\beta$; $E_{\alpha }=$ Energy of state $\alpha$

T = Absolute temperature in Kelvin; $\quad$ k=Boltzmann constant

If the energy of state $\beta$ is greater than that of state $\alpha$, then fewer nuclei will be present in the higher energy state $\beta$. However, since the energy difference between these two states is small the population difference is quite small and the macroscopically observable net magnetization vector for the entire population will be the vectorial sum of the magnetic moments of all the nuclei present in the sample. At equilibrium, in a homogeneous magnetic field, although the orientation of the spin magnetic moments of individual nuclei is restricted to certain directions with respect to the applied magnetic field, there is no preference for any direction for the projection of the magnetic moment onto the x-y plane. The X- and Y-components of the net magnetization vector of the sample should be zero at equilibrium because the X- and Y-components of the spin magnetic moments of individual nuclei are randomly oriented and the net magnetization vector is the vectorial sum of the spin magnetic moments of individual nuclei. Therefore, the equilibrium magnetization will be

$M_{equil}=M_{z,equil}=N<\mu _{z}>={\frac {N\sum {g_{i}\mu _{zi}e^{-{\frac {E_{i}}{kT}}}}}{\sum {g_{i}e^{-{\frac {E_{i}}{kT}}}}}}$

where

N = total number of nuclei in sample; $M_{equil}=$ Magnetization vector (net) at equilibrium

$E_{i}=$ energy of state i; $\quad \quad \quad \mu _{zi}=$ z-component of magnetic moment of state i

$g_{i}=$ degeneracy of state i

For a two state system, the $\mu _{zi}$ are equal in magnitude but opposite in direction. However, there are more nuclei in the lower energy state at the temperature of the experiment T, hence the numerator in the sum above is nonzero. Therefore, at equilibrium, the net magnetization vector has only one non-zero component (the $M_{z,equil}$-component) when the system is placed in a static homogeneous magnetic field.

In the presence of homogeneous magnetic field (possibly time dependent), represented by the vector B, the magnetization vector experiences a torque and the time dependence of the net magnetization vector M (neglecting relaxation effects) is

${\frac {dM}{dt}}=M\times \gamma B$

The Bloch equations describe the time dependence of the different components of the net magnetization vector subject to relaxation, in the presence of a time dependent magnetic field.

Relaxation time

After a collection of nuclei in a magnetic field that is at equilibrium with its surroundings is perturbed in some manner (usually by a pulse of electromagnetic radiation) the system requires a certain amount of the time to return to equilibrium. If this process is exponential, the rate constant is called relaxation rate. The relaxation rate is inversely proportional to relaxation time . T1 relaxation time characterizes the return to equilibrium of the longitudinal component of the magnetization of the collection of nuclei. Similarly, T2 relaxation time characterizes the return to equilibrium of the transverse component of the magnetization of the collection of nuclei that are being studied. In a static homogeneous magnetic field, the transverse component of the net magnetization vector for a sufficiently large collection of nuclei is always zero, at equilibrium.

The dominant mechanism of relaxation is usually the dipolar interaction with the closest neighbor. For a pair of identical spins separated by a distance $r$, in a homogeneous magnetic field, undergoing isotropic rotational motion in solution, the relaxation rates are:

${\frac {1}{T_{1}}}={\Big (}{\frac {\mu _{0}}{4\pi }}{\Big )}^{2}\quad {\frac {\gamma ^{4}\hbar ^{2}}{10r^{6}}}(3J(w_{o})+12J(2w_{o}))$

${\frac {1}{T_{2}}}={\Big (}{\frac {\mu _{0}}{4\pi }}{\Big )}^{2}\quad {\frac {\gamma ^{4}\hbar ^{2}}{20r^{6}}}(9J(0)+15J(w_{o})+6J(2w_{o}))$

where $J(w)={\frac {\tau _{c}}{1+w^{2}\tau _{c}^{2}}}$ is the spectral density function

$\tau _{c}$ is the mean time required for rotational motion of the pair of nuclei by one radian

$\mu _{0}$ is the magnetic permeability of free space

$\gamma$ is the gyromagnetic ratio and

$\hbar$ is the Planck's constant divided by $2\pi$.

For a pair of nonidentical spins I and S, separated by a distance r, the time dependence of the expectation values of the z-components of magnetization are given by the Solomon equations:

${\begin{aligned}{\frac {d<I_{z}>}{dt}}=-\rho _{I}(<I_{z}>-<I_{z,equil}>)-\sigma (<S_{z}>-<S_{z,equil}>)\\{\frac {d<S_{z}>}{dt}}=-\rho _{S}(<S_{z}>-<S_{z,equil}>)-\sigma (<I_{z}>-<I_{z,equil}>)\end{aligned}}$

$\rho$ and $\sigma$ are the auto- and cross relaxation rates, respectively. $\sigma$ and is responsible for the Nuclear overhauser effect.

${\begin{aligned}\rho _{I}&={\Big (}{\frac {\mu _{0}}{4\pi }}{\Big )}^{2}\quad {\frac {\gamma _{I}^{2}\gamma _{S}^{2}\hbar ^{2}}{10r^{6}}}(J(w_{I}-w_{S})+3J(w_{I})+6J(w_{I}+w_{S}))\\\sigma &={\Big (}{\frac {\mu _{0}}{4\pi }}{\Big )}^{2}\quad {\frac {\gamma _{I}^{2}\gamma _{S}^{2}\hbar ^{2}}{10r^{6}}}(-J(w_{I}-w_{S})+6J(w_{I}+w_{S})))\\{\frac {1}{T_{2}}}&={\Big (}{\frac {\mu _{0}}{4\pi }}{\Big )}^{2}\quad {\frac {\gamma _{I}^{2}\gamma _{S}^{2}\hbar ^{2}}{20r^{6}}}(4J(0)+J(w_{I}-w_{S})+3J(w_{I})+6J(w_{I}+w_{S})+6J(w_{S}))\\\end{aligned}}$

A homogeneous magnetic field, isotropic rotational motion and negligible chemical shift anisotropy are assumed in the derivation of the equations described above.

Dipolar coupling and Nuclear Overhauser effect

Irradiation at the resonance frequency of one nucleus in the molecule may cause changes in the intensity of a signal at a different frequency corresponding to another nucleus - this is called the Nuclear Overhauser effect (Noe). Nuclear overhauser effect is due to dipole-dipole interactions between the magnetic moments of a pair of nuclei. Unlike J-coupling, this interaction is not mediated through bonds. Hence, it may be possible to observe the Nuclear Overhauser effect between pairs of nuclei separated by many bonds provided that they are in spatial proximity. The strength of the observable Nuclear overhauser effect for molecules in solution is proportional to the inverse of the sixth power of the distance between the two nuclei due to averaging caused by rotational motion. Both the magnitude as well as the sign of the nuclear Overhauser effect depend on the rotational frequencies of the pair of nuclei with respect to the applied magnetic field. The Noe enhancement, $\eta$, in the steady state, when the resonance frequency of spin I is irradiated and the intensity of spin S is monitored is $\eta$ = ($\sigma$ $\gamma$_I )/ ($\rho$_S $\gamma$_S). This indicates that considerable enhancement in the intensity of the S signal can be obtained by irradiation at the frequency of the I spin, provided that $\gamma$_I/$\gamma$_S is greater than 1, because $\sigma$/$\rho$_S is approximately 1/2 when w$\tau _{c}$ is much smaller than 1 .

However, when w$\tau _{c}$ is much greater than 1, $\sigma$/$\rho$_S approaches -1 and negative Noe enhancements are obtained.

The sign of $\eta$ changes from positive to negative when w$\tau _{c}$ is close to one and under such conditions the Noe effect may not be observable. This happens for rigid molecules with relative molecular mass about 500 at room temperature e.g. many hexapeptides.

For most molecules (in isotropic media, at room temperature) direct observation of dipolar coupling in solution NMR spectra (as a separation of the components of a signal) is not possible, because dipolar interactions are averaged over all orientations of the molecule. However, it is possible to obtain partial alignment of macromolecules using anisotropic media, leading to incomplete averaging of the dipolar interactions - in such cases there is an 'apparent' change in the measured values of J-coupling constants because the residual dipolar coupling has the same form as the weak scalar coupling Hamiltonian. For a pair of nuclei, at a fixed distance, the magnitude and signs of the residual dipolar couplings are angle dependent. Hence, measurements of residual dipolar couplings (RDC) can be used to obtain angular constraints (used for structure determination).

Pulsed (radio frequency) NMR spectroscopy

Spin echoes

A pulse of radio frequency (RF) radiation is applied for sufficient duration to tip the net magnetization vector into the x-y plane, this is followed by a short delay $\tau$, followed by another r.f.pulse that inverts the magnetization around the x-axis, followed by another short delay $\tau$. The magnetization at the end of this pulse-sequence is

$M(2\tau )=M_{o}e^{-{\frac {2\tau }{T_{2}}}}$

The transverse relaxation time can be measured by repeating the experiment with different value of $\tau$, and plotting $log_{e}M(2\tau )$ vs $2\tau$. The resulting plot is expected to be a straight line with slope $-1/T_{2}$.

If the above experiment is carried out with a magnetic field gradient during the two delays $\tau$ then

$M(2\tau )=M_{o}e^{-{\frac {2\tau }{T_{2}}}}e^{-{\frac {2DG^{2}\tau ^{3}}{3}}}$

where D is the diffusion coefficient and G is the magnetic field gradient. Repeating this experiment with different values of G enables us to measure the Diffusion coefficient D.

Fourier transform NMR spectroscopy

Fourier transform NMR spectroscopy (Ernst ) is performed in the following way: A pulse (or a pulse train) of electromagnetic radiation is used to cause a perturbation in the sytem. The time dependent response of the system is recorded (this time dependent response is often referred to as the free induction decay, abbreviated as fid). A fourier transform of the response gives information regarding the frequency response. In the case of a single pulse perturbation, the fourier transform of the time dependent response is equivalent to a 1D NMR spectrum.

fid		NMR spectrum
free induction decay	FT{FID} → spectrum.	NMR Spectrum

Multidimensional NMR spectroscopy

Multidimensional NMR spectroscopy (nD NMR; Jeener ; Aue et al.) The nuclei in a magnetic field are subjected to a series of pulses of electromagnetic radiation separated by delays. The time dependent response of the system is recorded. The delays between the pulses may be fixed or incremented systematically between different repetitions of the experiment. The number of variable delays determines the dimensionality of the experiment. A multidimensional Fourier Transform of the entire data set characterizes the frequency responses of the system and enables a correlation between different NMR parameters. Different combinations of the pulses and delays known as 'pulse sequences' enable us to correlate and measure different types of NMR parameters. Some common examples of multidimensional NMR spectroscopy experiments (2D, 3D and 4D): COSY, NOESY, TOCSY, EXSY, HSQC, HNHA, HSQC-NOESY, HNCA, HNCO, HNCACO, HSQC-NOESY-HSQC. For a longer list of NMR experiments see List of NMR experiments.

This 2-D COSY spectrum shows two diagonal peaks and two cross peaks. The observed cross peaks are used to identify (correlate) the chemical shifts of nuclei that are J-coupled.

Two dimensional correlation spectroscopy (2D-COSY): Correlates chemical shifts of J-coupled nuclei.

Two dimensional nuclear overhauser effect spectroscopy (2D-NOESY): correlates chemical shifts of nuclei that exhibit significant Nuclear Overhauser effect. For molecules that experience free rotation along all three dimensions, the Nuclear Overhauser effect is generally observable between nuclei that are less than 5 angstroms apart.

Applications of NMR spectroscopy

Biomolecular structure determination

NMR spectroscopy can be used to determine the structure of macromolecules and to obtain information regarding their dynamics. However, the NMR spectra of macromolecules are much more complicated than those of small molecules and it is usually necessary to use multidimensional NMR spectroscopy in order to obtain data that can be used for structural analysis. The equential resonance assignment method was developed in order to associate specific nuclei in a protein with the observed resonance frequencies. Subsequently, the information obtained from quantitative and qualitative analysis of Nuclear Overhauser effects, J-coupling and chemical shifts is converted into geometric restraints. These geometric restraints are then subsequently used to build a model of the molecule.

Drug design and development

NMR spectroscopy is a useful tool in drug design and development. At present the most extensive application of NMR spectroscopy is for structure determination of ligands and receptors. In addition, NMR spectroscopic methods are used to study the interactions of ligands with receptors even when the binding is weak and transient ; and to build structure activity relationships.. Stable Ligand-Receptor complexes can be studied by either isotope-edited or isotope-filtered NMR spectroscopy. Either the ligand or the receptor can be labeled with an NMR active stable isotope. In isotope-edited NMR experiments, the spectra of the labeled part of the complex are selectively obtained; whereas, in isotope-filtered NMR experiments the spectra of the labeled part of the complex are selectively suppressed to record the signals from the unlabeled components of the complex. 2D-EXSY (Two-dimensional exchange spectroscopy) or transferred-NOe spectroscopy are generally used to study weaker ligand-receptor complexes (intermediate exchange). Weak and transient binding leads to rapid exchange of ligand between the complex and free form (fast exchange on the NMR time scale); even this can be detected by monitoring the changes in the chemical shift of the ligand or the chemical shifts of the residues of the receptor at the active site. A new approach to drug development (labeled SAR by NMR) has been developed based on the ability to detect the weak binding of fragments using NMR spectroscopic chemical shift changes; subsequently the fragments identified to bind weakly at separate sites on the receptor can be joined together using an inert linker and the resulting compound would be expected to bind much more strongly than either fragment (due to entropic and additive enthalpic effects).

NMR spectroscopy of drugs with stable isotope labels may be used to monitor the rate of dissolution of an administered drug, to monitor its absorption, distribution and elimination and also to identify the products into which it is transformed.

MR spectroscopy is of particular use in physiological studies, because the system of interest can be studied in a non-invasive manner, and also because it is possible to quantify the changes in a large set of metabolites simultaneously. Metabonomic studies characterize the physiological changes by quantifying the entire set of metabolites in an organism or its components - the ability of NMR spectroscopy to provide quantitative information on complex mixtures is of particular use in such studies.

Information regarding spatial distribution of NMR parameters may be obtained by using magnetic field gradients. This is the basis of Magnetic Resonance Imaging (MRI) a tool that has found extensive applications in medical diagnostics.

Food technology

NMR spectroscopy, usually in conjunction with other techniques, is useful for qualitative as well as quantitative determination of the constituents of food material. A few examples of application of magnetic resonance techniques in food science/technology are:

• NMR spectroscopy has been used to assess the quality of wines.
• Magnetic resonance techniques have been used to monitor the ripening of cheese. In addition, spectroscopic information has been used to determine the time dependence of the degradation of packed cheese.
• Magnetic resonance techniques have been applied to obtain process information in the production of bread.
• Magnetic resonance has been useful to study the time dependence of hydration/gelatinization of starch grains, which has important consequences for technology of many foods.
• NMR spectroscopy has been used for accurate determination of k-value, which is an indicator of freshness of fish.

In such applications, the ability of NMR spectroscopy to resolve complex mixtures, its non-invasive nature and its applicability to matter in a wide variety of states is particularly useful.

Summary

NMR spectroscopy is a versatile analytical technique for the study of structure, composition and dynamics of disordered, partially ordered and highly ordered systems in pure substances, complex mixtures as well as in live organisms.

Further reading

Fundamentals of Physics. 6th Ed. (2001). D.Halliday, R.Resnick and J.Walker. Wiley.

Biophysical Chemistry. Vol.II. (1980) C.R.Cantor and P.R.Schimmel. Freeman

Principles of Nuclear Magnetic Resonance spectroscopy in one and two dimensions. (1987). R.R.Ernst, G. Bodenhausen and A.Wokaun. Clarendon Press. Oxford.

NMR of proteins and nucleic acids. (1986). K.Wuthrich. Wiley.

Quantum description of high-resolution NMR in liquids. (1988) M.Goldman. Oxford.

Principles of Magnetic Resonance (1996) Charles P. Slichter. Springer Series in Solid-State Sciences.

Protein NMR spectroscopy: Principles and Practice. (2006) J.Cavanagh, W.J. Fairbrother, A6G. Palmer, N.J. Skelton, M.Rance. Academic Press.

Understanding NMR spectroscopy. (2005) J.Keeler. Wiley

Spectrometric Identification of Organic Compounds. (2005) Robert M. Silverstein, Francis X. Webster, and David Kiemle.

BioNMR in Drug Research. (2006) O.Zerbe, R.Mannhold, H.Kubinyi and G.Folkers. Wiley-VCH.

NMR in Drug Design. David J. Craik. (1995). Crc Series in Analytical Biotechnology.