Nuclear magnetic resonance spectroscopy

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Nuclear_magnetic_resonance_spectroscopy 

A 900 MHz NMR instrument with a 21.1 T magnet at HWB-NMR, Birmingham, UK

Nuclear magnetic resonance spectroscopy, commonly known as NMR spectroscopy or magnetic resonance spectroscopy (MRS), is a spectroscopic technique based on re-orientation of atomic nuclei with non-zero nuclear spins in an external magnetic field. This re-orientation occurs with absorption of electromagnetic radiation in the radio frequency region from roughly 4 to 900 MHz, which depends on the isotopic nature of the nucleus and increases proportionally to the strength of the external magnetic field. Notably, the resonance frequency of each NMR-active nucleus depends on its chemical environment. As a result, NMR spectra provide information about individual functional groups present in the sample, as well as about connections between nearby nuclei in the same molecule. As the NMR spectra are unique or highly characteristic to individual compounds and functional groups, NMR spectroscopy is one of the most important methods to identify molecular structures, particularly of organic compounds.

The principle of NMR usually involves three sequential steps:

1. The alignment (polarization) of the magnetic nuclear spins in an applied, constant magnetic field B₀.
2. The perturbation of this alignment of the nuclear spins by a weak oscillating magnetic field, usually referred to as a radio-frequency (RF) pulse.
3. Detection and analysis of the electromagnetic waves emitted by the nuclei of the sample as a result of this perturbation.

Similarly, biochemists use NMR to identify proteins and other complex molecules. Besides identification, NMR spectroscopy provides detailed information about the structure, dynamics, reaction state, and chemical environment of molecules. The most common types of NMR are proton and carbon-13 NMR spectroscopy, but it is applicable to any kind of sample that contains nuclei possessing spin.

NMR spectra are unique, well-resolved, analytically tractable and often highly predictable for small molecules. Different functional groups are obviously distinguishable, and identical functional groups with differing neighboring substituents still give distinguishable signals. NMR has largely replaced traditional wet chemistry tests such as color reagents or typical chromatography for identification.

The most significant drawback of NMR spectroscopy is its poor sensitivity (compared to other analytical methods, such as mass spectrometry). Typically 2–50 mg of a substance is required to record a decent-quality NMR spectrum. The NMR method is non-destructive, thus the substance may be recovered. To obtain high-resolution NMR spectra, solid substances are usually dissolved to make liquid solutions, although solid-state NMR spectroscopy is also possible.

The timescale of NMR is relatively long, and thus it is not suitable for observing fast phenomena, producing only an averaged spectrum. Although large amounts of impurities do show on an NMR spectrum, better methods exist for detecting impurities, as NMR is inherently not very sensitive – though at higher frequencies, sensitivity is higher.

Correlation spectroscopy is a development of ordinary NMR. In two-dimensional NMR, the emission is centered around a single frequency, and correlated resonances are observed. This allows identifying the neighboring substituents of the observed functional group, allowing unambiguous identification of the resonances. There are also more complex 3D and 4D methods and a variety of methods designed to suppress or amplify particular types of resonances. In nuclear Overhauser effect (NOE) spectroscopy, the relaxation of the resonances is observed. As NOE depends on the proximity of the nuclei, quantifying the NOE for each nucleus allows construction of a three-dimensional model of the molecule.

Cutaway of an NMR magnet that shows its structure: radiation shield, vacuum chamber, liquid-nitrogen vessel, liquid-helium vessel, and cryogenic shims.

NMR spectrometers are relatively expensive; universities usually have them, but they are less common in private companies. Between 2000 and 2015, an NMR spectrometer cost around 0.5–5 million USD. Modern NMR spectrometers have a very strong, large and expensive liquid-helium-cooled superconducting magnet, because resolution directly depends on magnetic field strength. Higher magnetic field also improves the sensitivity of the NMR spectroscopy, which depends on the population difference between the two nuclear levels, which increases exponentially with the magnetic field strength.

Less expensive machines using permanent magnets and lower resolution are also available, which still give sufficient performance for certain applications such as reaction monitoring and quick checking of samples. There are even benchtop nuclear magnetic resonance spectrometers. NMR spectra of protons (¹H nuclei) can be observed even in Earth magnetic field. Low-resolution NMR produces broader peaks, which can easily overlap one another, causing issues in resolving complex structures. The use of higher-strength magnetic fields result in a better sensitivity and higher resolution of the peaks, and it is preferred for research purposes.

History

Credit for the discovery of NMR goes to Isidor Isaac Rabi, who received the Nobel Prize in Physics in 1944. The Purcell group at Harvard University and the Bloch group at Stanford University independently developed NMR spectroscopy in the late 1940s and early 1950s. Edward Mills Purcell and Felix Bloch shared the 1952 Nobel Prize in Physics for their inventions.

NMR-active criteria

The key determinant of NMR activity in atomic nuclei is the nuclear spin quantum number (I). This intrinsic quantum property, similar to an atom's "spin", characterizes the angular momentum of the nucleus. To be NMR-active, a nucleus must have a non-zero nuclear spin (I ≠ 0). It is this non-zero spin that enables nuclei to interact with external magnetic fields and show signals in NMR. Atoms with an odd sum of protons and neutrons exhibit half-integer values for the nuclear spin quantum number (I = 1/2, 3/2, 5/2, and so on). These atoms are NMR-active because they possess non-zero nuclear spin. Atoms with an even sum but both an odd number of protons and an odd number of neutrons exhibit integer nuclear spins (I = 1, 2, 3, and so on). Conversely, atoms with an even number of both protons and neutrons have a nuclear spin quantum number of zero (I = 0), and therefore are not NMR-active. NMR-active nuclei, particularly those with a spin quantum number of 1/2, are of great significance in NMR spectroscopy. Examples include ¹H, ¹³C, ¹⁵N, and ³¹P.

Main aspects of NMR techniques

Resonant frequency

When placed in a magnetic field, NMR active nuclei (such as ¹H or ¹³C) absorb electromagnetic radiation at a frequency characteristic of the isotope. The resonant frequency, energy of the radiation absorbed, and the intensity of the signal are proportional to the strength of the magnetic field. For example, in a 21-tesla magnetic field, hydrogen nuclei (protons) resonate at 900 MHz. It is common to refer to a 21 T magnet as a 900 MHz magnet, since hydrogen is the most common nucleus detected. However, different nuclei will resonate at different frequencies at this field strength in proportion to their nuclear magnetic moments.

Sample handling

The NMR sample is prepared in a thin-walled glass tube – an NMR tube.

An NMR spectrometer typically consists of a spinning sample-holder inside a very strong magnet, a radio-frequency emitter, and a receiver with a probe (an antenna assembly) that goes inside the magnet to surround the sample, optionally gradient coils for diffusion measurements, and electronics to control the system. Spinning the sample is usually necessary to average out diffusional motion, however, some experiments call for a stationary sample when solution movement is an important variable. For instance, measurements of diffusion constants (diffusion ordered spectroscopy or DOSY) are done using a stationary sample with spinning off, and flow cells can be used for online analysis of process flows.

Deuterated solvents

The vast majority of molecules in a solution are solvent molecules, and most regular solvents are hydrocarbons and so contain NMR-active hydrogen-1 nuclei. In order to avoid having the signals from solvent hydrogen atoms overwhelm the experiment and interfere in analysis of the dissolved analyte, deuterated solvents are used where >99% of the protons are replaced with deuterium (hydrogen-2). The most widely used deuterated solvent is deuterochloroform (CDCl₃), although other solvents may be used for various reasons, such as solubility of a sample, desire to control hydrogen bonding, or melting or boiling points. The chemical shifts of a molecule change slightly between solvents, and therefore the solvent used is almost always reported with chemical shifts. Proton NMR spectra are often calibrated against the known solvent residual proton peak as an internal standard instead of adding tetramethylsilane (TMS), which is conventionally defined as having a chemical shift of zero.

Shim and lock

To detect the very small frequency shifts due to nuclear magnetic resonance, the applied magnetic field must be extremely uniform throughout the sample volume. High-resolution NMR spectrometers use shims to adjust the homogeneity of the magnetic field to parts per billion (ppb) in a volume of a few cubic centimeters. In order to detect and compensate for inhomogeneity and drift in the magnetic field, the spectrometer maintains a "lock" on the solvent deuterium frequency with a separate lock unit, which is essentially an additional transmitter and RF processor tuned to the lock nucleus (deuterium) rather than the nuclei of the sample of interest. In modern NMR spectrometers shimming is adjusted automatically, though in some cases the operator has to optimize the shim parameters manually to obtain the best possible resolution.

Acquisition of spectra

Upon excitation of the sample with a radio frequency (60–1000 MHz) pulse, a nuclear magnetic resonance response – a free induction decay (FID) – is obtained. It is a very weak signal and requires sensitive radio receivers to pick up. A Fourier transform is carried out to extract the frequency-domain spectrum from the raw time-domain FID. A spectrum from a single FID has a low signal-to-noise ratio, but it improves readily with averaging of repeated acquisitions. Good ¹H NMR spectra can be acquired with 16 repeats, which takes only minutes. However, for elements heavier than hydrogen, the relaxation time is rather long, e.g. around 8 seconds for ¹³C. Thus, acquisition of quantitative heavy-element spectra can be time-consuming, taking tens of minutes to hours.

Following the pulse, the nuclei are, on average, excited to a certain angle vs. the spectrometer magnetic field. The extent of excitation can be controlled with the pulse width, typically about 3–8 μs for the optimal 90° pulse. The pulse width can be determined by plotting the (signed) intensity as a function of pulse width. It follows a sine curve and, accordingly, changes sign at pulse widths corresponding to 180° and 360° pulses.

Decay times of the excitation, typically measured in seconds, depend on the effectiveness of relaxation, which is faster for lighter nuclei and in solids, slower for heavier nuclei and in solutions, and can be very long in gases. If the second excitation pulse is sent prematurely before the relaxation is complete, the average magnetization vector has not decayed to ground state, which affects the strength of the signal in an unpredictable manner. In practice, the peak areas are then not proportional to the stoichiometry; only the presence, but not the amount of functional groups is possible to discern. An inversion recovery experiment can be done to determine the relaxation time and thus the required delay between pulses. A 180° pulse, an adjustable delay, and a 90° pulse is transmitted. When the 90° pulse exactly cancels out the signal, the delay corresponds to the time needed for 90° of relaxation. Inversion recovery is worthwhile for quantitative ¹³C, ²D and other time-consuming experiments.

Spectral interpretation

NMR signals are ordinarily characterized by three variables: chemical shift, spin–spin coupling, and relaxation time.

Chemical shift

Main article: Chemical shift

Example of the chemical shift: NMR spectrum of hexaborane B₆H₁₀ showing peaks shifted in frequency, which give clues as to the molecular structure

The energy difference ΔE between nuclear spin states is proportional to the magnetic field (Zeeman effect). ΔE is also sensitive to the electronic environment of the nucleus, giving rise to what is known as the chemical shift, δ. The simplest types of NMR graphs are plots of the different chemical shifts of the nuclei being studied in the molecule. The value of δ is often expressed in terms of "shielding": shielded nuclei have higher ΔE. The range of δ values is called the dispersion. It is rather small for ¹H signals, but much larger for other nuclei. NMR signals are reported relative to a reference signal, usually that of TMS (tetramethylsilane). Additionally, since the distribution of NMR signals is field-dependent, these frequencies are divided by the spectrometer frequency. However, since we are dividing Hz by MHz, the resulting number would be too small, and thus it is multiplied by a million. This operation therefore gives a locator number called the "chemical shift" with units of parts per million. The chemical shift provides structural information.

The conversion of chemical shifts (and J's, see below) is called assigning the spectrum. For diamagnetic organic compounds, assignments of ¹H and ¹³C NMR spectra are extremely sophisticated because of the large databases and easy computational tools. In general, chemical shifts for protons are highly predictable, since the shifts are primarily determined by shielding effects (electron density). The chemical shifts for many heavier nuclei are more strongly influenced by other factors, including excited states ("paramagnetic" contribution to shielding tensor). This paramagnetic contribution, which is unrelated to paramagnetism) not only disrupts trends in chemical shifts, which complicates assignments, but it also gives rise to very large chemical shift ranges. For example, most ¹H NMR signals for most organic compounds are within 15 ppm. For ³¹P NMR, the range is hundreds of ppm.

In paramagnetic NMR spectroscopy, the samples are paramagnetic, i.e. they contain unpaired electrons. The paramagnetism gives rise to very diverse chemical shifts. In ¹H NMR spectroscopy, the chemical shift range can span up to thousands of ppm.

J-coupling

Main article: J-coupling

Multiplicity	Intensity ratio
Singlet (s)	1
Doublet (d)	1:1
Triplet (t)	1:2:1
Quartet (q)	1:3:3:1
Quintet	1:4:6:4:1
Sextet	1:5:10:10:5:1
Septet	1:6:15:20:15:6:1

Some of the most useful information for structure determination in a one-dimensional NMR spectrum comes from J-coupling, or scalar coupling (a special case of spin–spin coupling), between NMR active nuclei. This coupling arises from the interaction of different spin states through the chemical bonds of a molecule and results in the splitting of NMR signals. For a proton, the local magnetic field is slightly different depending on whether an adjacent nucleus points towards or against the spectrometer magnetic field, which gives rise to two signals per proton instead of one. These splitting patterns can be complex or simple and, likewise, can be straightforwardly interpretable or deceptive. This coupling provides detailed insight into the connectivity of atoms in a molecule.

The multiplicity of the splitting is an effect of the spins of the nuclei that are coupled and the number of such nuclei involved in the coupling. Coupling to n equivalent spin-1/2 nuclei splits the signal into a n + 1 multiplet with intensity ratios following Pascal's triangle as described in the table. Coupling to additional spins leads to further splittings of each component of the multiplet, e.g. coupling to two different spin-1/2 nuclei with significantly different coupling constants leads to a doublet of doublets (abbreviation: dd). Note that coupling between nuclei that are chemically equivalent (that is, have the same chemical shift) has no effect on the NMR spectra, and couplings between nuclei that are distant (usually more than 3 bonds apart for protons in flexible molecules) are usually too small to cause observable splittings. Long-range couplings over more than three bonds can often be observed in cyclic and aromatic compounds, leading to more complex splitting patterns.

Example ¹H NMR spectrum (1-dimensional) of ethanol plotted as signal intensity vs. chemical shift. There are three different types of H atoms in ethanol regarding NMR: the hydrogen (H) on the −OH group is not coupling with the other H atoms and appears as a singlet, but the CH₃− and the −CH₂− hydrogens are coupling with each other, resulting in a triplet and quartet respectively.

For example, in the proton spectrum for ethanol, the CH₃ group is split into a triplet with an intensity ratio of 1:2:1 by the two neighboring CH₂ protons. Similarly, the CH₂ is split into a quartet with an intensity ratio of 1:3:3:1 by the three neighboring CH₃ protons. In principle, the two CH₂ protons would also be split again into a doublet to form a doublet of quartets by the hydroxyl proton, but intermolecular exchange of the acidic hydroxyl proton often results in a loss of coupling information.

Coupling to any spin-1/2 nuclei such as phosphorus-31 or fluorine-19 works in this fashion (although the magnitudes of the coupling constants may be very different). But the splitting patterns differ from those described above for nuclei with spin greater than 1/2 because the spin quantum number has more than two possible values. For instance, coupling to deuterium (a spin-1 nucleus) splits the signal into a 1:1:1 triplet because the spin 1 has three spin states. Similarly, a spin-3/2 nucleus such as ³⁵Cl splits a signal into a 1:1:1:1 quartet and so on.

Coupling combined with the chemical shift (and the integration for protons) tells us not only about the chemical environment of the nuclei, but also the number of neighboring NMR active nuclei within the molecule. In more complex spectra with multiple peaks at similar chemical shifts or in spectra of nuclei other than hydrogen, coupling is often the only way to distinguish different nuclei.

The magnitude of the coupling (the coupling constant J) is an effect of how strongly the nuclei are coupled to each other. For simple cases, this is an effect of the bonding distance between the nuclei, the magnetic moment of the nuclei, and the dihedral angle between them.

Second-order (or strong) coupling

¹H NMR spectrum of menthol with chemical shift in ppm on the horizontal axis. Each magnetically inequivalent proton has a characteristic shift, and couplings to other protons appear as splitting of the peaks into multiplets: e.g. peak a, because of the three magnetically equivalent protons in methyl group a, couple to one adjacent proton (e) and thus appears as a doublet.

The above description assumes that the coupling constant is small in comparison with the difference in NMR frequencies between the inequivalent spins. If the shift separation decreases (or the coupling strength increases), the multiplet intensity patterns are first distorted, and then become more complex and less easily analyzed (especially if more than two spins are involved). Intensification of some peaks in a multiplet is achieved at the expense of the remainder, which sometimes almost disappear in the background noise, although the integrated area under the peaks remains constant. In most high-field NMR, however, the distortions are usually modest, and the characteristic distortions (roofing) can in fact help to identify related peaks.

Some of these patterns can be analyzed with the method published by John Pople, though it has limited scope.

Second-order effects decrease as the frequency difference between multiplets increases, so that high-field (i.e. high-frequency) NMR spectra display less distortion than lower-frequency spectra. Early spectra at 60 MHz were more prone to distortion than spectra from later machines typically operating at frequencies at 200 MHz or above.

Furthermore, as in the figure to the right, J-coupling can be used to identify ortho-meta-para substitution of a ring. Ortho coupling is the strongest at 15 Hz, Meta follows with an average of 2 Hz, and finally para coupling is usually insignificant for studies.

Magnetic inequivalence

Further information: Magnetic inequivalence

More subtle effects can occur if chemically equivalent spins (i.e., nuclei related by symmetry and so having the same NMR frequency) have different coupling relationships to external spins. Spins that are chemically equivalent but are not indistinguishable (based on their coupling relationships) are termed magnetically inequivalent. For example, the 4 H sites of 1,2-dichlorobenzene divide into two chemically equivalent pairs by symmetry, but an individual member of one of the pairs has different couplings to the spins making up the other pair.

Magnetic inequivalence can lead to highly complex spectra, which can only be analyzed by computational modeling. Such effects are more common in NMR spectra of aromatic and other non-flexible systems, while conformational averaging about C−C bonds in flexible molecules tends to equalize the couplings between protons on adjacent carbons, reducing problems with magnetic inequivalence.

Correlation spectroscopy

Further information: 2D-NMR

Correlation spectroscopy is one of several types of two-dimensional nuclear magnetic resonance (NMR) spectroscopy or 2D-NMR. This type of NMR experiment is best known by its acronym, COSY. Other types of two-dimensional NMR include J-spectroscopy, exchange spectroscopy (EXSY), Nuclear Overhauser effect spectroscopy (NOESY), total correlation spectroscopy (TOCSY), and heteronuclear correlation experiments, such as HSQC, HMQC, and HMBC. In correlation spectroscopy, emission is centered on the peak of an individual nucleus; if its magnetic field is correlated with another nucleus by through-bond (COSY, HSQC, etc.) or through-space (NOE) coupling, a response can also be detected on the frequency of the correlated nucleus. 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 1976.

Solid-state nuclear magnetic resonance

Solid-state 900 MHz (21.1 T) NMR spectrometer at the Canadian National Ultrahigh-field NMR Facility for Solids

Further information: Solid-state NMR

A variety of physical circumstances do not allow molecules to be studied in solution, and at the same time not by other spectroscopic techniques to an atomic level, either. In solid-phase media, such as crystals, microcrystalline powders, gels, anisotropic solutions, etc., it is in particular the dipolar coupling and chemical shift anisotropy that become dominant to the behaviour of the nuclear spin systems. In conventional solution-state NMR spectroscopy, these additional interactions would lead to a significant broadening of spectral lines. A variety of techniques allows establishing high-resolution conditions, that can, at least for ¹³C spectra, be comparable to solution-state NMR spectra.

Two important concepts for high-resolution solid-state NMR spectroscopy are the limitation of possible molecular orientation by sample orientation, and the reduction of anisotropic nuclear magnetic interactions by sample spinning. Of the latter approach, fast spinning around the magic angle is a very prominent method, when the system comprises spin-1/2 nuclei. Spinning rates of about 20 kHz are used, which demands special equipment. A number of intermediate techniques, with samples of partial alignment or reduced mobility, is currently being used in NMR spectroscopy.

Applications in which solid-state NMR effects occur are often related to structure investigations on membrane proteins, protein fibrils or all kinds of polymers, and chemical analysis in inorganic chemistry, but also include "exotic" applications like the plant leaves and fuel cells. For example, Rahmani et al. studied the effect of pressure and temperature on the bicellar structures' self-assembly using deuterium NMR spectroscopy. Solid-state NMR is useful also for metal structure understanding in case of X-ray amorphous metal samples (like nano-size refractory metal ⁹⁹Tc). Recent studies have combined solid-state NMR with molecular dynamics simulations to resolve nanosecond-to-millisecond-scale motions in hydrated materials like amorphous calcium carbonate.

Biomolecular NMR spectroscopy

Proteins

Main article: Nuclear magnetic resonance spectroscopy of proteins

Much of the innovation within NMR spectroscopy has been within the field of protein NMR spectroscopy, an important technique in structural biology. A common goal of these investigations is to obtain high resolution 3-dimensional structures of the protein, similar to what can be achieved by X-ray crystallography. In contrast to X-ray crystallography, NMR spectroscopy is usually limited to proteins smaller than 35 kDa, although larger structures have been solved. NMR spectroscopy is often the only way to obtain high resolution information on partially or wholly intrinsically unstructured proteins. It is now a common tool for the determination of Conformation Activity Relationships where the structure before and after interaction with, for example, a drug candidate is compared to its known biochemical activity. Proteins are orders of magnitude larger than the small organic molecules discussed earlier in this article, but the basic NMR techniques and some NMR theory also applies. Because of the much higher number of atoms present in a protein molecule in comparison with a small organic compound, the basic 1D spectra become crowded with overlapping signals to an extent where direct spectral analysis becomes untenable. Therefore, multidimensional (2, 3 or 4D) experiments have been devised to deal with this problem. To facilitate these experiments, it is desirable to isotopically label the protein with ¹³C and ¹⁵N because the predominant naturally occurring isotope ¹²C is not NMR-active and the nuclear quadrupole moment of the predominant naturally occurring ¹⁴N isotope prevents high resolution information from being obtained from this nitrogen isotope. The most important method used for structure determination of proteins utilizes NOE experiments to measure distances between atoms within the molecule. Subsequently, the distances obtained are used to generate a 3D structure of the molecule by solving a distance geometry problem. NMR can also be used to obtain information on the dynamics and conformational flexibility of different regions of a protein.

Nucleic acids

Main article: Nuclear magnetic resonance spectroscopy of nucleic acids

Nucleic acid NMR is the use of NMR spectroscopy to obtain information about the structure and dynamics of polynucleic acids, such as DNA or RNA. As of 2003, nearly half of all known RNA structures had been determined by NMR spectroscopy.

Nucleic acid and protein NMR spectroscopy are similar but differences exist. Nucleic acids have a smaller percentage of hydrogen atoms, which are the atoms usually observed in NMR spectroscopy, and because nucleic acid double helices are stiff and roughly linear, they do not fold back on themselves to give "long-range" correlations. The types of NMR usually done with nucleic acids are ¹H or proton NMR, ¹³C NMR, ¹⁵N NMR, and ³¹P NMR. Two-dimensional NMR methods are almost always used, such as correlation spectroscopy (COSY) and total coherence transfer spectroscopy (TOCSY) to detect through-bond nuclear couplings, and nuclear Overhauser effect spectroscopy (NOESY) to detect couplings between nuclei that are close to each other in space.

Parameters taken from the spectrum, mainly NOESY cross-peaks and coupling constants, can be used to determine local structural features such as glycosidic bond angles, dihedral angles (using the Karplus equation), and sugar pucker conformations. For large-scale structure, these local parameters must be supplemented with other structural assumptions or models, because errors add up as the double helix is traversed, and unlike with proteins, the double helix does not have a compact interior and does not fold back upon itself. NMR is also useful for investigating nonstandard geometries such as bent helices, non-Watson–Crick basepairing, and coaxial stacking. It has been especially useful in probing the structure of natural RNA oligonucleotides, which tend to adopt complex conformations such as stem-loops and pseudoknots. NMR is also useful for probing the binding of nucleic acid molecules to other molecules, such as proteins or drugs, by seeing which resonances are shifted upon binding of the other molecule.

Carbohydrates

Main article: Nuclear magnetic resonance spectroscopy of carbohydrates

Carbohydrate NMR spectroscopy addresses questions on the structure and conformation of carbohydrates. The analysis of carbohydrates by 1H NMR is challenging due to the limited variation in functional groups, which leads to 1H resonances concentrated in narrow bands of the NMR spectrum. In other words, there is poor spectral dispersion. The anomeric proton resonances are segregated from the others due to fact that the anomeric carbons bear two oxygen atoms. For smaller carbohydrates, the dispersion of the anomeric proton resonances facilitates the use of 1D TOCSY experiments to investigate the entire spin systems of individual carbohydrate residues.

Drug discovery

Knowledge of energy minima and rotational energy barriers of small molecules in solution can be found using NMR, e.g. looking at free ligand conformational preferences and conformational dynamics, respectively. This can be used to guide drug design hypotheses, since experimental and calculated values are comparable. For example, AstraZeneca uses NMR for its oncology research & development.

High-pressure NMR spectroscopy

One of the first scientific works devoted to the use of pressure as a variable parameter in NMR experiments was the work of J. Jonas published in the journal Annual Review of Biophysics in 1994. The use of high pressures in NMR spectroscopy was primarily driven by the desire to study biochemical systems, where the use of high pressure allows controlled changes in intermolecular interactions without significant perturbations.

Of course, attempts have been made to solve scientific problems using high-pressure NMR spectroscopy. However, most of them were difficult to reproduce due to the problem of equipment for creating and maintaining high pressure. In  the most common types of NMR cells for realization of high-pressure NMR experiments are given.

High-pressure NMR spectroscopy has been widely used for a variety of applications, mainly related to the characterization of the structure of protein molecules. However, in recent years, software and design solutions have been proposed to characterize the chemical and spatial structures of small molecules in a supercritical fluid environment, using state parameters as a driving force for such changes.

Rayleigh scattering

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Rayleigh_scattering

Rayleigh scattering causes the blue color of the sky at large angles to the direction of solar rays and yellow or orange colors for light from the direction of the Sun.

Rayleigh scattering (/ˈreɪli/ RAY-lee) is the scattering or deflection of light, or other electromagnetic radiation, by particles with a size much smaller than the wavelength of the radiation. For light frequencies well below the resonance frequency of the scattering medium (normal dispersion regime), the amount of scattering is inversely proportional to the fourth power of the wavelength (e.g., a blue color is scattered much more than a red color as light propagates through air). The phenomenon is named after the 19th-century British physicist Lord Rayleigh (John William Strutt).

Due to Rayleigh scattering, red and orange colors are more visible during sunset because the blue and violet light has been scattered out of the direct path. This can yield dramatically colored skies and monochromatic rainbows.

Rayleigh scattering results from the electric polarizability of the particles. The oscillating electric field of a light wave acts on the charges within a particle, causing them to move at the same frequency. The particle, therefore, becomes a small radiating dipole whose radiation we see as scattered light. The particles may be individual atoms or molecules; it can occur when light travels through transparent solids and liquids, but is most prominently seen in gases.

Rayleigh scattering of sunlight in Earth's atmosphere causes diffuse sky radiation. Since blue light wavelengths scatter more, the diffuse sky seen in daytime is blue. At twilight the sunlight on the horizon is missing the scattered blue light wavelengths giving a yellowish to reddish hue to the low Sun.

Scattering by particles with a size comparable to, or larger than, the wavelength of the light is typically treated by the Mie theory, the discrete dipole approximation and other computational techniques. Rayleigh scattering applies to particles that are small with respect to wavelengths of light, and that are optically "soft" (i.e., with a refractive index close to 1). Anomalous diffraction theory applies to optically soft but larger particles.

History

In 1869, while attempting to determine whether any contaminants remained in the purified air he used for infrared experiments, John Tyndall discovered that bright light scattering off nanoscopic particulates was faintly blue-tinted. He conjectured that a similar scattering of sunlight gave the sky its blue hue, but he could not explain the preference for blue light, nor could atmospheric dust explain the intensity of the sky's color.

In 1871, Lord Rayleigh published two papers on the color and polarization of skylight to quantify Tyndall's effect in water droplets in terms of the tiny particulates' volumes and refractive indices. In 1881, with the benefit of James Clerk Maxwell's 1865 proof of the electromagnetic nature of light, he showed that his equations followed from electromagnetism. In 1899, he showed that they applied to individual molecules, with terms containing particulate volumes and refractive indices replaced with terms for molecular polarizability. It was this paper that established the basic scientific model for the color of the sky.

Small size parameter approximation

The size of a scattering particle is often parameterized by the ratio

$${\displaystyle x=\frac{2\pi r}{\lambda }}$$

where r is the particle's radius, λ is the wavelength of the light and x is a dimensionless parameter that characterizes the particle's interaction with the incident radiation such that: Objects with x ≫ 1 act as geometric shapes, scattering light according to their projected area. At the intermediate x ≃ 1 of Mie scattering, interference effects develop through phase variations over the object's surface. Rayleigh scattering applies to the case when the scattering particle is very small (x ≪ 1, with a particle size < 1/10 of wavelength) and the whole surface re-radiates with the same phase. Because the particles are randomly positioned, the scattered light arrives at a particular point with a random collection of phases; it is incoherent and the resulting intensity is just the sum of the squares of the amplitudes from each particle and therefore proportional to the inverse fourth power of the wavelength and the sixth power of its size. The wavelength dependence is characteristic of dipole scattering and the volume dependence will apply to any scattering mechanism. In detail, the intensity of light scattered by any one of the small spheres of radius r and refractive index n from a beam of unpolarized light of wavelength λ and intensity I₀ is given by $${\displaystyle I_s=I_0\frac{1+{\cos }^2\theta }{2R^2}{\left(\frac{2\pi }{\lambda }\right)}^4{\left(\frac{n^2-1}{n^2+2}\right)}^2{r}^6}$$ where R is the observer's distance to the particle and θ is the scattering angle. Averaging this over all angles gives the Rayleigh scattering cross-section of the particles in air: $${\displaystyle {\sigma }_{\text{s}}=\frac{8\pi }{3}{\left(\frac{2\pi }{\lambda }\right)}^4{\left(\frac{n^2-1}{n^2+2}\right)}^2{r}^6.}$$ Here n is the refractive index of the spheres that approximate the molecules of the gas; the index of the gas surrounding the spheres is neglected, an approximation that introduces an error of less than 0.05%.

The major constituent of the atmosphere, nitrogen, has Rayleigh cross section of 5.1×10⁻³¹ m² at a wavelength of 532 nm (green light). Over the length of one meter the fraction of light scattered can be approximated from the product of the cross-section and the particle density, that is number of particles per unit volume. For air at atmospheric pressure there are about 2×10²⁵ molecules per cubic meter, and the fraction scattered will be 10⁻⁵ for every meter of travel.

From molecules

Figure showing the greater proportion of blue light scattered by the atmosphere relative to red light

The expression above can also be written in terms of individual molecules by expressing the dependence on refractive index in terms of the molecular polarizability α, proportional to the dipole moment induced by the electric field of the light. In this case, the Rayleigh scattering intensity for a single particle is given in CGS-units by^[15] $${\displaystyle I_s=I_0\frac{8{\pi }^4{\alpha }^2}{{\lambda }^4{R}^2}(1+{\cos }^2\theta )}$$ and in SI-units by $${\displaystyle I_s=I_0\frac{{\pi }^2{\alpha }^2}{{{\epsilon }_0}^2{\lambda }^4{R}^2}\frac{1+{\cos }^2\theta }{2}.}$$

Effect of fluctuations

When the dielectric constant ${\displaystyle ϵ}$ of a certain region of volume ${\displaystyle V}$ is different from the average dielectric constant of the medium ${\displaystyle \overline{ϵ}}$, then any incident light will be scattered according to the following equation

$${\displaystyle I=I_0\frac{{\pi }^2{V}^2{\sigma }_{ϵ}^2}{2{\lambda }^4{R}^2}\left(1+{\cos }^2\theta \right)}$$where ${\displaystyle {\sigma }_{ϵ}^2}$ represents the variance of the fluctuation in the dielectric constant ${\displaystyle ϵ}$.

Cause of the blue color of the sky

Main article: Diffuse sky radiation

Scattered blue light is polarized. The picture on the right is shot through a polarizing filter: the polarizer transmits light that is linearly polarized in a specific direction.

The blue color of the sky is a consequence of three factors:

• the blackbody spectrum of sunlight coming into the Earth's atmosphere,
• Rayleigh scattering of that light off oxygen and nitrogen molecules, and
• the response of the human visual system.

The strong wavelength dependence of the Rayleigh scattering (~λ⁻⁴) means that shorter (blue) wavelengths are scattered more strongly than longer (red) wavelengths. This results in the indirect blue and violet light coming from all regions of the sky. The human eye responds to this wavelength combination as if it were a combination of blue and white light.

Some of the scattering can also be from sulfate particles. For years after large Plinian eruptions, the blue cast of the sky is notably brightened by the persistent sulfate load of the stratospheric gases. Some works of the artist J. M. W. Turner may owe their vivid red colours to the eruption of Mount Tambora in his lifetime.

In locations with little light pollution, the moonlit night sky is also blue, because moonlight is reflected sunlight, with a slightly lower color temperature due to the brownish color of the Moon. The moonlit sky is not perceived as blue, however, because at low light levels human vision comes mainly from rod cells that do not produce any color perception (Purkinje effect).

Of sound in amorphous solids

Rayleigh scattering is also an important mechanism of wave scattering in amorphous solids such as glass, and is responsible for acoustic wave damping and phonon damping in glasses and granular matter at low or not too high temperatures. This is because in glasses at higher temperatures the Rayleigh-type scattering regime is obscured by the anharmonic damping (typically with a ~λ⁻² dependence on wavelength), which becomes increasingly more important as the temperature rises.

In amorphous solids – glasses – optical fibers

Rayleigh scattering is an important component of the scattering of optical signals in optical fibers. Silica fibers are glasses, disordered materials with microscopic variations of density and refractive index. These give rise to energy losses due to the scattered light, with the following coefficient: $${\displaystyle {\alpha }_{\text{scat}}=\frac{8{\pi }^3}{3{\lambda }^4}{n}^8{p}^{2}k{T}_{\text{f}}\beta }$$

where n is the refraction index, p is the photoelastic coefficient of the glass, k is the Boltzmann constant, and β is the isothermal compressibility. T_f is a fictive temperature, representing the temperature at which the density fluctuations are "frozen" in the material.

In porous materials

Rayleigh scattering in opalescent glass: it appears blue from the side, but orange light shines through.

Rayleigh-type λ⁻⁴ scattering can also be exhibited by porous materials. An example is the strong optical scattering by nanoporous materials. The strong contrast in refractive index between pores and solid parts of sintered alumina results in very strong scattering, with light completely changing direction each five micrometers on average. The λ⁻⁴-type scattering is caused by the nanoporous structure (a narrow pore size distribution around ~70 nm) obtained by sintering monodispersive alumina powder.

Parabolic reflector

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Parabolic_reflector 

One of the world's largest solar parabolic dishes at the Ben-Gurion National Solar Energy Center in Israel

Circular paraboloid

A parabolic (or paraboloid or paraboloidal) reflector (or dish or mirror) is a reflective surface used to collect or project energy such as light, sound, or radio waves. Its shape is part of a circular paraboloid, that is, the surface generated by a parabola revolving around its axis. The parabolic reflector transforms an incoming plane wave travelling along the axis into a spherical wave converging toward the focus. Conversely, a spherical wave generated by a point source placed in the focus is reflected into a plane wave propagating as a collimated beam along the axis.

Parabolic reflectors are used to collect energy from a distant source (for example sound waves or incoming star light). Since the principles of reflection are reversible, parabolic reflectors can also be used to collimate radiation from an isotropic source into a parallel beam. In optics, parabolic mirrors are used to gather light in reflecting telescopes and solar furnaces, and project a beam of light in flashlights, searchlights, stage spotlights, and car headlights. In radio, parabolic antennas are used to radiate a narrow beam of radio waves for point-to-point communications in satellite dishes and microwave relay stations, and to locate aircraft, ships, and vehicles in radar sets. In acoustics, parabolic microphones are used to record faraway sounds such as bird calls, in sports reporting, and to eavesdrop on private conversations in espionage and law enforcement.

Theory

Strictly, the three-dimensional shape of the reflector is called a paraboloid. A parabola is the two-dimensional figure. (The distinction is like that between a sphere and a circle.) However, in informal language, the word parabola and its associated adjective parabolic are often used in place of paraboloid and paraboloidal.

If a parabola is positioned in Cartesian coordinates with its vertex at the origin and its axis of symmetry along the y-axis, so the parabola opens upward, its equation is $4fy=x^2$, where $f$ is its focal length. (See "Parabola#In a cartesian coordinate system".) Correspondingly, the dimensions of a symmetrical paraboloidal dish are related by the equation: $4FD=R^2$, where $F$ is the focal length, $D$ is the depth of the dish (measured along the axis of symmetry from the vertex to the plane of the rim), and $R$ is the radius of the dish from the center. All units used for the radius, focal point and depth must be the same. If two of these three quantities are known, this equation can be used to calculate the third.

A more complex calculation is needed to find the diameter of the dish measured along its surface. This is sometimes called the "linear diameter", and equals the diameter of a flat, circular sheet of material, usually metal, which is the right size to be cut and bent to make the dish. Two intermediate results are useful in the calculation: $P=2F$ (or the equivalent: $P=\frac{R^2}{2D}$) and $Q=\sqrt{P^2+R^2}$, where F, D, and R are defined as above. The diameter of the dish, measured along the surface, is then given by: $\frac{RQ}{P}+P\ln \left(\frac{R+Q}{P}\right)$, where $\ln (x)$ means the natural logarithm of x, i.e. its logarithm to base "e".

The volume of the dish is given by $\frac{1}{2}\pi R^{2}D,$ where the symbols are defined as above. This can be compared with the formulae for the volumes of a cylinder $(\pi R^{2}D),$ a hemisphere $(\frac{2}{3}\pi R^{2}D,$ where $D=R),$ and a cone $(\frac{1}{3}\pi R^{2}D).$ $\pi R^2$ is the aperture area of the dish, the area enclosed by the rim, which is proportional to the amount of sunlight the reflector dish can intercept. The area of the concave surface of the dish can be found using the area formula for a surface of revolution which gives $A=\frac{\pi R}{6D^2}\left((R^2+4D^2{)}^{3/2}-R^3\right)$. providing $D\ne 0$. The fraction of light reflected by the dish, from a light source in the focus, is given by $1-\frac{\arctan \frac{R}{D-F}}{\pi }$, where ${\displaystyle F,}$ ${\displaystyle D,}$ and ${\displaystyle R}$ are defined as above.

Parallel rays coming into a parabolic mirror are focused at a point F. The vertex is V, and the axis of symmetry passes through V and F. For off-axis reflectors (with just the part of the paraboloid between the points P₁ and P₃), the receiver is still placed at the focus of the paraboloid, but it does not cast a shadow onto the reflector.

The parabolic reflector functions due to the geometric properties of the paraboloidal shape: any incoming ray that is parallel to the axis of the dish will be reflected to a central point, or "focus". (For a geometrical proof, click here.) Because many types of energy can be reflected in this way, parabolic reflectors can be used to collect and concentrate energy entering the reflector at a particular angle. Similarly, energy radiating from the focus to the dish can be transmitted outward in a beam that is parallel to the axis of the dish.

In contrast with spherical reflectors, which suffer from a spherical aberration that becomes stronger as the ratio of the beam diameter to the focal distance becomes larger, parabolic reflectors can be made to accommodate beams of any width. However, if the incoming beam makes a non-zero angle with the axis (or if the emitting point source is not placed in the focus), parabolic reflectors suffer from an aberration called coma. This is primarily of interest in telescopes because most other applications do not require sharp resolution off the axis of the parabola.

The precision to which a parabolic dish must be made in order to focus energy well depends on the wavelength of the energy. If the dish is wrong by a quarter of a wavelength, then the reflected energy will be wrong by a half wavelength, which means that it will interfere destructively with energy that has been reflected properly from another part of the dish. To prevent this, the dish must be made correctly to within about ⁠1/20⁠ of a wavelength. The wavelength range of visible light is between about 400 and 700 nanometres (nm), so in order to focus all visible light well, a reflector must be correct to within about 20 nm. For comparison, the diameter of a human hair is usually about 50,000 nm, so the required accuracy for a reflector to focus visible light is about 2500 times less than the diameter of a hair. For example, the flaw in the Hubble Space Telescope mirror (too flat by about 2,200 nm at its perimeter) caused severe spherical aberration until corrected with COSTAR.

Microwaves, such as are used for satellite-TV signals, have wavelengths of the order of ten millimetres, so dishes to focus these waves can be wrong by half a millimetre or so and still perform well.

Variations

Focus-balanced reflector

An oblique projection of a focus-balanced parabolic reflector

It is sometimes useful if the centre of mass of a reflector dish coincides with its focus. This allows it to be easily turned so it can be aimed at a moving source of light, such as the Sun in the sky, while its focus, where the target is located, is stationary. The dish is rotated around axes that pass through the focus and around which it is balanced. If the dish is symmetrical and made of uniform material of constant thickness, and if F represents the focal length of the paraboloid, this "focus-balanced" condition occurs if the depth of the dish, measured along the axis of the paraboloid from the vertex to the plane of the rim of the dish, is 1.8478 times F. The radius of the rim is 2.7187 F. The angular radius of the rim as seen from the focal point is 72.68 degrees.

Scheffler reflector

The focus-balanced configuration (see above) requires the depth of the reflector dish to be greater than its focal length, so the focus is within the dish. This can lead to the focus being difficult to access. An alternative approach is exemplified by the Scheffler reflector, named after its inventor, Wolfgang Scheffler. This is a paraboloidal mirror which is rotated about axes that pass through its centre of mass, but this does not coincide with the focus, which is outside the dish. If the reflector were a rigid paraboloid, the focus would move as the dish turns. To avoid this, the reflector is flexible, and is bent as it rotates so as to keep the focus stationary. Ideally, the reflector would be exactly paraboloidal at all times. In practice, this cannot be achieved exactly, so the Scheffler reflector is not suitable for purposes that require high accuracy. It is used in applications such as solar cooking, where sunlight has to be focused well enough to strike a cooking pot, but not to an exact point.

Off-axis reflectors

The vertex of the paraboloid is below the bottom edge of the dish. The curvature of the dish is greatest near the vertex. The axis, which is aimed at the satellite, passes through the vertex and the receiver module, which is at the focus.

A circular paraboloid is theoretically unlimited in size. Any practical reflector uses just a segment of it. Often, the segment includes the vertex of the paraboloid, where its curvature is greatest, and where the axis of symmetry intersects the paraboloid. However, if the reflector is used to focus incoming energy onto a receiver, the shadow of the receiver falls onto the vertex of the paraboloid, which is part of the reflector, so part of the reflector is wasted. This can be avoided by making the reflector from a segment of the paraboloid which is offset from the vertex and the axis of symmetry. The whole reflector receives energy, which is then focused onto the receiver. This is frequently done, for example, in satellite-TV receiving dishes, and also in some types of astronomical telescope (e.g., the Green Bank Telescope, the James Webb Space Telescope).

Accurate off-axis reflectors, for use in solar furnaces and other non-critical applications, can be made quite simply by using a rotating furnace, in which the container of molten glass is offset from the axis of rotation. To make less accurate ones, suitable as satellite dishes, the shape is designed by a computer, then multiple dishes are stamped out of sheet metal.

Off-axis-reflectors heading from medium latitudes to a geostationary TV satellite somewhere above the equator stand steeper than a coaxial reflector. The effect is, that the arm to hold the dish can be shorter and snow tends less to accumulate in (the lower part of) the dish.

Off-axis satellite dish

History

The principle of parabolic reflectors has been known since classical antiquity, when the mathematician Diocles described them in his book On Burning Mirrors and proved that they focus a parallel beam to a point. Archimedes in the third century BCE studied paraboloids as part of his study of hydrostatic equilibrium, and it has been claimed that he used reflectors to set the Roman fleet alight during the Siege of Syracuse. This seems unlikely to be true, however, as the claim does not appear in sources before the 2nd century CE, and Diocles does not mention it in his book. Parabolic mirrors and reflectors were also studied extensively by the physicist Roger Bacon in the 13th century AD. James Gregory, in his 1663 book Optica Promota (1663), pointed out that a reflecting telescope with a mirror that was parabolic would correct spherical aberration as well as the chromatic aberration seen in refracting telescopes. The design he came up with bears his name: the "Gregorian telescope"; but according to his own confession, Gregory had no practical skill and he could find no optician capable of actually constructing one. Isaac Newton knew about the properties of parabolic mirrors but chose a spherical shape for his Newtonian telescope mirror to simplify construction. Lighthouses also commonly used parabolic mirrors to collimate a point of light from a lantern into a beam, before being replaced by more efficient Fresnel lenses in the 19th century. In 1888, Heinrich Hertz, a German physicist, constructed the world's first parabolic reflector antenna.

Applications

Antennas of the Atacama Large Millimeter Array on the Chajnantor Plateau

The most common modern applications of the parabolic reflector are in satellite dishes, reflecting telescopes, radio telescopes, parabolic microphones, solar cookers, and many lighting devices such as spotlights, car headlights, PAR lamps and LED housings.

Lighting the Olympic Flame with a parabolic reflector

The Olympic Flame is traditionally lit at Olympia, Greece, using a parabolic reflector concentrating sunlight, and is then transported to the venue of the Games. Parabolic mirrors are one of many shapes for a burning glass.

Parabolic reflectors are popular for use in creating optical illusions. These consist of two opposing parabolic mirrors, with an opening in the center of the top mirror. When an object is placed on the bottom mirror, the mirrors create a real image, which is a virtually identical copy of the original that appears in the opening. The quality of the image is dependent upon the precision of the optics. Some such illusions are manufactured to tolerances of millionths of an inch.

A parabolic reflector pointing upward can be formed by rotating a reflective liquid, like mercury, around a vertical axis. This makes the liquid-mirror telescope possible. The same technique is used in rotating furnaces to make solid reflectors.

Parabolic reflectors are also a popular alternative for increasing wireless signal strength. Even with simple ones, users have reported 3 dB or more gains.